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vi 

NATIONAL ADVISORY COMMITTEE 

FOR AERONAUTICS 

TECHNICAL NOTE 

No. 1195 

EFFECT OF FINITE SPAN ON THE AIRLOAD DISTRIBUTIONS 

FOR OSCILLATING WINGS 

II - METHODS OF CALCULATION AND EXAMPLES OF APPLICATION 

By Eric Reissner and John E. Stevens 

Massachusetts Institute of Technology 

GU···-.. .,,---:···...-, T •. - T IB""" 71 Y ;.t f ". " 1-' .... . - .... I .: 1.-\ ' 11 ...... J.J -.. • _ .... "l. L 

Washington 

October 1947 

l 

https://ntrs.nasa.gov/search.jsp?R=19930082169 2019-08-28T15:25:04+00:00Z
-

/ 
NATIONAL .AJlVI80~Y COl-.1l.'JfITTEE FOR AERONAUTICS 

TECRNICAL NOTE NO. 1195 

EFFECT OF FINITE SPAN ON THE AIRLOAD DISTRIBUTIONS FOR OSCIJ~TING WINGS 

II - METHODS OF CALCULATION AND E]JU~LES OF APPLICATION 

By Eric Reissner and John E. Stevens 

StJ1VIHARY 

Lift and moment distrUutione aTe calculated for oscillating wings 

of finite span on the basis of' the three-dimensional theory of part I of 

this report. 'I'he results ob7.ainei are compared. wi th the correspond.ing 

results of the tvlc-di mensional theory. 

Rectangular and. elllpt:i.cal "j nBs of aspect ratios 3 end 6 are con--

sidered and the ranee of val ues of the red.nced frequency k extends 

from 0 to about 1.5. The calculations are mad.e for various bending, 

torSion, and aileron deflection functions and the results are given in 

tabular and in gr aphj.cal form. It j.s fauna. that for a given wing and 

g1ven red.uced-fre Q.uency value the effect of finite span deI-ends appreciably on the shape of the wine deflection functtons. It is also found 

that for a given Vling and Given deflection function the finite-span 

effect decreases as the reduced frequenc:r increases. For wings of 

aspect ratio 3 an a:pllrecia:-Jle three-dimensional effect occurs for value s 

of k up to about 1.0 and for "lings of aspect ratio 6, for values of 

k up to about 0.5. 

A ]?raGtical scheme of calculatlons is descr::.bed and auxillllry ta·-

bles are given for the numeri cal analys~s of additionaJ, examples. 

Formulas are included which a2.low direct i ncorporation of the threedi~nsi~l results in flutter determinants of the kind describeQ in 

AAF Technical Report No. 4798. Examples of flutter calculations on the 

basis of these formulas are given in an appendix to this report. Change 

of the two-d.imensional into the three-dimensional ai r forces appears to 

be responsible for f lutter-speed changes of from 10 to 20 percent for 

wings, with the possibility of larger con'ections in tail flutter. 

Whj.le l n the examples analyzed so far the aerodynamic·-span effect 

increases the theoretical flutter speed, the possibility of the effect 

being in the oppos:L te direction in other examples has to be considered. 
2 NACA TN No. 1195 

INTRODUCTION 

The magnitude of the aerodynamic-span effect in wing theory 1s one 

of the problems for which no complete answer has yet been found. With 

attention re stricted to ''lings of aspect ratio that is not too small (say, 

AR ;? 3) it may be said that, for wings in uniform. motion, lifting-line 

theory represents a satisfactory solution of t he problem. Fo!' wings in 

nonuniform motion the problem is considerably more diffi cult and most 

attempts to analyze the aerodynamic-span effect aypear t o be either incomplete or in their applicability restricted to special cases of the 

problem. 

In part I of the p!'esent re~ort an aerodynamic theory of the oscillating wing of finite span has been given ,.,hich i s consldered to be a8 

inclusive a8 lifting-line theory is for the wing in uniform motion. The 

final results of this theory were formulas for the spanwise distribution 

of air force e. and moments for the four basi c types of motion of flexible 

wings, namely bending, torSion, ile~on, and tab ~eflection. In these 

formulas the effect of three-dimeneionality appears as a correction term 

a to the basic function C(k) of the two-dimensional theory. The calculation of a necessitates the solution of an integral equation which 

is similar to, but le~s simple tha.n, the integral equation of liftingline theory for the lift (and circulation) distribution on wings in 

uniform motion. 

1fith a method thus established for the systematic calculations of 

finite-span corrections for the customary two-dimensional theory it 

becomes possi ble to arrive at statements with regard to the quantitative 

importance of the aerodynamic-span effect. It has in the past 'been 

held by a number of investigators that this effect, while significant 

for wings in uniform motion, is no longer so for oscillating wings. Physical consideration of the wake pattern indicates that for a given wing 

the actual three-dimensional flow approaches more and more nearly the 

two-dimensional pattern as the frequency of oscillation increases. It is 

thus perm:issible to say that for sufficiently high frequencies the 

aerodynamic-span effect is of an insignificant magnitude. This, hOI-leVer, 

leaves the question as to what constitutes a sufftciently high frequency 

and what is the magnitude of the effect if the frequency is not sufficiently high. It was therefore considered desirable to establish, at 

least roughly, the range of frequencies for ~'Thich there is an appreciable 

aerodynamic-span effect and to indicate the nature of the effect in this 

range. The variables which are mainly involved are .)'Ej!ect ratio, 

wing deflection form, and reduced freque'ncy k = rnb/U, where rn is the 

circular frequency of oscillation, b the eemichord of the wing, and U 

the velocity of flight. 

The calculations i n this r eport are in part de8i nAd to p~Tmit a 

rapid e sti!llB.'lji0n ojf tn...? ill~gr:.~ t:;;."_e rJf t::-t8 E'.'2 .r-v!.j )J{!l:;:"C'- ::l;·an dT~~ '0 ty 
• 

NACA TN No. 1195 3 

providing for consideration the results of an appreciable number of sample cases for wings of aspect ratios 3 and 6 and rectangular or elliptic 

plan form and for a range of values of' the reduced frequency k. 

It is felt that a point of further interest of this work should be 

the presentation of a scheme for incorporating the three-dimensional air 

forces in flutter calculations, for those cases in which such an incorporation is d€Jemed worth while. AJ.though the authors agree that such a 

refi nement will be unnecessary in a good many cases for which great 

accuracy is not required, they also feel that if it is desired to obtain 

theoreUcal flutter speeds which are within, say, 10 percent of the 

actual flutter speed.s, then in most cases j. t will be found necessary to 

incorporate the aerodynamic-span effect in the analysis. Appreciably 

larger corrections \rill often be obtained, particularly in problems of .'. 

tail flutter. . 

This investigation, conducted at the Massachusetts Institute of 

Technology, wns sponsored by, and conducted with financial assistance 

from, the National Advi sory Committee for Aeronautics. 

u 

x, y 

p 

x7,( y) 

xt(y) 

b 

b o 

s 

(l) 

B 

SYMBOLS 

velocity of flight 

Cartesian coordinates in plane of projection of wing surface 

density 

coordinate of wing leading edge 

coordinate of wing trailing edge 

semi chord 

semi chord at midspan 

ratio of span to chord at midspan 

circular frequency 

amplitude of three-dimensional circulation functi on defined by 

equations (8) and (13) 

'c ) 1) " dimensionless spanwise coordinates; y* ;;: y/bo' T)* ;: 1)/'00 

zm dimensionl ess coordinate of midchord line; Zm:; (Xl + Xt) /2bo 
4 

k 

ko 

km 

IJ. 

- (8 ) (2 

ASn 

C(k) 

a 

a 

c 

e 

d 

f 

7" m 

h 

NACA TN No . 1195 

reduced rrequency; k = mbjU 

reduced frequency at midspan 

function defined by equation (10) 

amplitude of tvTO-dimensional circulation function given by equation (7) 

variable of integration 

function defined by equation (16) 

function defined by Theodorsen 

correction function dofined by equation (6) 

location of elastic axis in units of the semi chord b 

location of aileron leading edge in units of b 

location of aileron hinge line in units of b 

location of tab leading edge in units of b 

location of tab hinge line in units of b 

aileron and tab overhans in units of b; 7, = e - c, m = f - d 

bending deflection of wings; h = h(y)eimt 

-( ) imt angle of attack of wings; a = aye 

~ 1 d b d fl t 1 ~ -_ A eimt ~, 1 ai .eron an ta e ec ion ang es; ~ ~ 

L lift per unit of span; L = L e

imt 

Ma moment about elastic axis per unit of span 

Mf3 , My aileron and tab hi nge moments per unit · of span 

F function defined by equation (9) 

J, Y Bessel functions in customary notation 

• 

• 
f 

NACA TN No. 1195 5 

H Hankel functions in customary notati~n 

-

CL l ift coefficient defined by equation (12a) 

~ rolling-m€ment coefficient defined by equation (12b) 

P, Q functions defined by equations (32) and (33) 

phase angle functions (j = h , a ', 13 •. y) 

LIFT AND MOMENT FUNCTIONS OF TIIE TIffiEE-DIMTENSIONAL THEORY 

Accordi ng to part I of the present report the following formulas 

hold f or the di stribution of lif t LJ moment Ma J aileron hinge moment 

~J and t ab hinge moment ~ due to a bending deflecti on hJ _a torsional angular deflection aJ an aileron angular deflection ~J and a 

tab angul ar def lection r: 

L r k 2 . . 'I h --==---- = l{ , - - + lk[C(k) + uhJ l -- 2 ; f') I b 2pU bo ~ c: -' 0 

+ 1( r ~ Uk + k

2

a] + [I + i k (~ - a) 1 [C (k) + .u a, '] J a; ~o 
6 NACA TN No. 1195 

) 

Ma, = bb 3t r ak

2 

_ (1 + a ) "ik[C(k) + o . J"" ~ + (~)2 3t 

2pU 22 b ' 2 2 J bJb "' b · o 0 \. ,- . p 0 

r 1 2 11 \ [ ( 1 ) -, ' I X '\ - [ikA - k A ] - (-. + a / "1 + ik -2 - a GC(k) + 0a ] ~ a: I 2 0".:2 a~ " \2 / J 

"- (~ + ) . [ ~ + ~~ E2 J[ C( ' + o~ 1 } (3 +. (~O )2 """ . " 

1 _. + 0, ] J ' 

~[ikB - k2B, ] +!E l~ 1 + ik (! - a)l [ C(k) + ON] J1 a: 

l_ 2 0".:2 0.1 2 :3 " \ 2 ,I VJ 

(

b )2 r 3t 2 - 1 [E ik J [ ( ) Jl - + - \ - [BQ + ikBQ - k BI3 J + - E :3 ", ~ +- E2 C k + 0'/3 j ~ b

o 

1_ 2 ~3 ~2 1 23t 2 

---- -" -- -
NACA TN No. 1195 5 

H Hankel functions in customary notati~n 

CL l ift coefficient defined by equation (12a) 

~ rOlling-mement coefficient defined by equation (12b) 

P, Q functions defined by equations (32) and (33) 

phase angle func ti ons (j = h , cr·.> !3 .'. ')') 

LIFT AND MOMENT FUNCTIONS OF TilE THRF~-DIMENSIONAL THEORY 

Aocording to part I of the present report the following formula.s 

hold for the distribution of 11ft L, moment M~, aileron hinge moment u. 

~, and tab hinge moment My due to a bending deflection h, a torsional angula.r deflection ~, an ' aileron angular deflection ~, and a 

tab angular deflection y: 

+ 1t f ~ [l k + k

2

aJ + [1 + ik (~2 - a) 1 2 

'I _ b 

[ C(k) + (J J \. (l, -

(l, J bo 
6 NACA TN No. 1195 

, 

r 1 '1 \ [ 1 x , - [ikA - k

2 A ] - - (- + a - 1 + ik ( -2 - a ) l 2 ~ 0.1 2 ) 

- (1 ) [ ' ik J - I : )2 - ~ + F'. - E1 ' + 2 E2 [C(k) +- °13 ] -) "i3 + (~o --

X r ~ [A 'kA ~2 A __ ] (J \ [E -( )- ik - ( ) 1 () ' 2 ~'3 + 1 'Y2 - _n - '1 - \ ~ + a) ' 1 d + 2 E2 d _ [C k 

'1 -- + or] ) r 

( 1t [ . 2 ] 1 l~ , I 1 ) ]- () '( - ~ - lkB - k -B, + - E 1 + lk ( - -- a [C k + 00.] ,I a. ! 2 ~ 0.1 2 3 - \ 2 ~ 

+ ( E....)2 r ~ [ B + ikB - k2B ] + ~-, E [E 1 ( d) b ) 2 r3 r2 71 21t 3 

o . 

x [C(k) + Or] } r (3 ) 

-- 
f 

NACA Tli No. 1195 7 

. (4) 

Equat lons (1 ) to (! ) emp'.oy t he notation of reference 1 and the 

terms A, B, C, and D are defined there . The values of the terms E 

are 

J 

and the terms Tare aJ 80 defined j n r eference 1. 

The dimensions 'a, c; d} G , f , I , and. m (all a s defined i n reference 1 and i n fig . 1) are in units of the loca l semichord b (b = bo 

at rni.ds:pan) and k = wb/U stands for the local reduce d freq.uency . 
8 NACA TN No. 1195 

The terms C1j(j::: h, 0" f3: ,) which alone represent the effect of 

ftn:i te 8 ()Eln are g1 ven by 

( 6) 

The t'vo-dimensional circul ati on functions 

are defined oy 

in eCluation (6) 

-- (2) 41C(ld :Lkm [ ik . l b -- o.'(.l = - --- e ~ + - E2 J'- (3(y) .., (2) 2 - b

o n'kll~ (k ) 

b - ) 

- o,(y b . o 

- (2) __ ~j (k)_ e

ikm fE1(d) lk ()J b -( ) 0'1 (2) + . ....::. E2 d - Y Y 

L 2 "0 0 7tkII 1 (k) -

In order to take into account the possibillty of phase Yariation in 

span cirection the function h(y), -;;:( y ), [3(y), and r(Y) in eCluations 

- i¢h (y) - i¢,o,(y) 

(1) to (4) and (7) must be multiplied by te rms e e 

-i¢f3 (y) -j,¢y (y ) 

e , an.d e ,respe c tively . In eCluati~ns (7) the Clv.antity 

k

m 

= k (x t + xL) / 2"0

0 

r epresents the effe ct of sweep of the midchord line 

of the wing. 

The three--<ijmensional c :l.rculatlon functions n j are solutions of 

the followi !'.g · integral eCluation: 
c 

NACA TN No. 1195 

- 8 

In equati on (8) y * indicate c the l ocation of poi nts along the span i n 

units of tho semi chord. a t mids pr.n DO. 

The functi on F in t h6 socond i ntegr a.l 1s defi ned as 

F(x) = r co e -i A. r 1. + 1. _ fA. 2 

+ :x:

2 1 d A. 

•. / L X), XA. I 

o 

The function ~ is defi ned i n t erms of Bessel functi ons a s 

9 

(8) 

~(k) ::; (10) 

Numerical values of the functions C(k), ~(k), 

C(k) i C(k") 

tH1 2 )(k) , 

and F(x), whiGh are of importance f or t he evaluati on of the general 

formulas J are given in 'Gables I t o V. 
10 NACA TN No. 119 :i 

Lift and moments at 'points 0f_~o wi~_fle?tion .-- At poJnts of no 

wing deflection equati ons (J.) to (4) cannot be used directly 'be cause one 

or more of the (1 j terms becomes lnfinj. te ~ whUe the Gorrespondtng deflection is zero. In view of equa t icns (6) and (7) there may be written 

for a point where a def lecti on funcUon j(b, d" ~) ')') b~coln s zero, 

iJJ(k ) "(' ) . {. K . + "'-. -.. - -.---- , '. 'J '{'1 t .' J (, ) . . .. . '. ci . {} _./. 1 1 .i.C -

-,---.----.. -~ .. 0 

lC (k) :USn j 

)-1 -r;"\""" - E! 

kHi;e ) (kl 

(11) 

C(k) 

i J 1(lc) 

+ 

Ml3,j E b .. Jo·{k) - tJ1(k) :3 

') U2b 2 = - tklll .. OJ 2 bo f.P f..> 1 .iC{::'i ) ~.- --- e 

klIf 2 )(k) 

C(k) 

i a'l (k) 

My, j E ('1.) + Jo(Jc) ·- :l. J1(k) =-3 b - - o J ?pU2b2 2 b o iC(k) ik .- 0 4-'-- e 1)1 

kHl 2 )( k ) 

Expressions for lift and rolling-moment coefficients .- The c;oeff'icient of wing lift w~y be defined as . 

-

bo d (G~o) ( 12a) 

• 

j 
NACA TN No. 1195 

For a rectangular wiP3 

For an elliptical -wing 

8sb 2 

o -- = 2 

Sw 

8Si) ~ 8 --:: _. Sw Jt'. 

The wing rolling-mcment cooff'icient may be defined as 

r S b O 

11 

_: -sb

o 

L y dy 

1 

- oU2 (2Sbo)Sw 

! • L ( Y- d (Y-) (12b ) Sbo ) sbo i 

I 

I , I 

2 o 

Corresponding formulas may be wri tten down for a pi tching .. -moment coefficient and for control-surface hinge--m<'lment coefficlents ._ In the ~rt)~t .. 

paper, however, use i s made only of the defini t i ons of CL and~. ' 

SOLUTION OF TEE INTEGRAL EQUATION FOR THE CIRCULATI,ON Ftmc'.h01i 

For the solution of equation (8) a procedure is chosen which i~ 

analognus to one of the known me thods f~r the soluti on of the liftingline equati on f or unif orm moti on. 

Define new variables ¢ and S by the r elations, 

y oe = 8 cos ¢, 1) -)(' :: 8 cos G 

and write 

(13) 
; 

12 NACA TN No. 1195 

In terms of the new variables equation (8) becomes , 

.t ¢ ~ rt \' K j ' cos n e 

-) , K S.l.n n + -b ~(k) l I' L n - - de· __ nj n b o ~ S(COB ¢ -cos e) 

o .. ~. 

--~--- COB ¢ - cos e ' " ) )' F(koB COB ¢ .. ~ ICOS ¢I K ' ,os 119de 

Icos ¢ - cos 91 . nJ 

( 14) 

Introduce the known formula, . 

I 

o 

COB nede = _ 1C sin n ¢ . 

cos ¢ - cos e sin ¢ 

and write equation (14) in. the form, 

sin n ¢ b ' 1C ~ sin n ¢ 

---+- -~ 

n bo S sin ¢ 

1'( 

.r COB ¢ - cos e l' 

;. FCkos I cos ¢.-:- COB 81 ) cos nede Jt I COB ¢' - cos e I ',' - o 

( 15) 

.J 
NACA TN No. 1195 13 

N01v) definf;" a set of functions Sn by the relation, 

k s . 0 

+ l1( 

sin n ¢ :::----

sin ¢ 

~t.....:: cos e F(kos lcos¢ _ cos e l )cos n Bde 

1 cos ¢ - ' cos e I 

Equation (15) may then be written 

( 16) 

Equation (17) is solved approximately by satisfying it ~t as many suitably chosen points ¢m as there are coeffic lents I~. Write 

. sin n do rc b () ( ¢) Anm. = --n-=::..E + -s - 11 k S k s, b n 0 ill 

o 

(18) 

and 

-- (2) ¢ D j ( s cos ill ) 

Th0n the system of equations t o be solvRd is 

(20) 
14 NACA TN No. 1195 

Note, that if the ,,,ing deflection is symmetrical about mid.span, only odd. 

values of n OCcUl~, while for e.ntisymmetri cal deflections only even . 

values of n occur. If this is taken into account, all points ¢m may 

be chosen wi thin a semispan. If the number of points ¢ "Thich are 

m 

selected is N, then equation (20) is a system of N simultaneous complex linear equations to be solved for the N complex qu.antities ~J' 

Note further that the left-hand side of equ8:tlon (20) depends only on 

wing plan form and on reduced frequency, but not on the fOl1m of the 

d.eflection fvnction which detel~unes the right-hand. sido. 

The functions S (k s)¢) which occur in equation (18) have been 

n 0 ill 

calculated for a number of values of n, k B, and. ¢ and thE> results 

o m 

are contained. in t able VI . The functions sin n¢mJn are also calculated. 

for the same values of nand ¢ and the reBults are given in table 

m 

VII. Figure 2 contains a serios of plots of Sn agatnst koB, so that 

also for intermediat9 values of k s t.he values of S are available. 

o n 

Calculation of the functions S according to equation (16) inn 

volves the evaluation of an integral containing the f~unction F . defined. 

by equation (9). Vith tabular values' of J!' . available, the integration 

can be carried. out graphically or by approximating F in the rangE> of 

interest of the variable x b;V a function which penni ts explicit integration . Both theRe procedures have been used. (See appendix A.) Use 

has also been made of calculations carried. out in England (reference 2) 

in' connection with a theory of the aerodynamic-span effect which differs 

from the one evaluated here. 

SUMMARY OF PROCEDURE FOR CALCULATING THRF..E-DlMENSIONAL EFFECTS 

(1) Choose spanwise stations at "'hich integral equation (8) is to 

be satisfied. (Fresent range of calculat:Lons of auxiliary 

functions restricts choice tb the sjx stations, 

cos ¢m = 0, 0.2, 0.4, 0.6, 0.8, and. 1.0 

in units of the semispan.) 
, 

NACA ~~ No. 1195 15 

(2) Tabulate the values of -'·J /bo ' KJ Y'JIl a, :'..td , . jf nSt, derl . of' 

C J d J fJ J f, 1. and - m at thef,e stations. Take from t·able rr 

the values of the function 4j.C/lrH1 (2) at these stations. 

- (2) Calculate, according to equatJ. ons (7) the values 0 at 

these stations. j,m 

(3) Calculate by means bf taDles II, VI, VII the values of A of 

of equation (18). run 

(4) Solve the system of e~uat1on (20) for the coeff1cientc Knj , 

preferably by the Crout me thod for the solution of systems 

of simultaneous equations with complex coefficients. 

(5) Calculate the va"Lues of n J at the selected stati ons 

by means of e quati on (13). f/.ake USB of tA.ble VII for factors 

sin n¢m/n. 

(6) - / _(2) Calculate the val.uea of D J D j at the aelected stations. 

(7) Culculiit\a- t enur> 

(8) Substitute terms 

. moments. 

from e quation (6) making use of t!ible III. 

in eguaUons (1) to (4) for lift and 

It is this ~rocedure that has been used to obtain the results which are 

d.i scu6sed in the follo.ring se c t:l.on of thl s r e po:ct. 

COMPARISON OF TWO-DIMENSIONAL AND THREE-DIMENSIONAL LIFT DISTRIBUTIONS 

In order to ju~e the effect of the three-dimensional correction 

terms in the expressions for lift and moment,s; a number of cases have 

been analyzed numerically and the results aTe shown in t able VIII, and 

by means of' figures 3 to 21 which ~on ain the two-dimensi onal a s well as 

the three-dimensional distributions. 

Calculations have been made for wir..gs of elliptica.l plan form and 

for .Tines of rectangular plan form. In both cases the aspect ratios 

chosen are 3 and 6J Gnd a range of value s of the red.uced fre~uency ko 

is covered. As w.J.ng deflections the following "I-,'8re chosen; 

(1) Translation of r i gid wing (~ ::: 1) 

\ 0 

(2) Pitching of rigid wing about midchord (a = 1, a = 0) 
16 

( 3) RO~ing of ,rigid ,Wing (~~, = ! / sb 0)-., 

': ''') < 

(4)· LineEl:r aptisymmetrlcaJ- tor~i on (,0: =, y/sPo' 

(5) Linear symmetric'o.l torsion (d:";' h;l/sbo, a 

a 

= 

(6) Linear symmetri;ca1 bending (~o ::' ·1 yl/sbo ) 

(7) Parabolic symmetricalbendi"ng (-~o. = (Y/Sbo )2) , . 

(8) Deflect::"on of 'full-span ai leron ([3 , :::. 1) 

(9) Def16ction o~ pa£~ial~s~an aileron _ 

(.I y! <~sbo : 13 .= 0; lyl>A.soo : 13 = 1) 

NA~A TN No. 1195 

!' , ' 

~. 0) 

0)' 

Figures 3 to 21 contai n the re sul ts of these calculatione. (Some of 

the .results were first ootained in .reference 3.) The :p.oints at which too 

integral equation for the circulation has teen satisfied a.re indicated in 

the, figures; the number of these points de termines the number of simulta:-

neous equations to be solved. The following conclusions ~ be stated: 

The aerodynamic-span effect for a given aspect 'ratio and given wing 

deflection decreases noticeably as the values of the reduced frequency 

Ko i ncrease . There is substantial agreement between the r esults of the 

three-d'mensional theory and the r esults of the two-dime sional theory 

when the value of leo is sufficiently large. The rate at which this 

approach between the results of the two theories takes plnce depends on 

aspect ratio and ving .deflectton function. The lover the aspect ratiO, 

the slower the approach. The finite-span effect' perSists to values of 

k which are higher for linear spanvTise variation of deflection than 

a 

they are for no spani-Tise variation of deflection, and higher for parabolic variation than for linear variation. While these facts are what 

,wuld be eXI'8cted, the p!'f.lsent calculations should . furnish the qusmti tati ve infoIT..?, tj.'on .yo:'ch i is need.ed. in ord.er t o knm'! when to neglect the 

aerr.I :::';Y £lanllC·-Gl\9.D bffect and when not. to neglect it. Roughly, it may be 

state d, then that "rhen J\R = 6,. there is no need to consili.er the 

aerodynamic- ·span effect when 

t~ consider the effect when 

k >1' and when AR = 3 there is no need 

o ' , 

ko > 2. For SLlB.ller values of ko the 

the effect should be g~ven great er consideration. Impbrtant corrections 

certainly do 'occur when ko < 0.5 and AR;:; 6 and when ko < 1 and 

AR ;:; 3. Corresponding conclusions '>"ill hold for v':!.lues of aspect ratio 

between 3 and 6 and for values beyond these limits, Ai together the 

numerical r e sults obtained may also serve as l;U1 indication of the magnitude of the effect to be expected in cases which 'are somewhat different 

from those analyzed. 

, 
• 

L 

NACA TN No. 1195 17 

It appears from the present calculations that the aerod,ynamic--epan 

effect :i.s appreciably less for hinge-moment distributions. It is thought 

that this fact :was , not knmm previously. 

In connection with present resuJts for ~rtial-span control .sur..-

faces, it should be said that greater accuracy than has been obtained is 

desirabJ.e. The d.iscontinuous spanwise variation of deflection should be 

taken into account ej.ther by using :more term.."l in the series for n or by 

using a spE;;cial function 0 disc; whl,ch accommodates the discontinui tie s 

in a manner analogous to that in uniform-motion wing theory. 

It 1s noted that for the rectan.'3uJ.ar wings' the lift distribution is 

not zero at the tip as woul d be expected. 'rhls may be explained as follows: While i n an exact lifting-·-surface 'theory the ch~rdwise pressure 

distribution at the tip would vanLsh, this condition cannot be satisfied 

in the present approximate theory. In the approximate theory only an 

average condition can be prescrJbed and the natural choice of this average 

condi tion is to make the circulation vanish at the tip. For not tot) 

large values of k this circulation condHion is effecUvely equivalent 

to the condition of vanishing tip-1ift i.ntensity . With 1ncreaslng value 

of k these two conditions become less and less similar. The same observation can be made regarding the moment intoIlsHies at the tip for 

rectangular wings. In an exact theory the moment functions would vanish 

at the ~ip while the present theory is unable to ensure this. 

It must be said that to the extent that the tip condi tions are not 

completely taken care of,the present theory doe s not fully account for 

the effect of finite span. H~wever, even when this difficulty occurs, 

corrections are obtained which are ln the right directlon and which 

therefore may be applied. It should be emphasized that no such dif'fi.-

culties occur for wings with zero tip chord because for such wings lift 

and moment intensities approach zero near the tip in the two-dimensi onal 

theory. 

In addition to the d.1aGrams shm"ing 8panwi/?e lift distributions 

there are given in fi gures 15, 16, and 17 curves showing the variation 

of wing lift coefficient for rigid translation and pi tchlng, and of wing 

rolling-moment coefficients for rigid rolling and linear antisymmetrical 

torsion. These figures show again hQ"· with the increasing ko the results of the two-dimensional and of the three-dimensional theorY approach 

each other. 

AERODYNAMIC-SPAN CORRECTIONS IN FLUTTER CALCULATIONS 

The use of the finite-span corrections of this report in flutter 
18 NACA TN No, 1195 . 

oalcul~tions may be explained by use of the procedure descri bed in reference lL 

In the case of combi ned torsion and bending aileron flutter with 

geared tab, . flutter speed and fre quency are determined from a determ:J.-

nantal equation of the form 

A B C 

= = -

D E F = 0 (21) = :::: -

G -n I 

= ,- == 

The terms of this determinant i nvolve structural components and also 

aerodynamic component s which are designated in the followi ng calculations by the subscript A. The rodyn~lc component s only are considered here . According to and using the notati on of r eference 11- , with 

a dimensionless spanHisc coord.ina te z, defined by 

, 
NACA TN No. 1195 

SA = L:2 (~J[4J -LZ(C - ·U [fh(Z)] [f (Z)Jd(Z) 

+ n( 4 (~S [Le] [fh(Z)j I!~ (Z~ d(z) 

~ = J~~'O (~r)3 ~ -Lh (~ + a)] [fh(Z)] [faCz8 d( z) 

~ = tfo 1.0 (~r) 4 [- ~ (~ + a) + Ma - La (~ + a) 

x (~H)] [feL( Z)] [f~(z~d(z) + n fZ: (~S [Me - Le 

x (~ + a)] [fa(z)] [ff3 (z~J d(z) 

~ = fZ:2 (~;) [Th - Ph(c - eU [rh(Z)] f!',, (zD d( z) 

n = l' Z2 (E...)4 [T - Th (1 + a) - P (0 - e) + Ph(c - e) =A zl br a 2 . a 

x (~ + a)] [fc;t(z)] lrl3( zn d(z) 

~A = fz: 2 (~rY [TI3 + pz(c - e)2 -(Pp + Tz)(C - e)] 

x e:13(Z)]2d (Z) + n Jz:4(~rY [To - PoCo - e) 

X (c - e)] [f,, (Z)J2 d(z) 

+ Q - Q 13 z 

19 

(22) 
20 NACA TN No, 1195 

Introducing for the I~ J MJ T) p) and Q functions their equivalent 

values in the notation of' the pr Gsent. repo:rt) the aerodyna.m:ic determinant 

terms become : 

, '.' 

' .. '.! . 

. ~ 

,.' 

. ! 

, . 

. (." 

:.'.~ : . 
NACA TN No. 1195 

~A = J :.0 GY [1- ;iC(k) J(fh(Z)]2 d(z) 

. . . .. . . ". '.: t'.,: I '. , . 

~~ :: j 1., 0 (~) 3 r .~ ~ + a) ' " 2i~ (~_ 'a)' ) .. 0 O . . L ' .k . k 2 k 2 . , . . r .. . .... ' . . ' . - .. .. ", . "~ '. , .. 

, '.' X [ fa,(z)J.d(z) -. -.... ~., 

J 

1.0 

~A = 

o 

21 

: .' 

(.23) 
22 

According 

as corrections 

C(k) b0comes 

to equations (1) to (4) the 

to the Theodorsen function 

C(k) + 0.. By substi tut.ing J . 

· , 

NACA ~~ No .. 1195 

finite-span corrections appear 

C(k) in such a manner that 

C(k) + OJ for C(k) in the 

determinant t erms, the d.eterminant terms developed using two-dimensional 

aerodynamic considerations become three--dimensional terms. With the 

thr~e-dimensional aero~rnamic series indicated by a superscript 3 there 

may be wri t t en: 
NACA TN No. 1195 23 

( :3 ) J 1. 0 ( E-)2 I~ 21 -; - a ~ = ~ + - k ah !C fh ( z) 'j dz 

o . b~ L •. l 

(:3 ) .' . 0 (b \ :3 r r 2 2i ( 1 \ j \ ~_A = ~I\ + J . - - I -- + - I - - a ) j aa. l :=on o . \ br J '_ L k 2 k ', 2 i.J ' j 

~i:3) = E + Jo~'O ( 'bD 

) 4 ( (:!:2~ + a ) r 2" -t- k

21 (! - a ) 1 a If =A r . 1_ k " . , 2 I CJ.. •• 

( :3 ) 1. ° (b 4 (I 1 ) ( 1 " ,. 2El iE2. 1 1 F :::F +r -.. ( -+a - / -+- a l =A =A Job ) I \ 2 \ 1C J ; k 2 k i 13 ( r - '. .. ./ 

1 . 0 / b . 4{(1 ) x [fa,(z)] [f13 (z) ] d(z) + n J \ )\ .. -;- + a 

o , ;)r ... \. 2 
NACA TN No . 1195 

By rewri tlng tho fjnHe- r.lpan correction terms a nd. inr'U.ca t i ng 

them by a [), prefix, the ~ n.i. teo -erpan c or:t'ec tions for t he bendingtorsion-ailer on flut-Ger de Germirumt of reference 4 a re then as 

follows: 

-.- -.----~-. ------~~-~----' 
NACA TN No. 1195 25 

1 . 0 

6A::J 

:: 0 

1 . 0 

[). B :: J 

o 

x [ f ~ ( z) ] d( z) 

1 . 0 

6 "Q= j' - 0 

1 . 0 

6 :g; = J - 0 

( b "4 ,, J. ' \ ~ 2 2i '1 \ i 8 -_ ) ( -+ a ) I -+- ( -- a), , ucx,[fa,(Z)] d(z) 

\. Dr \. 2 L k 2 k \2 :' 

6 G = 

6R = \ r -:-E.3 _ ! r -

l . + ~ G- -' a ) 1 cree, \l ' 

1_ rc L ~ k ' .J ) 

r E 3 ,r 2E 1 i E 2 , I l 2 '\ --- - -+- I VA) [ f A(z) ] d( z) , 2 ' 2 k J I-' I-' 2rr ,. k 
26 NACA TN No. 1195 

These integrals for the finite--span corrections for the terms of 

th", flutter determinant cannot in general be integrated explicitly. It 

is neceDsary in all but very special cases to integrate these corrections graphiCl'9J.ly or by numerical means. 

Bendi~1g and torsion of ~tang~~ wlng.- This case may be considered individually as an example in which explicit integration is 

possi ble . By taking the val ue s of 0h and 00, from e~uation (6), 

there may n .rst be w:ri t t en 

6A 2i ::::--

k 

1 r 

6B 1 I 2 21 = j -+ (.~ -a) = '- k 2 k \ 2 i 

1 

!C j J 1 '] (l ( rt o, '\ fh(z)fa,(z)dz x + 

. .1 -1 ) I J o - i J 1 ;=;- (2 ) '- / 

0 ~ a, 

1 

~+ a) 

2i [ C 

; T "I n 

( ° h 6D -· '-'1 I 

~.J . - + ---1 

= \ 2 k J o - i J 1 0( 2) ) 

0 "h . 

x fb ( z) f (z) dz ' ! a. 

'\ r 

!'-a~ 1 tiE (~+ ' 2 21 = a ) 

= / l k 2 + k · 2 / 

1 

I C 

iJI 

l 

.r (Da, ) r 12 dz ' J I , fo,(z) x + _ (2 ) - 1 I 

Jo - 1 1 J 00, ~ ..J 

0 

In view of the constancy of k for rectangular wings and the fact 

that the circulation f unc tions occur as quoti ents only, it is permissible to normalize the two-dimensional c i rculations in such a way that 
NACA TN No, 1195 27 

an.d to indicate a correspond1.ng change in the three-dimensional circulation functions by writing 

Introduction of equati ons (27) and (28) into equation (26) gives 

J. 

6. A = - 2i. r C + iJ

1 1 Jr [{ V,>_ ' Ktili Si~ n¢ : fh(z) = k L Jo - iJ1 J _ _ J 

o 

1. 

6.B [

2 2i ( 1 ) l [ i J =- -+-

1 1 r { (\' I - --a c+ / ) K..: 

k2 k 2 ~ J o - iJ1 ..J . ~ ~ lJa ' 0 

1 

( 1 ) 2i r iJ1 6.D= -+a - C+ . = 2 k J o - lJ1 ] J {> ~ o 

x f~(z) - f~(z)fh(z) ] dz 

6. ~ = '(; + a ) [ ~2 + !i ( ~ - a ) J [c 

1 

+ lJ1 1 In (( \' ~~ si~ n¢) f~(z) J o - iJl..J . \.. L ' n 

o 

sin E2. 1 

n j 

By assuming linear tot'sion and parabolic bending, that is, 
28 

f (z) = z, Cf., 

NACA TN No. 1195 

(30) 

the remaining integrations in equations ( 29 ) may be carried out and 

the result is 

l 51\ (K~.h + ~ ~h ) j· J l 16 , + J 0 - i J 1. ~ -----~--'O--- I:::.A=--2i [ C 

= k 

- 1 

4 (' 1, 1, 1, ) "3 Klh + 5" IS h - 35 % h + 105 K7h - ... - 1 

x ~~----~-----~~----~~--------~--- 4 

I:::. ~ = ~2 + a \) LI k

2 

2 + 2i ( ~ - a ') 1 [ c + iJ l J k 2 ., _ J o - i J l 

Kl + - K - -- K + - Ie - -1 ( ' l' l' l' ) 

Cf., 5 ..1.0, 35 50, 105 7 Cf., • • • 

x ----~~----~~----~~---------~--- 3 

Equations (31) will be ap·plied in apperidix B of this report for 

the flutter analysis of a rectangular uniform wing. Equations (25) 

will be applied for the analysls of a tail-flutter l'roblem. 

THE RESULTS OF BlOT AND BOEHNLEIN 

In reference 5 theoretical expressions were given for lift and . moment at midsp&l1 of a r i gid elliptical ~Ying with motion consisting 

. , 
NACA TN No. ,1195 29 

of translat.ion and. pitching about the ,m:ld.chor ,:l line . T~le resulta of 

r e ference 5 may ·be comparc:: d wi th c orre sponding rt's12l ts obta:l.ne cl on the 

basis of the formulas of' the present report. 

The relevant expre ssions of efe n~ e 5 may be written :i n the ::0-::';'. 

:: 1t 

' M Co) r ik 

0 o - l 11 :: 1( ' _. -- Q1'.,R(ko ) : , 2pU2 b

o

2 \ , 2 J b , ~ 0 ' 

+ ' 1( ( ! ( iko A~ 2 " 

I - ko Aa, ) 2 \ J, / 

It may be seep t hat the effect of three-dimensi ona li t :l :m th(?se formulas 

i13 responsi 'ble for the change of the f ugct:Lon ' C (;So) of the two'':'' , 

dimensional th~ory i n to the functions PAR and QAR 1-Thich are tabulated in reference 5 for a r aDGe of aspect ratio and of values of k

o . 

. Comparison with 6<luutlons (1) and. (2) of the pre sent ' report show's 

the following r~latl on8 between t he r e sults of r eference 5 and the 

pre s ent results: 

(34) 

Calculati ons sho,\,1, that for the lUlifoTm defl e ction fu..71ctions which 

are considered here, 
30 NACA TN No. '1195 

As a formal difff"r ence bet10leen t he two sets of r esults there r emains the fact that in the preE?ent work the carre cticn terll1.S for lift 

and moment are the same} while in the work or r efer ence 5 this i s not 

so. The authors of the present paper are not able to decide "Thich one 

of the two sets of results is nearer to the actual facts in this 

respect. It may} hOvTever) be noted that the differences between the 

p- and Q-functions are Quite sI\lall. The ordf3 r of magnitude 

of the differences between the two appoars to be no gr8ater than the 

order of magnitude of the deviat::'ons of tIle approximate results from 

~he results of an as yet not existing exact solution of the problem. 

A numeri:::al comparison bet,veen tho P- and Q-functions and 'the 

function C + cr has been made for the four values of aspect ratlo, 

6, 3, 2, and 1.5 . Figure 22 e~roduces the r eal and imaginary parts 

of the various functions} together "\"ri th the function C of the tvrodimensional theory. It is seen that agyeement between the ti'TO kinds 

of three-dimensional correcti ons is Quite close. A theoretically 

important di ffer ence is the fact that f or i ncreasjng ko the results 

of the present work converge toward th8 r esult of the wo-~ime nsional 

theory while} for aspect ratios 3) 2, and 1.5} the results of 

reference 5 do not converge toward the result of' the two-dimensional 

theory. As ko approaches zero the results of the present work approach the results of l ifting-line theory . It is noted that the same ' 

is very nearly true for the functi on Q} "hile the function P is 

consistently somevThat smaller than would. follow from lifting-line 

theory. 

Determination of terms cr.- It "Tas found that,for the tyro forms 

of motion considered previously, sufficiently accurate r esults were 

obtained from a one-term approximation for the circulation function 

n. By taI;:ing 

(36) 

and determining the coefficient Ie1 by satisf;ying the integral eQuation (8) at midspan (¢l:::'~) there follows from eQua.tion (17): 
NACA TN No. 1195 31 

and hence 

n (0 ) , 1 

----- = ------~------------------ _ (2) 

n ( 0) 

Then) according to e 9,uation '(6) ther e rollows for the correction terms 

r ' iJ~(ko) -, 

0h (ko) 0a,(ko) I C(ko ) + I 

= ;::; I Jo(ko) - J~(ko) -' 

r 1 

J X (38) I -1 

It 

~(ko)S~ ( k s, 

It L \ 1 + -- I 

S \ 0 ' 2 " ) 

Equation (38) :l,s evaluated by means of tables II, III, and IV. Numerical values of the func ions a so obt ained are l isted in table IX. It 

may be noted tl~at equation UG), in addition to giving :t11e correction 

term for translation and pitchi ng, also gives the proper correction for 

the deflection of a full-span aileron on an elliptical wing as long as 

the dimensionless aileron-hinGe- line coor~ina e e and leadin ~dge 

coordinate c are constant along the span . 

As a conclusion to thi s discussion i t may De stated that from a 

pra,ctical point of vi elT there seeIl',S, to be little to choose between the 

results of reference 5 'and the present results i n those cases where the 

results of refer ence 5 apply. A practical disadvantage of the results 

of reference 5 appears to be that no prov _sion is made to obtain corresponding' results "Then other ,than Uniform d.eflec"ti on functions are to be 

taken into consideration. 

The present authors \-I'ould have l iked to include also a comparison 

of their r esults vTith' t he ' r e sults of W. P. Jopes in r eference 2; however ) notational clifficulties could not be overcome ::n time . 

Massachusetts Ipstitute of Technology; , 

Cambridge , l-1ass. ; ' .Tanuer y 4,. 1946 . 
32 NACA TN No, 1195 

.A1?PENDIX A 

NOTES ON TEE EVALUATION OF THE FUNCTIGrJ Sn 

According to equation (16) 

n 

= sin n ~ + ikos 

sin ¢ "Ii 

..... 

J cos ¢ - cos e 

J 1 cos ¢ - cos 81 

o 

X F(kos I cos ¢ - cos e I ) cos nade 

and according to equation (9) 

F(x) 

= / 

. o 

co 

-1A. 

e J x

2 

+ A.

2

1 dA. :: R(x) - H(x) 

XA. 

Tile function F is given numeriGally in table V. It can be shown that 

it possesses the foIl wing limiting behavior: 

x ~ co 

x -?> 0 

F(x) ~ 1 i 

F(x) 

: , 

i1t ~ - In x - o. 39 In 2 - -2 

It can further be sho .. ffi (reference 6)" that the following representation 

holds: 

F(x.) 

( . 

= i .I ./ 

x 

G(U") :.:.: ~ du 

u2 

where G(u) satisfie s the differential equation, 

d2G _! dG _ G i 

= - du2 U du u 

with the initial conditions, 

--- 

(a) 

(b) 
NACA TN No. · 1195 33 

G(O) = 1, .G' (0) = -5. (c) 

Equations (a) to (c) are li st e for future r eference. So far, i t 

has not been found possibl e t o <ieri ve computational advantaGes f rom 

t hem. 

Graphical integra ti on.- The integral, 

n: 

r cos, ¢ - cos !- I(kosl cos ¢ - cos rJ l )cos n8(lG .) I cos ¢ - cos e! 

o 

may r eadily be evaluated by graphical or numer:ical (Si mpson's rule) 

m0 ns as the ~ntegrand r emai ns, fini t e throughout . 

The i ntegral, 

n: 

~ 

.J cos ¢ cos e ( . ¢ I ) '"I--J

·-- ---8 I R kos! cos - cos e cos r 11e 

lcos ¢ cos 

o 

is subject to the difficulty that the fuIilltion R is logarithmically 

infi ni t e when cos ¢ = cos e,. The r e may be written, for E: ~ ¢ S 1'(-(, 

1'( ¢ -e: ¢ +( 1'( 

/ = 

~ / 

.( + . / 

/' +.! 

0 0 ¢ -( ¢+f. 

and the first and thi rd integr als evalua t e d graphically. \-lhen ¢ 0, 

,·rri t e i nstead 

1'( e: 1'( 

/ I ;- = 

.J + / . J 

0 0 e: 

The second integral may, for sufficiently small € ) be estlmated 

analyti cally as follo"Ts: 
34 NACA TN No . 1195 

de = dT) 

cos ¢ - cos e ;:; 11 sin ¢ 

cos ne :::: cos n¢ -' T) n sin ~ 

He nce 

¢+ c € 

( - () 

./ / ¢-€ - € 

- 1f~ sin n¢ ) dT) = - 2n s:'n n¢ i 

1 " .... 1.\ 

1 1 

2 

This ap:pr-oxlmation is. satisfac t ory when si n ¢ kos-,: ,< 0 .05 . 

In this manner the functic~8 Sn have been calculate d for 

( d) 

kos = 0 . 5 and vari ous check c''i u.lations have been ca rried. out for the 

va lue s of S obtained from n:.f0rrJllCe 2 _ n 

wher e 

Approximate an,a lyti cal i nt'" atio .- 1'/1'1 te Sn in the for m 

T( 

II r:!: _ 7x I F ( ! x I ) ~ x I . . j . X I .' 

cos n8d.e 

o 

x = kos(cos e - cos ¢) 

(e ) 

The r ange of integr ation in }~ i s I xl < 2kos . In this range ther e 

may be writte n approximately :. 
NACA TN No. 1195 

N 

\-' n = - In x + ;' x 

n=o 

Iv: 

=) ~f1 L _ 

n=o 

1 

J 

For a given kos &, arbitrarily close approximation for Sn may be 

35 

(f) 

be obtained by makinG the number of terms, N and M, large enough. 

For kos :s 2, (x < 4), the followIng approxima.tions have been 

used: 

Ra(x) = - In x - 0.270 + 0 .764x - O.129x2 + 0 .01lx3 

I (x) 

a 

1.319 - 0.757x + 0 .20?A2 - 0.020x 3 

(g) 

The coefficients of thase expressions were obtained by making Ra and 

1a ~ree vdth R and I, respectively, in fo~ pOints of the interval 

o < x < 4. 

By introducir~ equations (g) i nto equation (e) the following integrals occur: 
,., 

NACA TN N~. 1195 

/ cos e - cos .¢ 

i cos e - cos ¢I cos nede = 2 sin n2 

n I 

i 

o 

'j) ~os G - cos ¢)cos nBQG = 

rc 

(co~ - cos ¢) 3 

-- ' cos neQe = 

" j I cos e - cos ¢ I 

o 

I 

J' (cos e - cos ¢)3 cos neQe = 

o 

r 

, rc 

) 2' ,., . 

I 

I 0, 

L 

n 1 

n ;: 2, 3, 

cos 

. = 1 

n = 2 

sin 2¢ 

\ 

2 )' 

2 sin 2¢ sin2¢ 

3 

( sin(n - 2)0 2 sin n¢ 

'.n(n - 1) (n - 2) n(n2 - 1) 

,": _ _ ·1 7 n ~ "! 

+ (" n;:3,4, ... n(n + 1 ) (n + 2) ) 

3 rt 

8 

3 

4 

rt cos ¢, n ;: 2 

n ;: 3 

A some"That less simple calculation leads to a recursion formula for the 

integral, 
NACA TN No . 1195 

1! .n 

_c_o_s_ e _ _ c_o_S-..L-¢¢_1 ( In I cos e ,- cos ¢I )'1 cos nede 

Icose cos 

c 

~1e recurs~on formula is 

/ 

- ;)fn_ 1 _2 (1 1\ .L' ( 1 cos ¢ fn + + "- Ii) .!.n+1 

2 sin (n - 1')¢ sin Cn + ::: l)¢ I n ::: 2, ~ 

J. 1 .J) 

n I- n- n + -' 

vJi th the ini tial conditions) 

f 1 = 2 sin ¢ i 2 In I sin ¢! + In 2 - 1 1 - 2 ('.os ¢ ( ¢ - ~ ) 

::: s:i.n 2¢ j 2 In I sin ¢I + In 2 

L 

sin 2¢ 

2 

37 

On the basis of these formulas a n~ber of values of the fun8tions 

Sn have been calc·..llated and the results compared vi th the results of 

the graphical sol ution . It "Jas found that there w'asvery good agreement 

for the imacina~' parts of Sn: differences between the two value s obtained by the hlO methods occurrj.::1g only in the thir d. or fourth decimal. 

Somewhat les8 satisfacto:cy agreement was found for the r eal pa.:cts of 

Sn} with deTI.ations of from 1 to :3 percent. \\1h1le for practical purposes an uncertainty of this magnitude with regard to the values of Sn 

is of no importance J it :1.s bGlieved that the results of the graph:Lcal 

analysis are the more reliable. It still appears to be desirable to 

set up an analytical scheme by means of wh i ch the val ues of Sn may be 

calculated accur ate to, say, three or four Qecim~ls . 

A transformation of SJ,(kos, 1!/2) ,- It may be shOlm that tbis func- I 

tion can be written as a single integr al when use is made of comj?let e 

elliptical integral s . Fron equation (16) 
NACA TN No, 1195 

rc/-j 'l 

I 

2} .I 

o 

with 

, . 

. J 

o 

= 

co 

e

-D .. ( 1 ]. J (koS cos 8 )2 + )...2) 1 dB 

\ --- + - - ----------- cUv ,. kos COS A )", l;:os (cos B) ... .J 

cos e r • J 

/ 

:: 

rc/2 co 

e- i )... r /' j ekos cos 6) 2 ~ )...2 "- '\ 

J (~+ ; de l d.)", 

0 }cos )... kos/- I I 

' '> 0 

rc/2 ' rc 

,Y J)...2 + (kos)2 -1)...( 2 1 r: /, (kOS)2 I . 2 I J e 1 - + -- .J j 1 

Sl n e de J d). 0 " kos )... k 8 )... )...2+ (koS) 2 

0 

0 

:: 

:: 

there fol lows then 

Q:) 

{ l 

I 

--rc J -ikos)... ( rc 1 Ie 1 - + - I 1 '~ 2 )... ,_ 

o 

- ,.r;-;-)... 2 E / ,-1_\-11 dr-

\. \jl -:- )...2 }JJ 

where E designates the complete e:liptical i ntegral of the second kind. 

It appear s that this r epresentation might serve as the starting point 

of further analy tical .Iork . It is probable that corre sponding, l ess 

simple expr essions may be obtained for Sn(kos, ¢) also when n f 1 

and ¢ f ~ . 2 
NACA TN No. 1195 39 

A:rPENDIX :8 

APPLICATION OF TEE THEORY TO WING FLUT'.I'ER PROBIEMS 

DISCUSSION OF THE THEO~Y 

It "'as shown in reference 4 that the equati ons of motion of a ,.,ring 

in bending and torston aileron flutter with a geared tab can be wr;itten 

i n the form or a determinant 8,S 

A B C 

= ::: ::: 

D H' .1" F :::: 0 (Bl) = ::: := 

G H I :::: 

In t he pre sent r eport tile el ements of the flutter determinant of reference 4 are r epre sented as t he sum of a. struc t ura l term) a 

two-dimens: onal aerodynamic t erm] and a three-di mensional corr0ction 

t erm a s f olJ.ows: 

A :: 

~s + ~ 

B = ::: ~s + ~A 

C = Sa + SA == 

D 

::: = ~s + ~A 

E :::: ~8 + Ell ::: = .... ""1-

F Fs + !A ::: :: 

G G + r 

::: =8 '.;,.A 

H ~8 + H ::: ::;:A 

I :: 

:= ~s + ~ 

+ A 

= 

+ b. 11 

= 

+ l::. C 

== 

+ b. D 

::: 

+ b. E 

+ ~ F 

:::: 

+ b- e 

::: 

+ 6 H 

:::: 

+ 6 I 

::: 

l 

I 

I 

I 

) 

~ i 

1 

J 

(:82) 

In these teros the subscript s refers to the structural terms; 

the subscript A refe rs to the aero~ynanUc terms calculated by convent i onal two-dimens:onal methods int egrated along the Spfu.: and the prefix 

6 . r efe:;:'s to the fim te-span correction to the aerod~;: namic parts of the 

elements of the ·ieterminant. 
BR = ='-' 

Ss = 

40 NACA TN NJ:,. 1195 

The aerodynamic parts of e~uati n (B2) are given by equations (23 ) 

and ( 25) of this r epm:·t . The structur al t erms aTe listed. be low n the 

notation of reference 4. 

1. C S ' 

(~xa.)r j' a. . [ fh (z)J . .'. f ' ( ") 1 L •. a.. Z ..; d.z 

.J (Sa..)r 

XI3 )T J 

Z2 ~ [:b( z)J [ f 13 (z)J dz . 

z· (S(3) r ... 

~s = ( ~ ['~132 + (e -- a)x!3] ( 

-T 

2 ( (_1.) .2 . ( (J)(J. \ :2 (~rp )r! l--

\ (l.)a.. / (l.)!3 / 

+ n f ~ [r52 (2 +. n) \~ 

"" + 2 (f - e )xc J ~ I 

-r 

r 

J Z 

:> 

1 0 (2 + n) + 2 ([ - o )bSo ; i 

[10 ( 2 + n) + 2 (f - e)bSc) , r 
NACA TN No. 1195 

In addition to the general structural terms of eql,18 M.on (B3 ), the 

structural t erms for tl1e £Jpecial cass of the unHorm rectangula r ,.,ing 

tn bending and. torsion flutter with a pa r abolic bending I"!nd. a linea.r 

torsion mode are list er. aepara t e1y as fol1mTs. The corresponding aero·-

dynamic terms are given in equations (23) and (31). 

M ( O'a. )2 ( 1.1), ,2 ( -I 

\ 1 1Gb , I ~s = 1 - - + ) \ I 5:rrpb ./ 

o lJ.) / \. (1)a. I 

"I , 

~s Sa. 

= 

4:rrp·oo :; 

(B4) 

~s 

Sa. 

= 4:rrpbo 

:.: 

Ia. , '" (1)tL \ 8 p' = ---- I 1 \ - . (1 + -i (-1 ) -;0 '1 -ua. 

31tpbo L '-ill )' J 

where 

~1 mass of the wing per uni t span 

Sa. static moment of wing pe r uni t spun. about t he e l astic axis 

lex. moment of inertia 01' the wing per urD.t span about the e lastic axis 

In order to apply fini t e- span correct·· ons to f lutte r analysis a 

procedure should be outline d. A. possi ble method. ts to 6ina1y ze first 

t he problem in question by using the convt-o n t i onal t.!o-d.::'mensiona1 a e rod.ynamic terms i ntesrated. along ·t,ho span . If the result r~3 speed occurs 

in a range deemed cri t:i cal . the finite-span corrections ma~r -be intr o:""" 

duced. 

Si nce the analysis '...lsing the t"t1'0-a.ime:asi ona.l values of the a e rodynamic parts of the terms of <~ f..L..~tter de -:'e:::';:ui nant SiY,,'-) s the approximate range of fre ~uencs :~ara:n0te rs i n whicll flutter may be expected .• tl"'e 

range of frequency parameters "Thich mast be inve stiga te for an analy sis 

using finite- span correc tions is mini:m:i.zod . In general) the m<mber of 

r'requency parame ters invest igate d by us~n<.: three-d.i:nensional aerodynam.i.c 

'considerations may be l imited. to thrOE; '.ralues when thi s method is used. 

In the ana l y siS conducte d it has been f ound t hat the calculated flutter 
42 NACA TN No. ll'f~ 

speed f'ccurs a t a somewhat lower value of the f'reqwmc: para;.nete r '.-lhBrr 

the finite-span correct,ions are applied . This sl1ou.ld 'oe kert in rnlnd. 

"\orhen conducting flutter analysis which takes into account t.he effbct 0:-

a finite span. 

The proced.ure outlined will be used. :;'n the tvro eX.9Illples -pr esented. 

in this appendix . 

EK.4NPlE I 

This example is presented pr::m.a.:ril;,r to 5.l1ustr ato tt.e methocc of 

ap:plyj.ng finite- span correc:':ons to a simple .rir..{3 in be::J.ding and. t orsion 

flutter. The wJ_ng Sf' .lected. for the analYBi ," was t.he N-':5 ~riIlG o~ r efere nce 7 . In order ·to SUPl)ly rr:.ore accurate data on 1,hl.) charc.cter1bt:!.CS of 

tho wing] it was again t e ste d. in the 1'1.1. T. flutte r la'cor atory 'vi th 

equtpn:rnt which was not avail.a.ble when the or~" g:i.:iJ.al t osts werb conducted.. 

The N-75 ~Ti118 is a rectangular wing of aspect ratic 6 wi th the 1'01-

low'ing characteristics: 

28'::>0 = 60 in. 

2bo 10 in. 

a = -0 . 30 

}II 0 .ocr"6 sl ug /rt 

Sa, :::: 0.00068 slug-:-t!ft 

Ie, 0.00059 slug-f't~/ft 

illa,(measured LlllCOUpleQ gro~ 1 

freql;.ency in torsion) ::: 8 .9 ::::ps 

(L'l'lJmeasure d. uncoupled ground 

freqt:.ency i n. bend.:i.ng) = 3.9 cps 

Ch = 0 .062 
NACA . 'rl~ No . 9~: 

.... I 

' . .. .: I 4 . _ '" 43 

.' Si nce .all ex:pe:r:imental te'ets were .:cG.n d,ucttid at , approximately stand-

~d s E;8.":leve l ~ond.i tions .. the a.n8.lysiS will be conduc t0d. · ~s:J:ng the' 

standard s ea-level val ue for the densi ty, 

. 3 0.002373 slug/f t 

. "" 

, , 

.. .~ 

No data on the tYue flutter mode shape s we r e pr ov ided!. . 20 the anal~ 

ysis vriJ.l be ca::ried out vrl t.h 0. l :·'.near t or Sion and. a p~abo': c b..:mdi ng 

mode a bsume d . The terms of th~ flutte r determinant for this spe c i al :-

caSl; ha ve be en i ntegrated expLcit.ly a..T1 l are roy G. d. in e'luati ons (2'3), 

( 31) ) and (134). The analysis w~ II be pdrf oy:mecl and.. th.ese 5.ntesrat ed 

flutter terms uEted . 

Flutt."lr .\nal:e ~ R wi th ' ,,"JCi-:Dj .::J.0I:sional Value s 

f or th", rody l'l ~ c Po.rtG 0:' the Dete rminant 

Whe n tho 8uppE':hl data is i ncor por ") .. ~ed. j.n the structuroJ. perts of 

the determinant "te :::ws fr~T:1 e quati ':In (BJ+) , t he s.::. t e rms 0':; c omo J 

M r 

( 2 ( (J.) . 2 , 

s == !l ~ h " (1 + igh ) 

51!pb o 

2 ' (J.) / , ;- ) 

u 

/ (J.) )2 .., 

:: 1 -, ~~" I 1 o ~O?J ( - ~ (1 + iBh) I • ).:;'. j.L . _ .... - I 

1 \ (J.) ...J 

~s ~8 Sa. :: :: ---- 0. 3154 

41!pbo 

3 

The cor responding ~er od~TIEL~c par ts of the de terminant tellnS ~A' 

} l?[-U and A f r om a qua t i ona ( 23 ) f rom tvro -CJ.imensi ona1 considera ti ons 
NACA TN 'No. 1195 

ma~" 'be cc.lculated di!'ectly for severa·l values' of the' frequency parameter 

'01 th~ use of 'Cable :;:. 

'llh8 two-,d.i.mensional aerodynamic terms for the rectangular wing with 

a lineal' torsion mode and a parabolic bend'ng mode a re: 

= :::. 1 ! :- 1 _ : 2'C1 (' K, 'I 1 

:5 1 k .. ~ -

B := ;.' -r -' (1 "" a.) - C(k) r 2 _ + i (-21 - a 'J:! _1 jl =A. ' .. 4 I k L ~ k . : 0" f ~. 

E 

=.0.. 

1 r :. /1 '\ :0- 3 1 - k !\ '2 -' a -' + 

'I 

X C(k) ( 

' 

-

I

I 2 9i ./ 1 \ , 'r -+'::""/--a 

l. k 2 k \, ~~ ,J 

Sjnce the aerodynam:i, c terms of the fl'ltter dcterminaJlt for this 

special case may 'De calculated by simple formula sub stitution of the 

values of C(k) a) and k) and s~ nc e C(k) is tabulated ae a fu.."lction 

of k i n t!3.'''le I ) the actil.aJ calculation of the aero.d.yna.:nic tErms ~s 

not perform0d here. The d.e"A:Hllli nant 4 ems hO',18ver) are tabulate d in 

table X a s a function. of the frequenc:' ~')e.r8.!ll.ete r k. 

Irhe f lutter de terminant thus evalua'tGd and' tabulated. in t able X 

wili be solved by the method. outlined. in reference 4. This method 1:1:1, 11 

not ba illustrated) 'but. the t .... ro solutions to e ach of the resultant COll:-

plex quadratic equa i:-:..ons aloe ta: ulated. a s a fu.."1ction of t?e frequenc;r 

parameter k in the foL ovlin~ ta'bJ,e.: 

o 

-------- --
NACA 'l'N No. 1195 45 

RO()'fS 0]' 'IID~ FIJUTr.rER EQUATION 

The flutter speed. by this methcd is d.e termined graphically in figure 23, 

and is e qual to 28.6 m~h . 

]'l1.1tter Ana l ,ysis wi th Ftn.i te .. .span Corree tions 

Sinee the priI1L9.ry pu:!:"pose of the r'eport j s t o i ilustrato a means of 

incorporating fini t o-spEm correctione in flutte r analysis, the Illeans .of 

calculaUng the corroc c;on functtons cTj(h, . 0" /3, 'Y) will bE:: illustrau.' d 

and the s e functions applied in this simple flutt er probl em. 

It has been shO'l,m in 8Q.lW,t. on ( 31) f or this special case that the 

fj.ni t e-span corre ctions may be oxpressed l n the following form: 
--~~-~~---~~,~.-.~~ ---~~~-------~ 

46 

f (z) ::; z 

0, 

NACA TN No . 1195 

DA ::; 

1 i ",11: / , K3h ,. ( 

- ~ K1h + - - ::; 16 \ 3 / 

, 2 2i ( 1 , 1. 0 , 3 , 

J 6 B - + - \ - - a , LJo,z dz == k 2 k \ 2 : 

'. ! / {.} 

1 1 i / 1 . i r 

{ iJ1 == - i_ - + \ 

_. a I ie + 2 ~2 k 2 : T :'.. J 1 / 1- "'0 

/ , K rr ,. K Gn, ' I - \ 1 0, + I - 1 I 

L 4 \. 3 / _: 

2' ( 1 

1 . , 3 

~k ( 

1 t-> D == - + a J LJhZ ~ == - + a == k . 2 / 2 / 

C + 

~ 4 lI' ~ ~, 4. -' ~ ~ L 3' ·""'1}] + 15 .. '",!1 - 105 Ksi.1 + 315 ,h 

I 

1! I 

::; 

,/ 1 i 2 2i / 1 (a + I - + - 1 -- a . \ 2 ·J k 2 k \ 2 

1. 0 

J 6E 

1J1 

'1 

l C + I 

J o - iJ1 j 

, 

1 ! 

I 

_ I 1 _, l ' 1 , 

X i_ K1 0, + 5' K3a, - 35 K =a. + 1;5 Y.. ;'1'0, -

Si nce t he a lgebraic t erms deve loped r t hi s special example cannot 

be developed easi l y or t he gene ral case , t he ~e ti nn t ermR wi l l 

usu~lly have to be deterrrd-ned by graphical or numb r i cal i n tegrati on . For 

t his exampl e , then J t he correcti nns will be dE:term.i.ne(l by gr 3.phi cal integrati on and by formula substi tution i n the r e lations l i sted pr ev iously . 

It can be shown f rom equati on (27) that t he correcti on func tion LJj 

may be wrj. t ten for the r ee t anglL-ar "'ing as, 
NACA TN No. 1195 47 

.,here f; (y) is the deflect:'on at the st ation under consideration and <.J 

,-, is a function whjch is proIJo~tional to the circulation and may be 

expressed as, 

w ~ )' K' . sin n¢ ~. nJ n 

The K' 

nj terms of this Fourier series may be found from the set of 

e~uations, 

The relation, 

w = ---

f (y) 

j 

holds only to the special case of the rectrD1Bular wing ; but the general 

method of finding w is the same as that for find.ing n j' 

In ordor to illust~ate the method the correction terms will be calculated for kc = 0.4. Since the method is similar for any value of kQ, \ 

the procedure w~ll not be repeated for ~he other values of the frequency 

parameter investigated . The results, however, will be tabulated for 

other values of Ko' 

The parameters and tabulated functions ne cessary for the determination of OJ for the selected value of the frequency parameter, 

ko = 0.4, are 
48 

k :;: k ::: 0 . 4 

(\ 

b 

- = 1 

"bo 

S = f 

k 8 :;: 2 . 4 

o 

NACA TN No. 119~' 

The Sn(¢, kos ) functions YstEl(l :'.n "tr_e foEOvring table e obtained from "the plots of Sn(¢' kos) a;3ainst kos ::'n figur") 2: 

I :. 

, -

'I sr. I '0 

I 

I 

10 

I 

I 

I 

I 

i 0.4 

j 

I , 

i 

, 

n 

7 

J! ~ r ~ r 1 

I 

I 

i 

I 

s:i,n n ¢ 

n 

1 

-0 ,~ ~, 

. . L').) 

. 2000 

-. 1429 

0.9165 

-. 1099 

-.0>35 

.1330 

0.600 

.31~0 " 

-. 0151 

-.1393 

o 

o 

o 

o 

1 

I 

! 

i 

I 

I 

! 

--'11 

Sn(¢' k3) I 

0. 230 _. 

-. 493 + 

.6-,0 -

-.715 + . 310if 

! 

, 

0 . 280 - 0. 300::.1 

-. 204 + .11511 

-. 326 + .174il 

. 7"'f'-'''' - .31511 

0 .588 - 0 .280il 

Q - "l <; ~ I . -,85 • ...,3./.1-

- .no + .137i 

-1 . 330 + .3601 

1.665 + 1 ·310i 

3.il:; + . 772~. 

5.050 + . 4E35i 

7 .025 + . 344i 
NACA TN No. 1195 49 

iv'l1en the se funct:i.ons have been tabulated, the integral equation 

must be solved to fix the circulation distribution along the span. It 

is thought that the circulatl on can be approximated satisfactorily for 

a r ectangular .,ing by the use of a fo~term Fourier series approximation. For this examp1B the integral equation will be satisfied at 

:dSbo = 0, 0.4, O.B} and 1. 0 . 

The integral equation, 

, I 

\ K .A = fj (Y)m ~ __ nJ run 

tben becomes a set of f our simultaneous linear equations to be solved 

for the Knj coeffi cients. Since the modes assumed are s~etrical: 

only .dd valu ~ of n will occur. 

The ~ functions are equal to 

A = sin n¢ + ~b ~(k)Sn(¢ ) ks) 

nm n sbo 

and when these terms are evaluated for n = 1, 3; 5, 7 at 

yl sTIo = 0, 0.4, 0. 8, 1.0, the -'\w functions become ) for y/sbo = 0, 

2f(0 . 2641t - 0.09641) (0.230 _ 0.2951) 

6 

= 1 + (0.1384 - 0.0505 ' )( 0.230 -- 0.2951) 

= 1.0169 - 0.05241 

A31 = -0 · 3333 + . (0.1384 - 0.0505i)(-o.493 + 0.3541) 

= -0. 3837 + 0.0739i 

%1 = 0. 2000 + (0.1384 - 0.050·5i)(0.630 - 0.337i) 

= 0.2702 - O. O~85i 
50 NACA TN No. 1195 

~l = -0.1429 + (0.1384 - 0.05051)( -..() .715 + o. 310i.) , 

= -0.2262 + 0.01901 

, " 

A~2 = 0.9165 + (0.1384 - 0.05051)(0.280 - ,O.300i) , 

= 0 .9401 - 0.0557i 

A32 = -0.1099 + (0.1384 - 0.0505+)(-0 .204 + 0.1151) 

= -0.1323 + 0.02621 

A52 = -0.0936 + (0.1384 - 0.0505i)(-O~326 + 0.1(41) 

= -0.1299 + 0.04051 

A72 = 0.1380 + (0.1384 - 0.0505~)(0.T12 ~ 0.315,1) 

= 0.2289 - 0.08261 

A13 = 0.600 + (0.1384 - 0.05051-)(0.588 - 0.280i) 

= 0.6672 - 0.06841 

,A33 = '0;3120 + (0,1384 - 0.05051){O .985 ~ 0.5351) 

= 0. 4213 - 0.12381 

A53 = -o.ol51 + (0.1384 - 0.05051)(-0.110 + 0 .1371) 

= -0.0234 + 0.02451 

A73 = -O.lJ98 + (0.1384 - 0 .05051)(-1.330 + 0.3601) 

= -0.3057 + 0.11701 

. ~ 
NACA TN No. 1195 31 

and for y/sDo = 1.0; 

A14 :;:: (0. 1 384 - 0.0505i)(1 .665 + 1. 3l0i ) 

0.2966 + 0.0972i 

A34 :;:: (0 .1384 - 0.0505i)(3.115 + O. 772i ), 

:;:: 0.470:, - 0.0505i 

A54 = (0 .1384 ~ 0.0505i)(5.050 + 0.485:1.) 

:;:: 0. 7234 - 0.18791 

A74 :;:: (0.1384 ,.- 0 oc:;o~;) (7 or."'-, • '.,1 ,.... • _ ... ~ 4- 0. 344i) 

:;:: 0.9896 - 0. 3C721. 

:4i th t!1e valuea of A determined. and sJ.nce the modes i n Cluesnr.t 

tion are knm-m, the S '3 t of s:=;'mul taneous equations , 

....., 

,> K~,j Anm :;:: f j ('Y)m 

may be solved . Th."_s solution is carried out in table XI by the use of 

t he Crout method o~ r Gi'e:;:'6nce 8. 

ldhe n the j terms have been calculated, the fu..'1c tion w which 

i s proporti onal to the ci rculation i s s~abl shed ~'1d the corr ection 

functions C1~(j = 0" h) ma;;r ba calculated. . The value s cf w at vari- ,/ 

ous stations along the s:pe..n are calcuJ..a.t0d in table XII and the 

OJ( j :;:: ~,h) functions are calcula~0d in t able XIII. 

By usins the cor=ect ion f~.ction , the correC~lons to the determinant t erms m~y be evaluated graphi cally . I n table Xlv the i nto grands 

of the f ollovTing per tinent integrals are tabulated. for this graphical 

integration: 
r - -

I 

I 

I 

I 

NACA TN ·No. 1195 

, '. 1 • 0 <l. /} 1.0 3 (11.0 r ; • 0 

/ (}h Z'dZ () z dz 0h7.. 3 dz a z 2 dz 

a. ' ' , I a. i 'J o 0 ':) J .J '0 0 

These i n tDGr als a r e evaluated gr aphically in figure 24, (a), (0), (c) } 

and (0,). 

For this 81le cial caSE) it is possible to compare the va 2.u0s of the 

Gra;Jhi cal i ntegrat.ion .rl t h t hose obtained by 8ubstJ. tuting values :L n the 

alge brai c e quations of e quation (::)1) ae is -ione in the follm-ring t able : 

Fo:' k ',) :: o. 4. 

Te rm : 

1. :: 

.r 4 

O'hZ -dz 

0 

By graph:ical nt ~3.t:i.on·- -~J' clire ~~-~ 

----. -.-'-' - "--1--' I 

I -0.039 + O.OlE I 

I "-0 . 040 + O. OlO:~ 

1. 0 

f Ga.z3dZ 

;) 

I -. 042 + . 014i -. Oi 2 + 

1. 0 

f 3 

(}hZ d z 

(i 

- .043 + .0121 -. 043 + .013i 

1. C 

f o () z2 dz 

(J, 

-.04J + . 0J.I'i - .047 + .018:;. 

._--_._ ._---

The a erod,yna7"-; c correct.:.ons themse lves are tabulated for a ran€;'e 

of ko in table XV along \'li tl: t!:e cal cul9. ted. de t e rminant tey1.llB c()~ 

r e cted for f~nit8 61an e ffec ts. 

The flu tter detcrmj,nar~ts o~rec ted. f or f~ ~. t e-s11an eff e cts, the 

t e rms of which a::-e E s:;e d. i n 1,a' le )CV J can be expanfLed ant 801 vcd,. Thi s 

was done, anc~ the esul~' ir..g · oItp~e x c:.u3::l.rati c e quat::..on Vias solved by 

use of the metho~ of ~fe nc e 4. The se r e sults are ~abulated as a 

flli"1ction' of ko i n the f oi'l m,,:i.nc ta.'tle: 
NACA TN No. 1195 53 

ROOTS OF THE FLUTTER EQUATION 

First solution Second. sol ution 

ko Vl V2 

(mph) gl (mph) 82 

0.333 1 20 .8 -0.565 33.3 0.035 

.40 1'17.2 -.428 29.7 - . 058 

I 

.50 J 13.5 I - . 316 25.8 I - .120 I I 

The actual d.etermination of the calculated. flutter speed. is carried. out 

graphically in figure 23 and the flutter speed is equal to 34.2 mph. 

mscuesion C?f the l~esults of example I.- When the N-75 wing of reference 7 vIi th a 10-inch chord was tested. for flutter, the wing was 

observed to flutter at about 34 mph for low angles of attack. The same 

vnnG was ~gain tested by the staff of the M.I.T. flutter labor atory, , 

when thj s ,,,ing wes selected. for analysis in the present report, to provid.e the measured uncoupled ground frequencies in bending and torsion 

and. to c.etGrmine the decay curves of the unco1A.pled bending and. torsion 

modes 8(, the damping coefftcients could be calculated.. In add.ition, 

the wing ',Tas again tested in flutter and was observed to flutter at 

34.2 mph. 

Since the calculated value of the flutter speed with finite-span 

corrections was found. to be 34 .2 mph for a value of the frequency 

parameter ko of about 0.3, the check between the theor etical and 

measured values is remarkable. It should be noted, however, that this 

close check was entirely "unexpected since the flutter modes used in 

the analysis were not found by conducting a ground-vibration-mode 

analysis, nor were they observed in actual flutter tests. It is believed, however, tl1at the mod.es used were reasonable. 

As an afterthought, the M.l.T. flutter laboratory installed. end. 

plates in the .Tind tunnel as near the ti,s of the wing model as was 

practicable (t he gap was about 1/16 in. ) in order to approximate twodimensional flow condi tiona as nearly as possible. When this was done, 

the measured flutter speed was 30.7 mph. The calculated two-dimensional 

value was 28.6 mph for a value of ko of about 0 . 41 . This too is a 

remarkable check, conSidering that a truly two-dimensional flow was not 

possible under the test conditions because a slight gap was left between the wins tips and the end plates . From this analysis it seems 

that the extra vrork of a three-dimensional analysis is justified. 
54 NACA TN No. 1195 

EXAMPLE II 

In th~s example the tail-flutter problem of Example No. 7 of reference 4 will be analy zed with the benefi t of finite-span corrections. 

The tail in ~u stion fluttors in fuselage torsion and fuselage bending. 

For the sake of the analys s the surfaces themselves are assumed to be 

r igid . 

v!hen this example ",as analyzed in reference 4, a method of e~uivalent aerodynamic chords was useQ to calculate ~~e aerodynamic 

coefficients. Thi s method is lli.sui table for more exact analyses and is 

replaced in this problem by a two-di mensional method of calculating the 

aerodynamic co ffic ents which depends on the plan form of the surfaces 

to be analyzed. and in which the tvlo-<iimensi onal aerod.ynamic coe:'ficients 

are integrated along the span. In this problem the structural parts of 

the determinant tonns used wi.l.l ::'6 those oi' Example No . 7 (reference 4). 

The vertical and hori zont al tail surfaces used i n this problem are 

shown i n fi u~ s 25 and 26 . Various other physical char acteri stics of 

the surface s are listed i n tables XVI and. XVII. The values of the uncoupled f r e.-:.uencie s to be used in the analysi s are 

ma(assumed lUlcoupled fuselage side-bending frequency) = 46.1 radians/sec . 

~(ass~~ed uncoupl ed fuselage torsion frequency) = 65.0 radians/sec 

m~(fr e~rudder fre~u ncy) = 0 radian/sec 
NAeA TN No. 1195 55 

From Example No. 7 of reference 4, the structural terms used are 

r 

~ == 67.27 ! 1 L 

~s == 100.1 ft 

c = 1. 731 ft -8 

D == 100 .1 ft 

-8 

r (ill ' 2 "l 

E == 1891 I l - \ ~ ) (1 + . ) I 

~ga. ft ' -8 L , ill J 

F == 23.8.; ft 

-s 

G = 1. 737 ft 

-s 

H = 23.9l6 ft 

-s 

I = 5.992 ft 

-s 

The fuselage torsion and side-bending modes assumed in this example 

give the tail surfaces the followIng modes: The horizontal tail will 

flutter with a rigid rolling mode. The vertical tai l .nll have a rigid 

rolling mode and a uniform pitching mode about the fuselage bending 

axis. The rud.cier will have a uniform deflection mod.e and. since a geared. 

tab is prov3. ded on the rudd.er, the tab will also have a uniform deflection mode. It is for these modes that the a erodynandc parts of the 

flutter determinant must be calculated. . necause of the di fference in 

the root chord.s, the value of ko for the vertical tail inll be 1.16 

times the ko of the horizontal tail. 
56 NAr-A TN No . 1195 

Flutter Analysis with Two·-Dimensi onal Values 

for the Aerodynandc Parts of the De terminant 

By rearranging the relations for the t wo--dimensional aerodynamic 

terms in equation (23) to fit the spec i al condHi ons of th1s example 

(,.,here the H subscript indicate s values :pertai ning to the horizontal 

tail and the V subscri pts, to t he vertical tall), the terms become 

f

Lo ( by ', 2 

!:.A = (sbo ) - ' . ) V \ 0 3. 33 J 

2i ( 1 

+ - I - - a 

kv \ 2 

~A = (sb ) 

o V 

2E1.(d) 

\ I 1 I I C (kTT ) r zdz 

/ " 

/ 'tv ' 3 - - ] 

( - - )' ·f, -l ~ C(.l - C(.l \ 3.33 . kv ~2 ~l 

.'-_., + iE~ dl J C(~) } zdz 

k 2 

V 
NACA TN No. 1195 57 

f

l. O ( bV \ 3 ( ( 1 \ 2i J 

DA = (ab ) -- : ~ - a + - + a J - C(kV) ).. zdz .- 0 V 0 3.33 J l 2 / kV J 

2i ( 1 "\ l ") + - - - - a ) C (ky) ~ dz 

kV 2 -' ) 

f l. O C.bv_ \ 4 { _ 

FA = (sb ) .. , ) - 0 V ' oj. 3.) 

1 \ 1 \ i 2E1 iE2 , ' l ( " ( " r- l"l 

+ \ - +- a I - )'! - + - C (kV) , d z 2 / l1 L kV 2 kV J j 

GA = (sb ) - °v 

H = (so ) -- ) ~ .. ?:.. B -! B I f l. O ( bV 

"4('l-' l 

"':'A 0 V 0 3.33 / l k ~ 0.1 J 

,l!;3 .,." ~ 1 i ( 1 '\ \ l } -- l - + - - - a ) J C(kV) dz 

T( ~ kV 2 
NACA TN No. 1195 

f

l.O (b \ 4 { r B(3 iBf3 ] = (sb ) ~ 1 _ ! ~ + _ _ 2 _ B 

o V 0 3 . 33 J L k~ kV 13 1 

( \ l "" i E2 :i), ( ) I i G kv J dz 

kV -

The se inteGrals may be evaluated graphically for a given value of 

the fre~ nc y parameter . By noting that the fre quency lldrameter s are 

di ffer ent for the vertical and horizontal ta"Ll surface s, the function 

C(k) can be found from the tables . The funcLons A, B, and C can 

be fO~~Q i n r eferenc e 1 . The func tion E i s defined i n equation (5 ). There is, then, no obstacle to this graphical integration . 

The r e sults of evaluating these i ntegrals for a range of ko ar e 

tabulated in table XVIII along with t he e lements of the determinant. 

The determirants themselvee are solved by the method ?f r eference 4 and 

ar e tabulated in the f ollowing table : 

ROOTS OF FLUTTER DETERMINANT FOR VARIOUS ko 

I I I I I I I 

koH koV I Vl V2 I 

(mph) I ('1 (mph) I g2 

I J , 

0.686 0 .795 172 0.004 268 -0 .176 j I 

, 

I I 

.600 .696 194 ,011 314 - .248 ! ! I I 

. 436 .506 257 .052 431 j 

-. 303 J i 
RACA TN ;0 . 1195 

The ca1cul~ted. flutter speed i s dete rmined ra:?h: calJ.y in f :lguT6 27. 

The true airspeed calculated by this met.hod is 2!~3 mph. 

Flutter .~alysi8 with Aerodynamic Terms Corrected 

for Finite-Span Effects 

The r e lations for the finite-span c orre cti ons f rom equation (25) 

modified for this example become 

r 1. 0 ( ' 'by " ,2 ;' 2i '~ 

6 A = (sb ) -- , - - ah T ..II z2 d.z 

o 3 .., I ! k V 0 . j)' L V 

6 B = (sb ) 

°v 

f' 1. 0 ( 0v 

I "_ J 0 3.33 

, 3 

\ 

I 

/ 

~ r2i ( 1 "- ' 2 \ - '., + 

L ~' - - - 13.1 

kv \ 2 / 

-I 

-' 

a 1. zdz aV J 

.6 C ;

(' 1. 0 \/ bv \ 3 f I r 2El ,E2 -: ': := (sb ) --- } I - - L - + - Ie Q t,. zd.z 

o V J 0 . 3 .33 / , ' 1t k~ ky- -1 ! 

ti D 

0,E 

+ n( ab ) 

0y 

:= (sb) '[ 1 . 0 ( ,9v )3.r ( :1: + a 

o V 3. 33 / '- 2 ) 

2i I - a t,.. zdz 

k ' hV : V " 

<= ( 

./:' 
60 

6 Q: 

6R 

6. I 

::: 

= 

x as ~ dz . V ) 

; 

(sbo ) 

V 0 

(' 1. 0 

(sb ) I 

a V Jo 

;~ 1. 0 

(sb ) / 

J °v c 

3.33 / 

( bv ",\3 

\3 .33.J 

/ bv ', 4 

( \ 

\ 3.33 ) 

I bV 4 

\ i 

) 3. 33 .I 

Fl.':" ( 'Lv n( sbo + ) 

v 10 3 . 33 l 

[ 

. 

-t 

'. 

.~ 

I 

./ 

I 

\. 

4 (' 

-< ! 

'-

..,. l ~3 i I ZG.z - °h7 

It k:.. J 

V 

E2. ( 1 

:!:....+ '1 

- , - - a 1 

1\ l. 'jr2 ky \ 2 .' 

" / 

-y 

E3 ?.E l iE2 ! I 

L --- + a~v ;-

2 

}~ kv i 

2n: i 

Es r ?I?, (d) ! ...-...L:JJ,. iE2 (d) 

i + --- 2 2 2 kv rr c. 

kV 

i.E 2 ( d.) " 

+--- 1 

kV -.i 

"\ 

a·'{,V :. dz 

dz 

-1 I , a ~ dz 

lV .1 

vJhen the se terms are in estigated, it i s obse rved. that five types 

of OJ appear in th~ d;i.ffer~n'L r elaUons. This means t hat five sets 

of simultaneous equatlons must De so2.ved to d.et0,rmine the c 'i rculation 

distributi ons over thes€: surfa ces and thus tte correction funct :~ on8 a .. j 

for each value of m/u selected. . 

For the sm'face s provided i n this problE:; ~; it is f e lt that. a thr eeterm' Fourier ser ies approxiw~tion f or the c i rculat ion will be satisfactory. This means t.hat five eets of' three si multaneous 8cuations must be 

solved . 

- ------ ---- - -- --- - 
NACA TN No . 1195 61 

I n this :pr oblem there are three basi c t:,'pes of solution in each of 

w'hi ch the Anm terms are the same : the rolling of t he hori zontal tail 

surfaces; tl:.e rol]_ing of the verti c al tail surfaces, and the pitching 

of the vertJcal tail sID'faces, rudder d_eflection, and tab deflection . 

Of the five solutions to" be conducte d, on~ j the solution for the tab deflection is problematical, because of the discon~inuou8 angles of attack 

at t e tab extremi t::'e s which maKe it impossible t.o obtain a sa ti sfac tory 

Cir culation distr ibution with a finite number of terms in the Fourier 

series arproxiI!1.at1on . The stand i 8 t~~en that some corr ection is better 

than none thoU{:;h; so the thre0- term approx:;.rnation for the c i rculation i s 

~e d for thi s mode too . 

For the t .. lO r olling modes the inteGra). equation will be satisfied 

at y/sbo ; 0 . 4, O.b, and 0 .8, an~ for the three symmetrica l des 

Jis-bo = 0, l~J and 0.8. For the :"'olling modes } n = 2, 4; 6, and for 

the symme tr-' ca2. mod.s s n = 1 ; 3, 5'. 

In the analy si s the vertical tail is handle d as if .: t .. lere half of 

a wing i·Ji th a full-spa:q. a ileron. This is assumed -w be satisfactory 

since the hori zontal tai.l acts as an end plate at the root chor d. 

In order to illustrate the prdcedure a value of 

l e cted "Thi ch wt ll £5i-,8 ~or t!1e verti cal tail a -:alue 

w/u i?ill be seof ko = 0.506 and 

for the l,orizontal -cail a value of k o ; 0.436 . 

For the rolling of the hori zontal t ail tbe correction functions 

0h may be calcu~ated as follows: 

The ne ces -ary parameter s and fcIDctions ~or the solutions are first 

Hsted i n the follOi.ring tables: 

(k = 0.436 ) o 

I 

~T •. , I 

iC(k) i Jl(k) I y ! \J- (k) --, j k _. C(k)+·---

SOo ' bo kHi 2

)(k) J ~(k) iJl (k) I \j • 

, I 

! . I 

0.4 10.375 0 .860 0.9126 - 0.1496i 0 . 5982 + 0 . 01511-1 0 . 2L,63 -- 0.08131 

.6 1 .336 L770 . 9451 - . 1747i . 6205 - .0088i .2834- - .10381 

.8 i . 286 . 655 .9953 - .2054i 652~ - .0394i 1 .3007 - .1089i 
62 NACA TN No. 1195 

k a ; 1.101, y/sb = 0.4 

0 - a 

In! sin n¢ Sn(¢' koS) n 

I -

2 1 0.3667 0.494 - 0.278i 

4 - -.2493 -.850 + .293i ! 61 .1132 .575 - .150i ~- ! 

I ... dsb o ::: 0. 6 

I - -

I 

I 

0.4800 o . 'YDo "- O. 390i ! 

I 41 -.1344 -.547 + .16Oi i ~ - -.1098 -.710 + .176i 

I 1 I I 

y/sbo :: 0. 8 I 

I I ; 

f 

I-;T 

i --------j 

0.4800 0.788 -- 0.4 33i r 

! 4 I .1344 .525 - .1701 ! l!l -.1098 - 1 -.710 + .203i : ____ _ 1 

\-lhen this i s done, the ~ coefficients of the i ntegral equation, 

may be evaluatod. 

By takinc t he necessarJ values f r om the tables and noting that 

the Anm coeffi cients become) 

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

NACA TN No. 1195 63 

for y/sco = 0.4, 

rr(0.860) A22 == 0.3667 + (0.2463 - 0.08(31) (0.~ 94 - 0.2(81) 2.525 

== 0.3667 + (0.2635 - 0.0934i) (0.494 - 0.278i) 

== 0.4709 - O. 1194i 

A42 = -0.2493 + (0.2635 - 0.0934i) (-0.850 + 0.2931) 

= -0.4459 + 0.15661 

. A61 == 0.1132 + (0.2635 - 0.0934i) (0.575 - 0.150i) 

== 0.2507 - 0.09321 

for Y/Sbo = 0.6. 

A22 == 0.4800 + 1I(0'17C) (0.2834 - 0.1038i) (0.790 - 0.3901) 

2.525 

== 0.4800 + (0.2715 - 0.09942) (0.790 - 0.390i) 

== 0.6559 - 0.18441 

A42 == ~,). j! 4+ (0.2715 - 0.09941) (-0.547 + 0.1601) 

== -0.2670 + 0.0978i 

A62 == -0.1098 + (0 .2715 - .0~94 ) (-0.710 + 0.178i ) 

== -0.2849 + 0.1111i 
64· NACA Tl~ No . 1195 

1((0.655) 

= 0 . 4-300 + - --- (0 . 3007 - 0 .1089i)(0 . 788 - 0.4331 ) 

2 . 525 

= 0 . 4800 + (0.2450 - 0.08Q7i) (0 . 738 - 0.433i) 

0.631~7 - 0 .176i 

A43 = 0 .1344 + (0.21+~ 0 -- 0 .08871)(0 . 5:25 - 0 .1705. ) 

= 0.2479 - 0.0382i 

A63 1098 + (0 .2450 - 0 .0887i)(- 0.710 + 0.203i) 

= 2657 + 0.1127i 

~lhen the Amn cOefficients have oeen calculat€d, the tvrodimensional circulation functions n(2) mua~ be eval~ated . For the Jm rolL.ng mode J 

where i n th~s case, 

ilc.."1 

In this example the sweepback function e appear s as ther e is 

some sweep in the surfacos . The tra~linc and leading-edge coordinates 

of the s~faces can be found in figures 25 and 26 . By using the previously tabulated parameters and functions and introduci ng the slveep 

functions; the value s of n~2) may De dete rmined as follows: 
r--------- ~----~ 

NAqA TN No. 1195 

for y/sbo = 0.4, 

Xl + Xt -0.733 + 0.988 ~ = = = 0.128 

2bo 2 

k = k z = 0.0558 

m om 

ikm 

e = (0. 9985 + 0.05581) 

~ 0.0591 + 0.551Si 

for Y/SDO ~ 0 .6, 

- 0.600 + 0 .930 6 

zm = ---- ---- = 0.1 5 

2 

k = 0 00 m 1 5 . J 

ikm 

e = (0 .9965 + O. 1~~. \ 

= 0.0768 + 0.77091 

for ;r/sbo = 0.8, 

r\ 1..t75 . 0 ""'5 Z = -v . '. -r . '-'/ ::;: 0.190 

m 2 

i km 

e = (0:9954 + 0.0960i) 

65 

D~2) = 4(0. 9953 - 0.2054i ) (0. 99)4 + 0.0960i)(0. 286i )(0 .8) 

= 0.0996 + 0. 92451 
66 NACA 'rN No, 1195 

The SGC of equations, 

.\' K .A : ":'(2) L nJ nm = Ohm 

may now be solved . This is done in table XIX by the Crout method, 

When the value s of Knh have been determine d, the vall,les of 

can be found and thts is done .i n table XX. 

- The values of the correction functi ons can nOlT be calculated with 

i t not ed that 

Thi s is done as fol1ow·s, 

( 88 c; ) . ) l' -0.0594 + 0.4.3421 I 

uhH = 0 .59 l + O. Ol:;l+l ~- ------- - l J 

0.0591 + 0.55181 

for y/sbo = 0, 6 

= -0.1) ~2 7 + O.l l Oi 

r -0. oB95 + 0. 58. O.I'.i _, 1-;1 

= (0. 6205 - 0. 0088i) I 

.. 0.0768 + O~ 7709i . ..i 
NACA TN No. 1195 

for y/ sbo = 0.8 

[ 

-0.1057 + 0.66861 '- lJ = (0.6523 - 0.0394i) 0 . 0996 + O. 92~·5i 

An exactly similar.procedure must be followed to f~nd the corrections, crhV' cr~V' cr~VJ and crlV for a Given value of kov ' It should 

be remembered though that the value of k is dHfenmt for the vero 

tical tail. 

When all the correction functions have b'3en evaluated) the -values 

of the finito-span corrections can be found by a grapbical or numerical 

1ntegration of the integ:cals listed in the beginning of this sect:'on of 

Example II. This integration was carried out, and. the values of the 

finite-span corrections found are listed in table XXI. The flutter determinant terms calculated with use of two-dimens:onal aerodynamic 

consj.derations and those co:'rected for finite-span effects are also 

listed in this table. 

The determinants can now' be solved. For this problem the solutions 

are found by the method of reference 4 and the results are tabulated in 

the follOlving table: 

r:- First solution Second solution 

koV VI V2 

(mph) gl (mph) g2 

0.436 0.506 267 0 .030 423 -0.212 

.600 .696 196 .012 308 -.151 

.686 . 795 173 .006 233 -.071 

The actual flutter speed is determined graphically in figure 27. The 

calculated value of the true airspeed at "'hich flutter occurs when the 

finite-span corrections are used is 299 mph. 

Discussion of Example II.- Example No.7 of r eference 4 indicates 

that the observed flutter speed for the tail analyzed in this problem 

fluttered experimentally at about 262 mph true airspeed.. The fact that 

the theoreti cal analysj,s, however, doe.s not check the observed flutter 

speed is not considered significant because the observed flutter speed 

was reported by a pilot some time ago and may not be accurate and the 
68 NACA TN No. 1195 

analys:s is approx .. mate in that no attempt was made to take into account 

the effective inertia due to wing motion or the elastic restraint and 

damping in the rudder control sy stem. 

It is considered significant that the three theoretical analyses 

give such wid.ely varying values for the calculated flutter speed. These 

values of the true airspeed eX3 220 mph for flutter speed calculated by 

the method of equivalent chords of r eference 4; 243 mph for a value of 

ko f 0r .t he flutter speed, of about 0.55, ;.,hen the two-dimensi onal aerodynamic terms are integrated along the span~ and 299 mph for a value of 

ko of about 0.45) when the aerodynamic terms are corrected for the ef--

fects of a fj.ni te span. 
NACA TN l\T 0 • . 1195 

REFEP-ENCES 

1. Theodorsen, Theodore, and Garrick, I. E •. : Nonstationary FlovT 

about a Wil1E,-Aileron-Tab Combina tion includinG Aerodynami c 

Bal ance. NACA Rep. No . 736, 1942. 

2. Jones, W. Pritchard: Theoretical Air-Load and Derivative 

Coefficients for Re ctanGular "Tines. Oscillat ion Sub-Com., 

A.R.C. 6505, 0 . 321, Ac. 2206, Feb. 194-3 . 

3. Estes, R. J., and Zart arian, 

of Oscill ating Airfoils. 

Technology, Feb . 1944, 

G.: On the Three -Dimensional Theory 

B.S .. Thesis, Massachusetts Inst. of 

4 .. Sm11g, Benjamin, and Wasserman, J4ee S .: Appl:i.cation of ThreeDimensional Fh~tter Theory to Aircraft Structures . Arm:y Air 

Forces Tech . Rep . No. 4798, July 1942. 

5. Biot, M. A. , and Boehnlein, C. T .: Aerodynamic Theor y of the 

Oscillating Ving of ]'inite Span . Rep . liTo . 5; for submHtal 

to the Army Air Forces, Sept . 1942. 

6. Reissner, E.: A Met hod for Calculation of Airforces on Oscillat:!ng 

F1nite I'TinG . Curtiss-itTright Res . Lab ., Rep . No . S:8 -76-S-2, 

Nov . 1943 . 

7. Rauscher, Manfred, lTtth BOBosia 1·, A., Lvnn, R ., end Gouzoule , T.: 

Report on the Design and Construction of Six Models Represent:!ng 

TvTo Desi@ls and Three Differ ent Scales. Navy Contract No . 74858, Items 2 and 3, M.LT . Flutter Res . Lab! , Sept . 25 , 1941 . 

8. Crout, P . D.: A Short Method for Eva l uatine; Determinants end 

Solving Systems of Linear Equations 1dth Real or Complex 

Coefficients. Trans. A.I.E .E. , vol . 60, 1941. 
-

70 

TABLE I 

VALUES OF FUNCTION C = F + IG 

k C ' 

0.00 1.0000 - 0.00001 

0.01 0.9824 - 0.0~821 O.02~ 0.9545,- 'Q.O 721 

0.04 0.9267 - 0.11601 

0.05 0.9090 - 0.J~q51 

0.06 ' 0.8920 - 0.1~251 

0.08 0.8604 '-: 0.16-041 

0.10 0.B320 - 0.1~231 

0.12 o .8,063 - 0.1 011 

0.16 0.7628 -' '0'.18~61 

0.20 0.72~6 :" ' 0.18 6i 

0.24 0.69 9' - 0.lB621 

0·30 - 0.6650. - 0. J 793i 

0·t4- o . 64-b.9 -' 0.1 73"81 

O. 0 - 0.625'0' . 0.16501 

0.44- 0.6130 - -0.1592i 

0·50 - 0,·5979 - O.lKO~i 0.56, 0.58~~ ~ O~i -2 1 

0.60 -- 0·57 '- ~i378i 

0.65 0.5713 - 0.1312i 

0.66 0.56K9 ... 0.1308i 

().70 0.'56 ' 7 ~ O~1265i : 

0,.75· O. 55~1 :-: 6 .. ~ 12'1'21 , 0.80 ....... " 0.5541 .: o~i165i 

0.85 ' 0. 5,°7 - '.11201 

0·90 0.5 60, - ,0.10781 

0·95 O. 542( ,- ,,0 .10L~0~ 1.00 0.539 - ,0.10031 , 1.20----' . 0.53 00 .;, '" 0.0877i 

1.50 -- 0.5r

l 0. 07361 

2 .00 ~ 0·51'J;' - o. 'la771 

t· OO 0. 506 3 - 0. 0 001 

.00 ....... o . 5037 - 0 . 03 05i 

6 .00 0.5017 - 0. 0206i ~o.oo 0 . 5006 - 0. 01241 

OC> 0 · 5';08 - 0 . 00001 

NATIONAL ADVISORY 

COMMITTEE FOR A£ ONAUTICS, 

, : 

NAOA TN No. 1195 

TABLE II 

VALUES OFFl!~CTION ~(k) 

k. 

.0.,00 

0.02 

o ~ 04 : 

0.06 .. 

0.08 

0.10 

0.125 

0.15 . 

0.:175 

0.20 " 

, ,0.225 

0.25' ' 

0 . 275 

0'·30 

g.,:'46 

0·45 

O~·50 ! 

0'·5'5 

0.60 

0.65 

0 .. 70 ' 

0·75 

0'.80 .. , 0·90 ' 

1,.00 

1.25 

1.50 , 1.-75:' 

2.00 

2.10 

2.15 

2 . 20 

2 .25 

2· 30 

2.(5 

2,40 

2 ~" 8 2 .54 

2 .60 

0.5000 - 0.,00001 

0'.4810 - ' 0.'04221 

0.4607 - 0.06651 ' 0".4410 - O. OB291 

0.4226 - 0.09421 ' ::D~405l - 0.10181 , 0.3857 - 0.10821 , ~ 3690 - 0.11201 

0.3522 - 0.11301 '0.3393 - 8.11391 

0'>3268 - 0.11321 .0.3154 - 0.11161 

0.3049 - 0.1 0991 

"0.2955 - 8.10761 , 0.2787 - 0.10231 , 0 .2'644 - o. 0~641 

0.2519 - p~09031 

,. 0~2408 - 0.08421 

b.2289 - 0.07751 

" 0.2220 - 0.07221 

0.2139 - 0. 06651 

0·.2062 - Gf.06101 

0 . 1991 - 0.05571 

0.i924 - 0.05071 . 0.1801 - 0.04131 

0.1688 - 0.03291 ' " O· .1 L~ 36 - El; 01591 

0.1218 - 0. 00421 

0.1027 + 0.00301 

0.0864 + 0.00661 

0.oB07 + 0.00701 

0.07BO + 0.00721 

0.0754 + 0.00721 

0.0730 + 0. 00721 

0 . 0706 + 0.00701 

0.u684 + 0.00691 

o. )663 + ") l06t):. , ,,' -. , J ~oJ:-l- T _.~ _ 

O.Oo!O T 0 . 0) 57i 

0 . 0591 + 0 ,00521 

I 
NAOA TN No. 1195 

k 

0.00 

0.01 

0.025 

0~04o 

O.O~ 

0.0 

0.08 

0.10 

0.12 

0.16 

0.20 

0.24 

0.30 

0.~4 O. 0 

0.44 

0.50 

0.56 

o.bo 

0.65 

0.70 

0.~5 O. 0 

0 .85 

0·90 

0 ·95 

1.00 

1 .20 

1.50 

2.00 

3. 00 

TABLE III 

VALUES OF FUNCTION 

1J (k) C(k) + J

o (k1-1J 1 {k) 

C + 1J 1 

J 0 - iJ1 

1.0000 - 0.00001 

0.9824 - 0.04221 

0.95t3 - 0.07

6

71 

0.92 R -.0.09 01 

0.908 - 0.105~1 

0.8911 - 0.112 1 

0.8588 - 0.12041 

0.8295 - 0.12241 

0.8027 - 0.12021 

0.756t - 0.10791 

0.717 - 0.08911 ·0.6845 - 0.06711 

0.6425 - 0.03101 

0.6180 - 0.00631 

0.58,° + 0.03101 

0.56 6 + 0.0554i 

0.5354 + 0.09131 

0.~0~3 + 0.12~91 O. 8 9 + 0.14 31 

0.4658 + 0.11531 

0.4424 +. 0.20121 

0.4187 + 0.22·621 

0.3945 + 0.24981 

0.370b + 0.27231 

0.3443 + 0.29351 

0.3179 + 0·31331 

0.2909 + 0·33191 

0.1747 + 0'R9091 - 0.0220 + O. 2451 . - 0.3550 + 0.279 71 - 0.1233 - 0.52291 

71 

TABLE IV 

VALUES OF FUNCTION 

1C(k) 

k Hl (2) (k) 

k 

0.00 

0.01 

0.04 

0.05 

O.-Ob 

0.08 

0.10 

0.12 

0.16 

0.20 

0.24 

0.30 

0·t4 

O. 0 

0.44 

0.5 0 

0.56 

0.60 

0.65 

0.6b 

0.70 

0.~5 O. 0 

0.85 

0.90 

0.95 

1.00 

1.20 

1.50 

2.00 

3.00 

lC, 

k H}2} 

1.5708 - O.OOOOi 

1.4

420 - O.O~~Ol 

1. 51~ - 0.1 001 

1.421 - 0.20121 

1.3932 - 0.21881 

1.3393 - 0.2429i 

1.2r1 - 0.25671 

1.2 51 - 0.26441 

1.1 71 - 0.26311 

1.1020 - 0.25071 

1.0473 - 0.2319i 

0.9795 - 0.1973i 

0.9413 - 0.1722i 

0.8921 - 0.1334i 

0.86~2 - 0.10751 

0.82 1 - 0.0691i 

0.78 9 - 0.03191 

0.7671 - 0.00771 

0.7411 + 0.02151 0.7~62 + 0.02731 

0.7164 + 0.0499 ... 0.6921 + 0.07721 

0.6b91 + 0.10~41 

0.6471 + 0.12 9i 

0.6238 + 0.1528~ 

0.6013 + 0.1758 

0'4791 + 0.1978~ O. 901 + 0.2755~ 

0.35~ + 0'R606~ 0.12 1 + o. 209~ 

- 0.2279 + 0.2799 i 

NATIONAL ADVISORY 

COMMITTEE fOR AERONAUTICS. 

_1 
72 NACA TN No. 1195 

TABLE V 

VALUES OF FUNCTION F(x) = /OOe- 1"'(1 + 1 _ ;xr~\d>. o x >- ~}o.) 

x F 

0.00 <)0 

- 1.a711 

.05 2.750 - 1. 681 

• 10 2.109 - 1.37~1 .20 1.490 - 1.24 i 

'RO 1.155 - 1.lt61 

• 0 0.93~ - 1.0 01 

·50 0.~7 - 0.9871 .bo o. s8 - 0·9221 

.~O 0'a64 - 0.8651 

• 0 o. 89 - 0.81~i .90 0.427 - 0.76 1 

1.00 0.376 - 0.726"1 

1.10 0.333 - 0.68

4

1 

1.20 0.297 - 0.65 1 

1.,0 0.26~ - 0.6241 

1.40 0.23 - 0·5931 

1·50 0.214 - 0.5671 l.bo 0. 19t - 0.5401 

1.~0 0.17 - 0·a1

l

1 

1. 0 0.160 - o. 9 1 

1.90 0.146 - o .47~1 2.00 0.134 - 0.45 1 

TABLE VII 

x 

2.2 

2·t 2 . 

2.8 

3.0 

3. 2 

3 ·t 3· 

R· 8 

.0 

4.2 

4.~ 

tt:8 

5.0 

5. 2 

5 ·t 5· 

5.8 b.o 

F 

0.113 - 0.4251 

0.097 - 0.3951 

0.083 - 0.3691 

0.072 - 0. 34a 1 

0.063 - 0.32 1 

o. 0a~ - 0.3051 

0.0 - 0~2841 0.043 - 0.27 1 

0.039 - 0.2601 

0.035 - 0.248i 

0.031 - 0.028 - 0.2,71 

0.2261 

0.026 - 0.21~1 

0.024 - 0.20 1 

0.022 - 0.2001 

0.020 - 0.1921 

0.018 - 0.185i 

0.017 - 0.1791 

0.016 - 0.ll2i 

0.015 - 0.1 71 

1/2x2 l/x 

NATIONAL ADVISORY 

COMMITTEE FOR AERONAUTICS 

VALUES OF FUNCTIONS sin ni AGAINST cos rI = r.. 

n s 

,,!-

Z:;cos ,5 0 0.2 0.4 s 0·5 0.6 0.8 0.9 1.00 

~ 1 1 0.9798 0.9165 0.8660 0.800 0.600 0.4355 a 

Si22p 0 0.1959 0.3667 0.4330 0.4800 0.4800 0.3921 0 . sin 2P -0.3333 -0.2744 -0.1099 0 0.1173 0.3120 0·3255 0 

3 

~ 0 -0.1803 -0.2493 -0.21 65 -0.1344 0.1344 0.2433 0 

s1n 2P 0.2000 0.1070 -0.093 6 -0.1732 -0.1994 -0.0151 0.1550 0 

5 

¥ 0 0.1558 0.1132 0 -0.1098 -0.1098 0.0707 0 

sin 7~ -0.1429 -0.0472 0.1380 7 0.1238 0.0295 -0.1398 -0.0022 0 

~ 0 -0.1249 0.0188 0.1083 o 113L~ -0.1134 -0.0558 0 

sin 2i. 0.1111 -0.0246 -0.0940 0 0.0980 -0.0525 -0.0882 0 

9 
NACA TN No. 1195 

TABLE VI 

VALUES OF FUNCTION 

73 

S (¢, k s) = sinnl + i kos /:s~- COS~IF cos ne de 

n a sin Tf Icos - cos 

o 

FOR VARIOUS VALUES OF n, kos, and y·:~/s = cos ¢ 

k 0 0.2 0.4 0.6 0.8 1.0 

1 1 1 1 1 1 1 

2 0 0.8001 1.2000 1.6000 2 

3 -1 ~.8401 -0·3593 0.4399 1.5601 3 

4 0 -1.0879 -0.6722 0.896, 4 

5 1 0.5461 -0.5105 -1.2464 -0.1259 5 

6 0 0.6792 -0.8234 -1·3723 6 

7 -1 0.3373 1.0542 0.2578 -1.6307 7 

8 0 0.1644 1.1335 ... 1·5113 8 

9 1 ... 0.2257 -0.923 2 1.1020 -0.7876 9 

NATIONAL ADVISORY 

COMMITTEE fOA AERONAUTICS. 

~ 0 0.4 0.8 

1 0.664 0.701 0.828 

-0·31 -0.293i -0.181i 

3 -0.851 -0·311 1.4.20 

+0.215j 0.063i -0.254i 

5 0.908 -0.465 -0.114 -0.1611 0.079i +0.030i 
74 

~COS¢' n . ___ 0 

1 . 0.468 

-0.3561 

2. 0.000 

0.0001 

3 -0.725 

0·30li 

4 0.000 

O.OOOi 

5 0.820 

-0.2441 

6 0.000 

0.0001 

7 -0.868 

0.205 

8 0.000 

0.0001 

9 0.895 

-0.1771 

~ 0 

1 0.271 

-0 . 3181 

2 0.000 

0.0001 

3 -0.545 

0.3551 

4 0.000 

0.000 

5 0.679 

-0.32li 

6 0.000 

0.0001 

7 -0.7a~ 0 .2 1 

8 I 0 .000 

I 0 . 0001 

9 I 0.803 i - 0 . 2591 

TABLE VI (contlnued) NACA TN No. 1195 

0.2 0.4 

0.479 0.516 

-0.3521 -0.34Oi 

0.516 

-0.2741 

-0.015 

0.2491 

-0.274 

0.0941 

-0.869 

0.2821 

0.451 -0.424 

-0.1301 0.1231 

0·584 

-0.1411 

-0.316 0.925 

0.0321 -0.2001 

0 .. 146 

-0.0321. 

-0. 201 -0. 83L~ 

0 .0661 0.154i 

kos = 2 

0.2 0.4 

0.284 0.327 

-0·3201 -0.3211 

0·351 

-0.2961 

-0.468 -0.221 

0.2941 0·1l3i 

-0.704-

0.3611 

0·376 -0.352 

-0·1721 0.1661 

0.505 

-0.1951 

-0.2~ 0.0 1 

0.814 

-0.2941 

O.lGO 

-0.0 71 

-0.178 -0.755 

0.1081 0 .2271 

0.6 0.8 l.0 

0.589 0.729 1. 250 

-0·3091 -0.2231 0.7271 

0.82, 1.2204 2.068 

-0.3821 -0.4191 0 .4761 

0.3R1 1.280 3. 021 

-0.1 01 -0.-384 i 0.3311 

-0.556 0.7610 4.009 

0.1531 -0.1621 0.249i 

-1.063 -0.1l6 5. 003 

0.2701 0.0631 0.2011 

-0.720 -1.269 6.003 

0.1671 0.1891 0.1661 

0.2~1 -1.498 7.002 

-0.0 2i 0.2261 0.1431 

1.026 -1.402 8.001 

-0.1871 0.1751 0. 1251 

1.008 -0.736 9·001 

-0.17li 0.Ob71 0.1111 

0.6 0.8 1.0 

0.421 0.622 1.551 

-0.3181 -0.2691 1.1731 

0'495 0.974 2.210 

-0. 291 -0.50 1 0·8771 

0.245 1.062 3.083 

-0.1751 -0 . 5051 0.647i 

-0.472 0 . 651 4.038 

0.2001 - 0.2031 0.4961 

-0·912 - 0.1l2 5 . 014 

0·3701 o .115t 0.4031 

-0.630 -1.179 6.012 

0.2381 0.2821 0·3321 

0.208 -1. 377 7 . 006 

-0.058i 0.3311 0.285i 

0.931 -1.301 8.004 - 0 .2751 0.252i 0.2501 

0.923 -0.68~ 9·003 

-0.2581 0.08 1 0 .2221 

NATIONAL ADVISORY 

COMMITTEE FOR AERONAUTICS. 
NACA TN No. 1195 

~ 0 

1 0.132 

-0.2341 

2 0.000 

0.0001 

3 -0.341 

0·3371 

4 0.000 

0. 0001 

5 0.486 

-0.3521 

6 0.000 

0.0001 

7 -0. 584 _ 0·3511 

8 0.000 

0.0001 

9 0.651 

-0.3371 

~ 0 

1 0.082 

-0.1891 

2 -0.000 

0.0001 

3 -0.240 

0·3001 

4 0.000 

0.000 

5 0.371 

-0.3321 

6 0.000 

0.0001 

7 -0.463 

0·3591 

8 0.000 

0.000 

9 0·531 

-0.3b71 

TABLE VI (concluded) 

k s = 4 

o 

0.2 0.4 0.6 

0.140 0.16~ 0.2~6 -0.2361 -0.24 1 -0.2 21 

0.19

4 0·390 

-0.25 1 -0.3901 

-0.300 -0.160 0.139 

0.2841 0.1201 -0.1701 

-0.488 -0·359 

0.3841 0.2321 

0.273 -0.2~0 -0.688 

-0,1911 0.1 41 o .1.t 351 

0.389 -0.487 

-0. 23 21 0.2961 

-0.27

4 0.6tO 0.176 

0.05 1 -6.3 71 -0.0711 

0.101 0.772 

-0.0631 -0.3 661 

-0.140 -0.623 0.774 

0.1731 0.3021 -0 ·3531 

k s = 6.0 

0 

0.2 0.4 0.6 

0.086 0.109 0.179 

-0.1871 -0.1781 -0.2401 

0.126 0.242 

-0.2171 -0·3331 

-0.215 -0.12§ 0.089 

0.2561 0.11 1 -0.1461 

-0.364 -0.284 

0.3601 0.2601 

0.210 -0.188 -0'436 

-0.1851 0.1691 O. 351 

0.308 -0 :383 

-0.2361 0.3061 

-0. 25t 0·511 0.154 

0.05 1 -0.3841 -0.0751 

0.077 0.64

-0.0681 -0,40 4 1 

-0.109 -0·517 0.649 

0.2251 0·3??1 -0.4001 

0.8 1.0 

0.484 2.0~5 -0.3151 1.7 21 

0.689 2'440 

-0.5251 1. 991 

0.766 3. 274 

-0.5721 1.2041 

0.468 4. 1t3 

-0.1841 0.9 31 

-0.1l4 5. 06

0.2221 0.80 4 1 

-1.O~7 0.3 81 

6.046 

0.6581 

-1.172 

0.4311 

7.023 

0.5651 

-1.l20 8.014 

0 .3141 0.4971 

-0.596 9. 008 

0.0881 0.4421 

0. 8 1.0 

0.387 2.474 

-0.3341 2.2371 

o·all 2.862 

-0. 971 1. 9891 

0·579 3.502 

-0.5691 1.6721 

0.341 

- 0 .0801- 4. 2§, 1.3 61 

-0.1l9 5.148 

0.3391 1.1901 

-0.928 6.099 

0.4481 0.9751 

-1.003 

0.4621 

7.046 

0.8391 

-0.952 0.31 1 

8.024 

0.7401 

-0·512 9. 022 

0.0571 0.6591 

NA TIO NA L AOV'SORv COMM IlTEE f Oq AlPO/l,A dT1G., 

75 
76 TABLE VIII JTACA TN No. 1195 

UNIFORM TRANSLATION OF RECTANGULAR WING OF AR = 3 

y -(2) L L 

k - C a

h bos 2pu2t>oh Ibo 2pU2t>oh /00 

0.1 67 0 0·757-0.1881 -0.105+0 .1291 0.055+0.3971 -0.013+0 .3421 

0.167 o.~ 0.757-0 .1881 -0.137+0.1251 0.055+0 .3971 -0.011+0.3251 

0.167 O. 0.757-0.1881 -0.285+0.1151 0.055+0·3971 -0.006+0.2471 

0.167 1.0 0.757-0.1881 -0.750+0.1 051 0.055+0·3971 -0 .CXXl2+0. 0041 

0·333 0 0.650- 0 .1751 -0.Ot3+0.0~81 0.008+0.6811 -0.073+0 .6361 

0·333 O.~ 0.650- 0 .1751 -0.0 9+0.0 31 0.008+0.6811 -0.079+0.6081 

0·333 o. 0.650- 0 .1751 -0.196+0. 0831 0.008+0.6811 -0.079+0 .4751 

0.333 1.0 0.650-0.1751 -0.622+0.0101 0.008+0.6811 -0.003+0 . 0291 

0.667 0 0.569-0.1301 -0.018+0.0301 -0.426+1.1911 -0.489+1.1551 

0.667 O.~ 0.569-0.1301 -0.036+0.0331 -0.426+1.1911 -0·m+1.116~ 0.667 O. 0.569-0.1301 -0.127+0.0091 -0.426+1.1911 -0. +0.9271 

0.667 1.0 0.569-0.1301 -0.459-0.1831 -0.426+1.1911 -0.042+0.2301 

UNIFORM TRANSLATION OF RECTANGULAR WIoo OF AR = 6 

0.167 0 0.757-0.1881 -0.020+0.0591 0.055+0.3971 o .02~+O .• 3e41 

0.167 O.~ 0.757-0.1881 -0.0~0+0.0701 0.055+0·3971 0.01 +0.37a1 

0.167 o. 0.757-0.1881 -0.1 9+0.1041 0.055+0.3971 0.003+0.31 1 

0.167 1.0 0.757-0.1881 -0.750+0.1 051 0.055+0 .3971 -0.0002+0.0041 

0·333 0 0.650-0.1751 0.001+0.0271 0.008+0.6811 -0.020+0.6811 

0·333 O.~ 0.650- 0 .1751 -0.013+0 .0391 0.008+0.6811 -0.032+0.6671 

0·333 o. 0.650-0 .1751 -0.100+0.0651 0.008+0.6811 -0.060+0·5761 

0·333 1.0 0.650- 0 .175i -0.622+0.0101 0.008+0.68li -0.002+0.0291 

0.667 0 0.569-0.1301 0.000+0.0071 -0.426+1.1911 -0.441+1.1931 

0.6 67 o.~ 0.569-0.130.1, -0.006+0.0141 -0.426+1.1911 -0.455+1.1801 

0.667 O. 0·569-0.1301 -0.058+0.0171 -0.426+1.1911 -0.461+1.0711 

0.667 1.0 0.569-0.1301 -0.459-0.1831 -0.426+1.1911 -0.042+0.2301 

UNIFORM PI~HING OF RECTANGULAR WING OF AR = 3 (a=O) 

-(2) - y C L L k Gelbos 2Pifb 2 pifboa oa 

0 0 1 - 0 ·313 3.142 2.14

8 

0 O.~ 1 -0·348 3.142 2.0 9 

0 O. 1 -0·505 3.142 1.554 

0 1.0 1 -1 3.142 0 

0.167 0 0.757-0 .1881 -0.105+0.1291 2.427-0.1301 2.063+0,2481 

0.167 o.~ 0.757-0 .1881 -0.137+0 .1251 2.427-0.1301 1.963+0.2271 

0.167 O. 0.757-0 .1881 -0.285+0 . 115i 2.427-0 . 1301 1.501+0.1571 

0.167 1.0 0.757-0.1881 -0.750+0 •1051 2.427-0 .1301 0.044+0.0031 

0.333 0 0.650- 0 . 1751 -0. 0t3+O .• 0~81 2 . .133+0 .3151 1. 957+0 .5Rfl 

0·333 O.~ 0.650 - 0 . 1751 -0.0 9+0.0 31 2.133+0 .. 3151 1.873+0.a 1 

0.333 O. 0.650- 0 . 1751 -0.196+0.0831 2.133+0 .. 3151 1.473+0 . 741 

0.333 1.0 0.650- 0 .. 1751 -0 .. 622+0 .. 0101 2.133+0 .3151 0.173+0.0221 

0.667 0 0.569-0.1301 -0.018+0.0301 1. 924+1. 2341 1. 837+1. 3101 

0.667 O.~ 0.569- 0 •1301 -0.036+0 •0 331 1.924+1. 2 341 1. 783+1.3011 

0.667 O. 0.569-0 .1301 -0.127+0 . 0091 1. 924+1. 2341 1. 517+1..12§1 

0.667 1.0 0.569-0 •13°1 -0.459- 0 .1831 1.924+1.2341 0.673+0.17 1 

NAT IONAL ADVI SORY COMMIT TEE FOR AE RONAUTIC!) 
NA~A TN No. 1195 TABLE VIII (cont1nued) 77 

UNlFORM PITCHING OF RECTANGULAR WING OF AR = 6 (a = 0) 

! 0 0 1 -0.17s-! ~:itt~ 2·291 

0 O.~ 1 -0.20.37 2.502 

0 O. 1 -0.3467 3.142 2.052 

0 1.0 1 -1 3·142 0 

0.167 a 0.757-0..1881 -0.025+0 •0591 2.427.-0.1301 2. )3~+1Y'0Lt~1 

0.167 0.4 0.757-0 .1881 -0.041+0.0701 2.427-0.1301 2.279+0.0.781 

0.167 0.8 0.757-0..1881 -0.158+0.0991 2.427-0..1301 1. 90.5+0 .1391 

0.1 67 loG 0.757-0..1881 -0.750+0.1051 2.427-0.13 01 0.045+0 . 00 31 

0.·333 0 0..650- 0 .1751 0.001+0.0271 2.133+0·3151 2.121+0.4011 

0·333 0..4 0.650- 0 .1751 -0.013+0.0391 2.133+0 .3151 2.0P+G.43G1 

0·333 0..8 0..650- 0 .1751 -a.l00+0.06a1 2. 133+0 .. 3151 1. 7 ,)+0.4681 

0.333 1.0 0.650- 0 .1751 -0.622+0 .10 1 2.133+0.·3151 0.173+0..0221 

0..667 0 0.569-0.1301 0.000+0.0071 1.924+1. 2341 1..~18+1..2581 

0.667 0.4 0 .569-0 .1301 -0..00.6+0. .0.141 1. 924+1. 2341 1. 91+1.2721 

0.667 0.8 0.569-0.1301 -0.058+0 .0171 1. 924+1. 2341 1.725+1.2261 

0.667 1.0 0.569-0.1301 -0. 459-0 .1831 1. 924+ 1. 2341 0..673+0..17741 

LINEAR SYMMETRICAL TORSION OF RECTANGULAR WING OF AR = 3 (a = 0) 

y 

-(2) L 

k C L 

boB 

Gcr. 

2 pU"b on 2pU"bon 

a a 1 00 a 

0 0.~5~ O.~ 1 -0.352 1. 257 o. 1 

0 O. 1 - 0 .596 2·513 1.01 

0 1.0 1 -1 3.142 a 

0.167 0 0.757-0.1881 cP-001 0 0.406-0.0201 

0.167 O.~ 0..757-0..1881 -a.l~+0.1201 0.971-0.0521 0.777+a. a8l 1 

0.167 o. 0.757-0.1881 -0.3 3+0.1411 1.4

42-0 .1041 1. 001 +0.17 1 

0.167 1.0 0.757-0.1881 -0.750+0.1051 2. 27-0.1301 0.044+0.0031 

0·333 a 0.650-0.1751 <X) - 001 a 0.340+0.0051 

0.333 O.~ 0.650-0.1751 -0.076+0.0821 0.353+0.1261 0.741+0. .2131 

0.·333 a. 0.650-0..1751 -0..262+0.0981 1.707+0 . 2521 1.007+0.3881 

0·333 1.0 0.650-0.1751 -0.622+0.0101 2.133+0·3151 0.173+0.0221 

0.667 0 0.569-0.1301 00 - 001 0 0 .235+0 . 0861 

0.667 O.~ 0.569-0.1301 -0.044+0.0361 0.770+0.4~41 0.699+0.~211 

0.667 O. 0.569- 0 .1301 -0.173+0.0121 1. 53~+0. 9 71 1. 095+0. 7~1 0.667 1.0 0.569-0.1301 -0.459-0.1831 1.92 ~+l.2341 0.673+0..17 1 

LINEAR SYMlvIETRICAL 'TORSION OF RECTANGULAR WING OF AR = 6 (a = 0) 

0 0 1 C>O 0 0. 42R 

a a.~ 1 -0..20.87 1. 257 a'K9 

0 o. 1 -0..4290. 2·513 1. 35 

0. loG 1 -1 3.142 0. 

0.167 0. 0..757-0..1881 ()O-ool 0. 0..292-0..10.21 

0.167 0.4 0..757-0..1881 -G. GL~9 +G .0.691 0..971-0.0521 0. · 90.2+8· 0.30.1 

0..167 0. .8 0..757-0.1881 -0.215+0 .1281 1'442-0.10.41 1. 373+0.1731 

0.167 1.0 0.757-0..1881 -0.750+0 .1051 2. 27-0.1301 0 .045+0 .0031 

0.333 0 0.650- 0 .1751 DO-oOl 0 0.230-0.0821 

0.333 o.~ 0.650-0.1751 -0..033+0..0401 0 . 853+0 .1261 0. . 304+0 .1691 

0·333 O. 0 . 650- 0 . 1751 -0.137+0.0931 1.707+0.2521 1.325+0.4291 

0.333 La 0.650.-0.1751 ,-a. 622+0 . 010i 2·133+0·315j 0.173+0.0221 

0 . 667 a 0 .569-0 .l}O1 00+001 a .1~8-a. aa~1 0 .667 O.~ 0.569-0.1301 - 0 .012+0.0151 0.770+0.4~41 0 . 8 1+0.51 1 

0.667 O. 0 . 569- 0 .1 3°1 .O79+0. 0~11 1. 539+0 . 9 71 1.301+0.9831 

0.667 loCi 0.5 69-0.1301 -0.459-0 .1 31 1.924+1.234.1 0 .673+0.1771 

NAT IONAL ADVISORy COMMI1 TEE FOR AERONAUTICS 
78 

k 

0.167 

0.167 

0.167 

0.167 

0.333 

0.333 

0·333 

0.333 

0.667 

0.667 

0.667 

0.667 

0.167 

0.167 

0.167 

0.167 

0.333 

0·333 

0·333 

0·333 

0.667 

0.667 

0.667 

0.667 

k 

0.167 

0.167 

0.167 

0.167 

0.333 

0.333 

0.333 

0.333 

0.667 

0.667 

0.667 

0.667 

TABLE VIII (continued) NACA TN No. 1195 

LINEAR SYMMETRICAL BENDING FOR RECTANGULAR WING OF = 3 

y L(2) L 

C (1h boB 2pU~oh/bo 2pU~oh/bo 

0 0.757-0 .1881 00 -001 0 0. 09 +0.06~1 

0.4 0.757-0.1881 -0.1~+0.1201 0.022+0.1591 -0.003+0.12 1 

0.8 0.757-0.1881 -0.3 3+0.141i 0.044+0.3171 -0.015+0 .16~1 

1.0 0.757-0.1881 -0.750+0.1051 0.055+0 ,3971 -o.~+o.o i 

0 0.650-0.1751 00 - col 0 0.017+0.1106i 

O.~ 0.650- 0 .1751 -0.076+0.0821 0.003+0.2721 -0.031+0.2381 

O. 0.650-0.1751 -0.262+0.098i 0.00~+0.5~1 -0.075+0 .325i 

1.0 0.650-0.1751 -0.622+0.0101 0.00 +Olb 11 - 0.00 3+0.0291 

0 0.569-0.1301 OQ-001 0 -0.005+0. ~ijl 

0.4 · 0.569-0.13°1 -0.044+0.0361 .0.170+0.4771 - 0 . 2m. +0. 01 

0.8 0.569-0.1301 -0.173+0. 0121 -0. R41 +0.9541 -0.361+0. 41 

1.0 0.569-0.1301 -0.459-0.1831 -0. 26+1.1921 -0.031+0.2301 

LINEAR SYMMETRICAL BENDING FOR RECTANGULAR WING OF AN :: 6 

0 0.757-0.1881 0() - 0() i 0 0.021 +0 . O~~i 

O.~ 0.757-0.1881 -0.049+0.069i 0.022+0.1591 0.007+0.14 i 

o . . 0.757-0.1881 -0.215+0 .1281 0.044+0.3171 -0.010+0.2271 

1.0 0.757-0.1881 -0·750+0.1051 0.055+0 .3971 - 0.0002+ O. 0041 

0 0.650-0.1751 00 -Q()1 0 0.039+0.0701 

O.~ 0.650-0.1751 - 0.033+0.0401 0.003+0.2721 -0.013+0.2591 

O. 0.650-0.1751 -0.137+0.0931 0.00~+0.5~1 -0.071+0.4301 

1.0 0.650-0.1751 - 0.622+0.0101 0.00 +0.6 li -0.002+0. 0291 

0 0.569-0.1301 00 - 001 0 0.032+0.0821 

0.4 0.569-0.1301 - 0.012+0.0151 -0.170+0 .477i -o~183+0 .~67i 

0.8 0.569-0.1301 -0.079+0.0311 -0. R41 +0.9541 .0.3&.3+0. 221 

1.0 0.569- 0.1301 -0.459- 0 .1831 -0. 26+1.1921 -0.0 2+0.2301 

PARABOLIC SYMMETRICAL BENDING FOR RECTAmULAR WING OF AN = 3 

Y C O'h 

boB 

0 0.757-0.1881 co - oOi 

O.~ 0.757-0.1881 0.035+0 .1061 

O. 0.757-0.1881 -0.392+0.1431 

1.0 0.757-0.1881 -0.750+0.1051 

0 0.650-0.1751 ~ - 1101 

O.~ 0.650-0.1751 0.0~9+0.0721 

O. 0.650 -0.1751 -0.2 7+0.0971 

1.0 0.650-0.1751 -0.622+0.0101 

0 0.569-0.1301 cP - bOl 

O.~ 0.569-0.1301 0.06§+0.050i 

O. 0.569-0.1301 -0.18 +0.0061 

1.0 0.569-0.1301 -0.459-0.1831 

L(2) -

L 

2pU~oh/bo 2pU~oh/bo 

0 0.006+0.0361 

0.009+0.06,1 0.000+0.0661 

0.035+0.25 1 .013+0.12~ 

0.055+0 .3971 -0.0002+0.00 

0 0.012+0.0591 

0.001+0.1091 -0.011+0.1221 

0.00~+0·t361 -0.060+0.2431 

0.00 +0. 811 -0.003+0. 0291 

0 0.0'05+0 . 0811 

-0.068+0.1911 -0. 085+0 . 2141 - 0 .272+0.7631 -0.280+0.5111 

-0.426+1.1921 -0.031+0.2301 

NATIONAL ADVISORY 

COMMITTEE fOR AERONAUTICS. 
.-

NACA TN No. 1195 TABLE VIII (cont1nued) 79 

0.167 

0.167 

0.167 

0.167 

0·333 

0·333 

0.333 

0·333 

0.667 

0.667 

0.667 

0.667 

k 

0 

0 

0 

0 

0.167 

0.167 

0.167 

0.167 

0·333 

0·333 

0·333 

0.333 

0.667 

0.667 

0.667 

0.667 

0.167 

0.167 

0.167 

0.167 

0·333 

0·333 

0·333 

0·333 

0.667 

0.667 

0.667 

0.667 

1.333 

1.333 

1.333 

1.333 

PARABOLIC SYMMETRICAL BENDING FOR RECTANJULAR WING OF AR = 6 

0 

O.~ O. 

1.0 

0 

o.~ O. 

1.0 

0 

O.~ O. 

1.0 

y 

bOB 

o·t o. 

0.8 

1.0 

o·t o. 

0.8 

1.0 

o·t O. 

0.8 

1.0 

o·t o. 

0.8 

1.0 

o·t O. 

0.8 

1.0 

o.~ O. 

0.8 

1.0 

o·t O. 

0.8 

1.0 

o·t O. 

0.8 

1.0 

0·757-0.1881 00 - 001 0 0.019+0.0231 

0.757-0.1881 0.105+0.0201 0.009+0.0631 0.007+0.0721 

0.757-0.1881 -0.239+0.1331 0.035+0 . 2541 -0.010+0.17~1 0.757-0.1881 -0.750+0 . 1051 0,055+0 .3971 -0.0002+0.00 ~j 

0.650- 0 .1751 • 00 -OQ1 0 0.023+0.03R1 

0.650-0.1751 0.091-0.0051 0.001+0.1091 0.002+0.12 1 

0.650- 0 .1751 -0.153+0.0951 0.005+0·t361 -0.059+0.3331 

0.650- 0 .1751 -0.622+0.0101 0.003+0. 311 -0.0002+0.0291 

0.569-0.1301 00 +ClO1 · 0 0.020+0.0361 

0.569-0.1301 0.0~7+0.0011 -0.068+0.1911 -0.068+0.2101 

0.569-0.1301 -0.0 6+0.0271 -0.272+0.7631 -0.309+0.6481 

0.569-0.1301 -0.459-0.1831 -0.426+1.1 921 -0.042+0.2301 

LINEAR ANTISYMMETRICAL TORSION OF RECTAIDULAR WING AR = 3 

r.;(2) -

L 

C oJ 2pU9JoJ 2pU9JoJ 

1 -'0.490 1.2~7 0.641 

1 -0.528 1.85 0.84

0 

1 -0·584 2.513 1.97 

1 -1 3.142 0 

0.757-0.1881 -0.2~3+0.1651 0.,71-0.0521 0.636+0.1291 

0.757-0.1881 -0.2 9+0.1621 1. 46-0.0781 0.887+0.1821 

0.757-0.1881 -0.342+0.1591 1., 2-0.1041 l'°ttK+O.2241 

0.751-0.1881 -0.750+0 .1051 2. 27-0.1301 0.0 +0.0031 

0.650- 0 •1751 -0.1~9+0.1301 0.8~3+0.1261 0.639+0.25~1 

0.650- 0 . 1751 -0.1 3+0.1281 1.2 0+0.1891 0.895+O'R7 1 

0.650- 0 . 1751 -0.233+0.1221 1.707+0.2521 1. 01)9+0. 611 

0.650- 0 . 1751 -0.622+0.0101 2.133+0 .3151 0.173+0.0221 

0.569-0.1301 -0 '.072+0 •0611 0'770+0.4~41 0.645+0 .5371 

0.569-0 .1301 -0.lo.8+0.0~41 1.155+0.7 11 o . 91 ~+o •. 77t1 

0.569-0.1301 -0.146+0.0 01 1. 53,+0. 9 71 1.13 +0.96 1 

0.569-0.1301 -0.459-0 •1 31 1.92 +1. 2341 0.673+0.1781 

RIGID ROLLING OF RECTANGULAR WING OF AR = 3 

0.757-0.1881 -0.2~3+0.1651 

0.757-0 .1881 -0.2 9+0.1621 

0.757-0.1881 -0.342+0.15.91 

0.757-0.1881 -0.750+0.1051 

0.650-0.1751 -O.1~9+0.1301 0.650- 0 .1751 -0.1 3+0.1281 

0.650- 0 .1751 -0.233+0 .1221 

0.650- 0 .1751 -0.622+0.0101 

0.569-0.1301 -0.078+0.0611 

0.569-0.1301 -0.108+0. 0~41 0.569-0.1301 -0.146+0.0 01 

0.569-0 .1301 -0.459-0.1 31 

0.526- 0 •0811 -0.oR4-0.0011 

0.526-0.0811 -0.0 2-0.0151 

0.526-0.0811 -0.053- 0 .0321 

0.526-0.0811 -0.090+0.4141 

0.022+0.1591 -0.01~+0.1061 

0.0G4+0.2381 -0.01 +0.1471 

0.0 +0.3171 -0.023+0.1741 

0.055+0.3971 -0.0002+0 .0041 

0.003+0.2721 -0.051+0.2101 

0.005+0 .4081 -0.076+0.2931 

0.00~+0.5~1 -0.094+0.3511 

0.00 +0.6 11 -0.003+0.0291 

-0.170+0.4771 -0.222+0.4111 

-0.245+ 0.~141 -0. R 1+0. 5 1 

-0. 26+1.1921 

-0·R2~+0.579i 

-0. 0 +0.7081 

-0.042+0.230 1 

-0'481+0.8811 -0'~7K+0. 8241 -1. 71+1. 3221 -1. 3 +1.2151 

-1.,61+1. 7631 -1. 53+1.~861 -2. 52+2.2031 -0.716+1. 251 

NATIONAL ADVISORY 

COMMITTEE fOR AERONAUTICS. 
80 

k 

0 

0 

0 

0.212 

D.194 

0.127 

0.42~ 

0.38 

0.254 

0.847 

0.776 

0.5°8 

0 

0 

0 

0.212 

0.194 

0.127 

0.42~ 

0 .38 

0.254 

0.847 

0.776 

0.5°8 

k 

0.212 

0.194 

0.127 

0.42~ 

0.38 

0.254 

0.847 

0.776 

0·508 

TABLE VIII (continued) NAOA TN No. 1195 

LINEAR SYMMETRICAL TORSION OF ELLIPTICAL WING OF AR = 3 (a = 0) 

r;(2) - Y C 

L 

C1a. 

0 0 6 2 pU2

0oa: 2 pU2

0oa. 

0 1 00 0 0 . 391 

O.~ 1 -0.419 1.152 0 . 669 

o. 1 -0·535 1.508 0·702 

0 0.719-0.1881 00 - 00 1 a O.2~ O.O121 

o.~ P .733 - 0.1891 -0.181+0.1141 0.865-0.0231 0.6 +0.0881 

0. 0.799-0.1811 -0.347+0.1411 1. 222- 0.1021 0.6 5+0.0781 

0 0.618-0.1621 00 - ooi 0 0 . 294+0 .0371 

O.~ 0.629 - 0.1671 0.024

+0.0 54 1 0 .762+0.1721 0.6RO+O.2~01 

O. 0.691-0.1851 -0.27 +0.1241 1.077+0 •0461 0.6 2+0 . 1 31 

0 0·551-0.1131 00-001 0 0 .175+0 . 1141 

o.~ 0.556 -0.1191 -0. 060-0.0111 .6 9t+o .5 ~91 0 . 6 30+0 ' 4 201 

O. 0 . 596 -0.150i -0.177+0.0661 0.95 +0 .3 61 0 . 664+0 . 181 

LINEAR SYMMETRICAL TORSION OF ELLIPTICAL WING OF AR = 6 (a = 0) 

0 

O.~ O. 

0 

.~ O. 

0 

O. ~ o. 

0 

o .~ O. 

Y 

-

OOB 

0 

O.~ O. 

0 

~ O. 

0 

0.4 

0.8 

1 +00 0 0. 388 

1 -o. 26t 1.152 0 . 849 

1 -0· 37 1.5°8 0.941 

0 . 719-0.1881 00 - 001 0 0 . 294-0.0711 

0.733-0.189i - 0 .080+0 .0761 0.865-0.0231 0.~64+0.0561 

0.799-0.1811 -0. 2J9+0. 1241 1.222-0.1021 0.93+0 . 0b5i 

0.618-0.1621 ()c> - 00 1 0 0.21 2- 0.0441 

0.629-0.1671 -0. OL1-6+0 . 0431 0.762+0 .172i 0.~00+0.2131 

0.691-0.185 -0.145+0.1121 1.077+0 . 0461 O. 37+0.1871 

0·551 - 0 .1131 00 - 001 0 0 .1 00+0.02R1 

0.556-0.1191 - 0 .026-0.0041 0.694+0 . 559 1 0.680+0. 04 1 

0.596-0 .1 501 - 0 . .088+0 .0641 0.95 6+O.33 6i 0. 800+0. 1-11 

LINEAR SYMMETRICAL BENDING OF ELLIPTICAL WING AR = 3 

. 

C C1h 

0.719-0.1881 - 1 

0.733-0.1891 

0.799 - 0 .1811 

- 0 .180+0.1191 

- 0 . 347+0 .1391 

0 .613-0.1621 - 1 

0 . 629 -0.1671 -0.115+0 . 0751 

0 .691-0 .1851 -0.277+0.11 61 

0 · 551 - 0.1131 - 1 

0.556-0.1191 -0.057+0. 0011 

0.596 - 0.15 0 1 -0.177+0 . 0631 

r;(2) -

L 

2PU~oh/bo 2 pif'boh/bo 

0 0.006+0.0741 

0.022+0 .1791 

0 . 038+0 . 2551 - 0 . 007+0 . 1R41 

- 0 . 007+0 .1 1 

0 0 . 016+0 . 0701 

-0.01 3+0'4471 -0. 050+0 . 2 511 

0.037+0. 21 - 0 . 037+0 . 2 641 

0 - 0.039+0 . 1651 

-0.26R+O.5t3i 27~ 497i - 0 .1 3 +0.7 2i -0. 21 +0.5361 

NATIONAL ADVISORY 

COMMITTEE FOA AERONAUTICS. 

• 
• 

NACA TN No. 1195 TABLE VIII (continued) 81 

0.212 

0.194 

0.127 

0.42~ 

0·38 

0.254 

0.847 

0.776 

0.508 

k 

0.212 

0.194 

0.127 

0.42~ 

0.38 

0.254 

0.847 

0.776 

0.508 

0.212 

0.194 

0.127 

0.42~ 

0.38 

0.254 

0.847 

0.776 

0.508 

LINEAR SYMMETRICAL BENDING OF ELLIPTICAL WING OF AR = 6 

0 0.719-0.1881 - 1 0 0.021+0.0611 

O.~ 0.733-0.1891 -0.078+0.0801 0.022+0.1791 0.003+0.1601 

O. 0.799-0.1811 -0.211+0.1231 0.038+0.255 1 -0.002+0.1881 

0 0.618-0.1621 - 1 0 0.034+0.0851 

O.~ 0.629- 0 .1671 -0.042+0.0411 -0.013+0 'E..471 -0.034+0.2871 

O. 0.691-0.1851 -0.146+0.1101 0.037+0. · 21 -0.034+0.3491 

0 0·551-0.1131 - 1 0 0.011+0.0961 

o .~ 0·556-0.1191 -0.021+0.0021 -0.26R+O.5~31 -0.266+0.5241 

O. 0.596-0.150i -0.087+0.0631 -0.13 +0.7 21 -0.216+0.6521 

PARABOLIC SYMMETRICAL BENDrnG OF ELLIPTICAL WING OF AR = 3 

y L(2) L - C a 

bos 2 pU2boh/b0 2pU~oh/bo 

0 0.718-0.1881 00 - obi 0 0.005+0 . 0 391 

o.~ 0.733- 0 .1891 -0.030+0.1051 0.009+0.0711 -0.001+0.0691 

O. 0.799-0.1811 -D.372+0.1441 0.030+0.2041 -0.007+0.1091 

0 0.618-0.1621 d)-1::I01 0 0.003+0.06~1 

o.~ 0.629-0.1671 0.024

+0.0541 -0.005+0 .1231 -0.016+0.12 1 

o. 0.691-0.1851 -0.27 +0.1241 0.029+0.3531 -0.034+0.2131 

0 0·551-0.1131 oO-ao1 0 -0.010+0.0851 

O.~ 0.55 6 - 0 •1191 0.039+0. 0121 -0.105+0 . 2171 -0.116+0.25~i 

O. 0.596-0.1501 -0.187+0.0611 -0.107+0.6101 -0.169+0.41 1 

PARABOLIC SYMMETRICAL BENDrnG OF EIJ..IPTICAL WING OF AR ::: 6 

0 0.719-0.1881 00-001 

O.~ 0.733-0.1891 0.04§+0.0381 

O. 0.799-0.1811 -0.22 +0.1281 

. 0 0.618-0.1621 ~ - 001 

O.~ 0.629-0.1671 0.049+0.0031 

O. 0.691-0.1851 -0.157+0.1131 

0 0·551-0.1131 00 - 0(') 1 

O.~ 0.556 - 0 •1191 0.0~8-0.0131 

O. 0.596-0.1501 -0.0 7+0.0591 

0 0.012+0.0301 

0.009+0.0721 0.005+0.0761 

0.030+0.2041 -0.003+0.1461 

0 0.021+0,0391 

-0.005+0.1231 -0.006+0.1331 

0.029+0 .3531 -0.028+0.2731 

0 0.012+0.0421 

-0.105+0 .2171 -0.101+0.2321 

-0.107+0.6101 -0.145+0 .5211 

NATIONAL ADVISORY 

COMMITTEE FOR AERONAUTICS. 
82 TABLE VIII (continued) NACA TN No. 1195 

k 

0 

0 

0 

0.194 

0.170 

0.127 

0.388 

0. 33

0.25: 4 

0.77~ 

0.67 

0.509 

0.194 

0.170 

0.127 

0.388 

0. 33

0.25 4 

0.77~ 0.67 

0.50 7 

1.553 

1.356 

1.017 

LINEAR ANTISYMMETRICAL TORSION OF ELLIPTICAL WING OF AR = 3 

Y 

bOB 

o·t 

o. 

0.8 

o.~ o. 

0.8 

o·t O. 

0.8 

o.~ o. 

0.8 

o·t O. 

0.8 

o .~ O. 

0.8 

O.~ o. 

0.8 

o·t o. 

0.8 

C 

L(2) L 

Cl'j 2pU~oj 2pifbo j 

ko = 0 

1 -0·571 1.152 0·t44 

1 -0.571 1.5°8 O. 7 

1 -0·571 1.508 0.647 

ko = 0.212 

0. 734-0.1891 -0.28

4

+0.1 521 0.865-0. 0231 0.515+0.1191 

0.75 -0.1881 -0.30 +0.1551 1.162-0.0591 0.b83+0.1361 

0·799-0.1811 -0.305+0 .1b21 1.222-0.1011 0.747+0.114i 

k = 0 0.424 

0.629-0.1671 -0.183+0.0961 0.762+0 .1721 0.530+0.2421 

0.647-0.1~41 -0.1'97+0.1141 1.021+0.1

4

91 0.b4

5+0.2801 

0.691-0.1 51 -0.200+0.1501 1.077+0.0 61 0.7 7+0.2331 

k = 0.847 0 

0.562-0.1241 -0.09~+0.0041 0.70G+O.5561 0.58~+0.5171 

0.569-0 .1301 -0.11 +0.0301 0.92 +0.6071 0·73 +0'4931 

0.569-0 .1501 -0.121+0.0931 0.957+0.3861 0.738+0. 801 

RIGID ROLLING OF ELLIPTICAL WING OF AR = 3 

. k = 0 0.212 

0.7§4-0.1891 

0.7 -0.1881 

0.799-0.1811 

-0.284+0.1 561 

-0.30 +0.1581 -0.30~.+0.162i 

k = 0 o .!~24 

0.629-0.1671 -0.182+0.10~1 

0.647-0.1~4i -0.196+0.11 1 

0.691-0.1 5i -0.200+0.1481 

k = 0 0.847 

0.562-0.1241 -0.098+0.016i 

0.569- 0 .1301 -0.113+0. 0371 

0.596-0.15°1 -0.120+0.0891 

k = 0 1.695 

0.520-0.0721 -0.005-0.0371 

0.525-0.0801 -0.030-0. 035i 

0·539-0.0991 -0.056+0.0261 

0.022+0.1791 -0.016+0.1081 

0.033+0.2411 -0.018+0.1441 

0.038+0.2551 -0.014+0.1581 

-0.013+0.4071 

0.003+0. 141 

0.037+0 .4411 

-0.064+0.2181 

-0.073+0.2881 

-0.058+0.313 1 

-0.258+0.549i -0.274+0.4~41 -0.268+0.7271 -0.31~+0.5 31 

-0.134+0.7621 -0.24 +0.6081 

-1. 376+1. 0151 -1·G04+1.00 61 

-1.527+1.343 1 -1. 38+1.265i 

-1.046+1.3771 -1.113+1.233i 

NATIONAL ADVISORY 

COMMITTEE FOR AERONAUTICS. 

.. 
I!OA TN No. 1195 TABLE VIII (oontinued) 

k 

0 

0 

0 

0.212 

0.194 

0.127 

0.42~ 0.38 

0.254 

0.847 

0.776 

0.508 

1.695 

1.553 

1.017 

0.212 

0.194 

0.127 

0.42~ 

0.38 

0.254 

0.847 

0.776 

0.508 

1.695 

1.553 

1.017 

1.553 

1.356 

1.017 

2.330 

2.034 

1.525 

UNIFORM PI'roHING OF ELLIPTICAL WIIC OF AR = 3 

-(2) - Y C L L 

boe OJ 2PU~oJ 2pnoJ 

0 1 -0.400 3.142 1.885 

o.~ 1 -0.400 2.878 1.72

O. 1 -0.400 1.727 1. OJ b 

0 0.719- 0 .1881 -0 .lt8+O .1181 2.322+0.0061 1.816+0.32~1 

O.~ 0.733-0.1891 -0.1 2+0.1221 2.161-0.0591 1. 6~~+0. 24 1 

O. 0.799-0 .1811 -0.227+0.1~11 1.527- 0 .1261 1. 0 +0.0931 

0 0.618-0.1621 -0.0~8+0.0631 2.044+0'4691 1·l60+0. 7161 

O.~ 0.629-0.1671 -0.0 3+0.0811 1.90 +0. 301 1. 20+0.6181 

O. 0.691-0.1851 -0.151+0.1181 1. 347+0.0571 1.°34+0.2441 

0 0·551-0.1131 -0.0,7+0 . 0011 1.880+1.7101 1.761+1.6631 

.o.~ 0.55l- 0 •1191 -0.0 9+0.0131 1.735+1·4~71 1. 5l 9+1. 3801 

o. 0.59 -0.1501 -0.101+0.0791 1.195+0 . 31 0.9 7+0 .5831 

0 0.518-0. 0671 0.005-0.0181 1.861+3.8291 1.925+3.7871 

O.~ 0.520-0.0721 -0.006-0.0221 1.657+3.1891 .1.687+3.1131 

o. 0.539-0 . 0991 -0.081+0.0201 1.110+1.2881 0.939+1.2471 

UNIFORM TRANSLATION OF ELLIPTICAL WING AR = 3 

0 0.719-0.1881 -0.1t8+0.1211 0.055+0 .4781 -0.026+0·3801 

O.~ 0.733-0 .1891 -0.1 2+0.1241 0.0

4

6+0. 4471 -0.020+0.3481 

O. 0.799-0.18li -0.228+0.1251 0.0 7+0.3191 -0.003+0 •2281 

0 0.618-0.1621 -0.0~6+0.0701 -0.067+0.8231 -0.160+0·l2li 

0.4 0.629-0.1671 -0.0 2+0.0361 -0.0,3+0 .7671 -0.1,8+0. 671 

0.8 0.691-0.18,1 -0.154+0. 1071 0.0 6+0.5521 -0.0 0+0.4291 

0 0·551-0.1131 -0.0,6+0.0081 ~0.828+1.4661 -0.84§+1. 3691 

0.4 0·557-0.1191 -0.0 7+0.0171 1-0.657+1.3581 -0.69 +1.2~31 

0.8 0.596- 0 .1501 -0.103+0.0651 -0.167+0.9521 -0.271+0.7 71 

0 0.518-0. 0671 0.00;-0.0121 -4-154+2.7571 -4.088+2.7731 

0.4 0.520 - 0 . 0721 -0.006-0.0161 - 3.436+2.5371 -3.357+2 '4061 

0.8 0·539-0.0991 -0.075+0.0081 -1.308+1.7211 -1. 333+1. 8li 

LINEAR ANTISYMMETRICAL TORSION OF AN ELLIPTICAL WJNj. OF AR = 3 

FOR HIGHER VALUES OF 

ko == 1. 695 

o·t 0.520-0.0~21 -0.001-0.0491 

o. 0.525-0.0 01 -0.030-0 . 045 i 

0.8 0.539-0 . 0991 -0.075+0.0161 

k o - 2.542 

o·t 0·511-0.0521 0.039-0.0001 

O. 0.513-0.0571 0.031-0.0311 

0.8 0·521-0.0731 0.037-0.0301 

ko 

0.66,+r.27~1 0.~06+1.2201 

0.87 +1.43 1 O. 76+1. 3401 

0.888+1.0301 0.763+0.9981 

o. 6t8+ 1. 9671 0.69

4

+2.0201 

0.8 1+2.2341 0.95 +2.2341 

0.869+1. 6391 0.847+1.5521 

NATIONAL ADVISORY 

COMMITTEE fOR AERONAUTICS. 

8~ 
84 

k 

0 

0 

0 

0.424 

0. 38

0.25 4 

0.847 

0.776 

0.508 

k 

0 

0 

0 

0.424 

0. 38

0.25 4 

0.84~ 0·77 

0.508 

TABLE VIII (e~tttlnued) NACA TN No. 1195 

UNIFORM SPANWISE AILERON DEFLECTION OF A FULL-SPAN AILERON 

OF ELLIPTICAL WING OF AR = 3 (e = 0, 1 = 0.2 b/bo) 

-(2) - y L L 

C °13 bos 2pU"'of3 2p~oi3 

0 1 -0.400 2.589 1.652 

O.~ 1 -0.400 2.373 1.514 

o. 1 -0.400 1.553 0:991 

0 0.618-0.1621 -0.080+0 . 0621 1. 664+0 . 3~91 1.411+0 .4811 

o.~ 0.629-0 .1671 -0.090+0 . 0741 1.554+0.2 21 L24t+0.4001 

o. 0.691-0.1851 -0.150+0.1211 1.111+0.0201 0.8 +0.1 671 

0 0·551-0.1131 -0.041+0.0011 1. 369+1.2001 1.26t+l·1411 

0.4 0.556- 0 . 1191 -0.050+0 •0 091 1.31~+0.9741 1.18 +0' 4321 

0·3 0.596-0.1501 -0.098+0 . 0831 0.95 +0·3291 0.761+0. 061 

UNIFORM SPANWISE AILERON DEFLECTION OF A FULL-SPAN AILERON 

OF ELLIPTICAL WING OF AR = 3 (e = 0, ! = 0.2 bIb ) 0 

y Mf3(2)(e) Mf3 (e) - C °13 boS 2pU9Jo 2'Jj 2pU9Jo 2'Jj 

0 1 -0.400 0 .187 0.124 

O.~ 1 -0.400 0.157 0.104 

o. 1 -0.400 0.Ob7 0.045 

0 0.618-0.1621 -0.080+0.0621 0.097+0 .1911 0.081+0.1971 

o.~ 0.629-0.1671 - 0 . 090 +0 . 0741 0.087+0.1551 0.073+0.1611 

O. 0.691-0.1851 - 0 .150+0 .1211 0.047+0.0371 0.037+0.0421 

0 0·551-0.11'31 -0.041+0. 0011 -0.023+0.4081 -0.029+0.4061 

O.~ 0.556-0.1191 -0.050+0 .0091 0 .003+0.3121 -0.005+0 .3101 

o. 0·596-0.15°1 -0.098+0 . 0831 0.023+0 . 0 841 0.021+0.0871 

UNIFORM AILERON HINGE DEFLECTION OF A HALF-8P AN AILERON OF AN ELLIPTICAL WING 

(e 1 

OF AR = 6 =- 2' 

y 

k - C °13 bos 

0 0 1 00 

0 o·t 1 00 

0 o. 1 - 0 .494 

0 0 .8 1 - 0.360 

0 0 · 9 1 - 0.239 

0.212 0 0 .719- 0 .1881 00 -oal 

0.194 O·t 0.73e-0.1891 00 - 001 

0.170 o. 0.75.-0 .1881 - 0.273+0.1261 

0.127 0.8 0·~49-0.1G11 -0.188+0.1301 

0.092 0.9 O . 3 - 0 .1681 -0.116+0.0911 

0.424 0 0 . 618-0 .1621 00 - aol 

0.389 O·t 0.62§-0.1671 OC) - 001 

0. 33

4 

o. 0.64 -0.1~41 - 0 .1 87+0 . 0871 

O.2~ 0.8 0.6

4

1-0.1 ~1 -0.107+0.12~1 

0.1 5 0 ·9 0.7 1-0.18 1 - 0 . 041~+0 .14 1 

NATIONAL ADVIS 

! = 0.1 ~) b o 

ORY 

COMMITTEE FOR AERON AUTICS 

-(2) 

L L 

2pU'1lo13 2p~..,13 

0 0.080 

0 0.2~5 1.554 0 .7 6 

1.10

4 0 .738 

o. 7 ~ 0.566 

0 0.036 - 0 . 0 561 

0 0 .175-0.121i 

1.187-0.1821 0·752-0.0101 

0.930-0.1481 0 . 707 - 0 . 0071 

0.629-0.102 1 0.541 - 0 . 0361, 

0 0.006-0.0571 

0 0 .108 - 0 .0261 

1.032-0.0701 

0 . 81)-0.1 001 

0 .726+0 . 0311 

0.677+0.0361 

0·557-0.099 1 0 . 519 +0 .0101 
NACA TN No. 1195 TABLE VIII (continued) 85 

UNll'ORM AILERON HINGE DELFECTION OF A HALF-8P AN AILERON OF AN ELLIPTICAL WING 

OF AR = 6 (e = 1., 7, = 0.1 ~) 2 b o 

k y C 

M/3(2) (e) Mj3 (e) 

bos 0(3 

2pU~o 2'j3 2Ptfbo 2'j3 

k = 0 

0 

0 0.6 1 -0.494 0.030 0.023 

0 0.8 1 -0.3 60 0.015 0.013 

0 0·9 1 -0.239 0.003 0.003 

ko=0.212 

0.170 0.6 0.754-0.1881 -0.2~3+0.1261 0.026+0.005i 0.022+0.0071 

0.127 0.8 0. 7

4

9-0.1811 -0.1 8+0.1301 0.014+0.0021 0.012+0.0031 

0.092 0·9 0.8 3-0.1681 -0.116+0.0911 0.003+0.0001 0.003+0.0011 

k 0-- 0.424 

0. 334 0.6 0.648-0.1~41 -0.187+0.0871 0.024+0.0131 0,021+0.0141 

0.2~ 0.8 0.64

1-0.1 ~1 -0.107+0.12~1 0.013+0.0051 0.012+0.0061 

0.1 5 0·9 0.7 1-0.18 1 -0.044+0.14 i 0.003+0.0011 0.003+0.0011 

- ROLLING-MOMENT COEFFICIENTS CR FOR A RECTANGULAR 'WING OF AR ". 3 

Due to rigid rolling Due to linear antieymmetrica.l torsion 

ko cRlii CR(2) /R CR/a C

R(2)/<i 

0 0 J 0.372 1.047 

0.167 -0.008+0.0621 0.018+0.1321 0.3~~+0.0781 0.809-0.0441 

0·333 -0.0,3+0.1261 0.003+0.2271 o.~ +0.1611 0.~11+0.1051 

0.667 -0.1 2+0.2621 -0.142+0.3971 o. 39+0.3451 o. 41+0.4111 

1.333 -0.676+0.6561 -0,817+0.7341 o. 20+0.8351 0.608+0.9801 

ROLLING--MOMENT COEFFICIENTS CR FOR AN ELLIPTICAL WING OF AR = 3 

Due to rigid rolling Due to linear antisymmetrical 

torsion 

ko ~R/4h l1CR(2) /4h ~R/4a: nCR(2) /4a: 

0 0 0 0.250 0.502 

0.212 -0.006+0.0581 O.013+0.09~1 0.276+0.0491 0.458- 0 '°tK1 

0.424 -0.026+0.1161 0.006+0.16 ~1 0.279+0.1091 0,404+0.0 1 

0.847 -0.111+0.2321 -0.082+0.2871 0.288+0.2111 0.363+0.1991 

1.64

5 -0.511+0.4861 -0.513+0 .5241 0.319+0 .4691 0·3!~0+o.4911 

2.5 2 -1. 231 +0.7531 -1.256+0 .7641 0.341+0.7561 0.333+0.7681 

NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS 
86 TABLE VIII (concluded) NACA TN No. 1195 

- LIFT COEFFICIENTS CL FOR AN ELLIPTICAL WING OF AR = 3 

-

Due to uniform translation Due to uniform pitching 

ko ~d-8h 

n:CL(2)/ -

8h rleL/ - 8a. ~L(2)/ _ 

8a. 

0 0 0 1.480 2.467 

0.212 -0.OOI+0. 29t1 0.048+0.3891 1.420+0.2121 1. 867- 0 .0691 

0.~24 -0.104

+0. 5 6 1 -0.0IR+0.6721 1.356+0.4~61 1.665+0 . 2971 

o. 47 -0.55 +1.0~61 -0.t9 +1.1721 1.337+1.0 91 1.495+1.07~1 

1.695 -2.637+2.0 01 -2. 65+2.1631 1. 385+2 .4211 1.431+2.46 i 

RECTANGULAR WING OF AR = 13.5 WITH INDICATED MODES (ko = 0.097) 

k 

0.097 

0.097 

0.097 

0.097 

0.097 

0.097 

0.097 

0.097 

0.097 

0.097 

0.097 

0.097 

0.097 

0.097 

0.097 

Mode Torsion First Second 

Y/bos bending bending 

0 0 0 o·t 0.~86 0.230 

o. o. 09 0.461 

0.8 0.951 0.725 

1.0 1.0 1.0 

Torsion mode 

Y/b s 0 C OJ 

0 0.837-0 .1701 oC- ~1 

o·t 0.837-0.1701 -0.Ot5+0.048i 

O. 0.837- 0 .1701 -0.0 9+0.0~11 

0.8 0.837-0.1701 -0.1091'0.0 91 

1.0 0.837-0.1701 -0.835+0.1221 

FIRST BENDING 

0 0.837-0.1701 DO - ~1 

o .t 0.837- 0 .1701 0.056+0.0011 

O. 0.837-0 .1701 -0.046+0.0641 

0.8 0.837-0 .1701 -0.124+0.0961 

1.0 0.837- 0 .1701 -0.835+0.1221 

SECOND BENDING 

0 0.837- 0 .1701 - 1 

o·t 0.837- 0 .1701 -0.153+0.1011 

O. 0.837-0 .1701 -0.176+0.1031 

0.8 0.837-0 .1701 -0.619+0 .309 1 

1.0 0.837-0 .1701 -0.835+0.1221 

0 

0.683 

0.589 

-0.068 

-1.0 

1:(2) -

L 

2pU~n} 2pU~oj 

0 O·R02-0.140i 

1.563-0 .1511 1. 72-O.070i 

2.1 5§-0.2091 1. 968-0. 04~i 

2-53 -0.24~1 2.190-0.00 i 

2.669-0.25 i 0.013-0.102i 

0 0.009+0. 013i 

+0.009+0.0581 0.008+0.062i 

+0.017+0 .1171 0.008+0.111i 

0.027+0 .1841 0.006+0.157i 

0.037+0. 25L~i +0.000+0.0011 

0 . 0 .011+0. 0261 

0.025+0.1731 0.004+0.142i 

0.022+0.1501 0.00~+0.1181 

-0.003- 0 . 01 71 0.00 -0.0051 

-0.037- 0 . 2 54i O.OOO-O.OOli 

NATIONAL ADVISORY 

COMMITTEE FOR AERONAUTICS. 

'"""------ - - -- -- - ---- - -----
ko 

0 

0.424 

0.847 

1.695 

3.390 

0 

0.318 

0.637 

1.273 

2.546 

0 

0.212 0.424 

0.847 

1.695 

2.542 

0 

0.106 0.212 

0.424 

0.847 

1.273 

TABLE IX.- SPAN CORRECTIONS AT MIDSPAN FOR RIGID ELLIPTICAL WING 

IN TRANSLATION· AND PITCHING. (ONE-POJNT APPROXIMATION) 

kos C 

0 1 + 01 

0.5 0.618 - 0.162i 

1.0 0.551 - 0.1131 

2.0 0.518 - 0.0661 

4.0 0.505 - 0.0361 

0 1 + 01 

0.5 0.657 - 0.1771 

1.0 0.573 - 0.1341 

2.0 0.528 - 0.0841 

4.0 0.509 - 0.0481 

0 1 + 01 

0.5 0.719 - 0.1881 

1.0 0.618 - 0.1621 

2.0 0.551 - 0.1131 

4.0 0.518 - 0.0671 

6.0 0.509 - 0.0471 

0 1 - 01 

0.5 0.824 - 0.1751 

1.0 · 0.719 - 0.1891 

2.0 0.618 - 0.1621 

4.0 0.551 - 0.1131 

6.0 0.528 - 0.0841 

C1h = C1a, = C1 

-0.571 

-0.196 + 0.0921 

-0.114 + 0.0081 

-0.018 - 0.0441 

+0.005 + 0.0071 

-0.500 

-0.176 + 0.1161 -0.103 + 0.0481 -0.042 - 0.0181 

+0.004 + 0.0171 

-0.400 

-0.154 + 0.1251 -0.079 .-.: 0.0751 -0.043 + 0.018i -0.010 - 0.0131 

+0.007 - 0.0051 

-0.250 

-0.108 + 0.0961 

-0.052 + 0.0771 

-0.021 + 0.0371 

-0.013 + 0.0091 

-0.008 - 0.0001 

C + (1 

0.429 

0.422 

0.437 - 0.1051 

0.500 - 0.1101 0.511 - 0.0291 

0.500 

0.481 - 0.0611 0.470 - 0.0861 0.486 - 0.1021 0.513 - 0.0311 

0.600 

0.565 - 0.0631 0.539 - 0.0871 0.507 - 0.0951 0.508 - 0.0801 0.516 - 0.0521 

0.750 

0.716 - 0.0781 0.669 - 0.1111 0.597 - 0.1241 0.538. - 0.103i 0.520 - 0.0841 

NATIONAL AD

COMMITTEE FOR A
88 NACA TN No. 1195 

TABlE X.- llE'l'ERM!NANT TERMS AS A FUNCTION OF k FOR EXAMPU: I 

Term Structural part of tenn 

k ~ 0.333 

~ 1.3331 r 1 - 0 .1920 (~ J (1 + 1~) 1 

~ 0.3154 

!? 0.3154 

~ 0.8789 r 1 - c;;= y (1 + 18,.) 1 

k ~ 0.4 

A 1.3331 r 1 - 0.1920 ( ~ J (1 + 1~) 1 = ala. 

B 0.3154 

D 0.3154 z 

E 0.8789 r 1 -(? )2 (1 + 1Sa.) 1 

k ~ 0.5 

A 1.3331 r 1 _ 0.1920 ( ~ )2 (1 + 1~) 1 

B 0.3154 = 

D 0 . 3154 = 

~ 0.8789 r 1 - (? Y (1 + 1Sa.) 1 

k = 0 .6 

A 1.3331 r 1 - 0.1920 (~ J (1 + 1S

h ) 1 = 

B 0 . 3154 = 

D 0.3154 = 

!: 0.8789 r 1 - (? J (1 + Sa.) 1 

Aero . part of tenn (tv<Hl.1m. ) 

-<>.0096 - O. 77991 

- 3.0592 - 0.74371 

0.1274 + 0.19501 

0.9075 - 0.80171 

0.0350 - 0.62501 

-2.0431 - 0.73441 

0.1163 + 0.15631 

0.6365 - 0.63751 

0 .0794 - 0.47831 

-1.2414 - 0.67691 

0.1051 + 0.11961 

0.4227 - 0.48621 

0.1081 - 0.38591 

-<>.8208 - 0.61121 

0.0980 + 0.09651 

0.3535 - 0.49271 

De term1nant terms 

f t.ro-d1.m. ) 

(S ~ Sn E sa. 

and 

( ~ y (1 + 1S) = n ) 

1.3235 - 0.77991 - 0.25600 

-2.7438 - 0 . 7437' 

0.4428 + 0 .19501 

1 .7864 - 0 .8017 - 0.87~ 

1.3681 - 0.62501 - 0.25600 

-1.7277 - 0 .73441 

0.4317 + 0.15631 

1 .5154 - 0.63751 - 0.8785D 

1.4125 - 0. 47831 - 0.25600 

-<> .9260 - 0.67691 

0.4205 + 0.11961 

1.3016 - 0 .48621 - 0.8785D 

1.4412 - 0.38591 - 0. 256m 

-<>.5054 - 0 .61121 

0.4134 + 0 .0961 

1 .2513 - 0.4927 - 0.8785D 

NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS 
NACA TN No. 1195 89 

TABLE XI 

SOli.lTION OF SIMULTANEOOS EQUATIONS :BY USE OF THE CROUT METHOD OF REFERENCE 8 (EXAMPLE I) 

L1near tors1on Parabol1c bending 

Fa(Y) Chsck Fb(Y) Check 

G1ven loBtru 

1.0169 -0.3837 0.2702 -0.2262 0 0.6772 0 0.6772 

-0.05241 0.07391 -0.07851 0.07901 0.02201 

.9401 -.1323 - .1299 .2289 0.4 1.3068 0.16 1.0668 

-.05571 .02621 .04051 -.08261 -.07161 -.05071 

.6672 .4213 - .0234 -.3057 .8 1.5594 .64 1.3994 

-.06841 -.12381 .02451 .il701 -.05071 -.05071 

.2966 .4701 .7234 .9896 1.0 3.4797 1.0 3.47971 

.09721 -.05051 -.18791 -.30721 -.44841 .44841 

Su~~lementary Matrix 

0.9808 -0.3837 0.2702 -0.2262 0 0.6772 0 0.6772 

.05051 .07391 -.07851 .07901 .02201 .02201 

4.3261 -. 3793 .4375 0.4 .6803 0.16 .4903 

.87461 .il501 -.15731 -.08711 -.08711 

.9a53 -1. 467a . -.~64 -.9840 .144

4 -.3782 

.3 701 .633 1 . 561 .37011 .03 31 .31871 

.3681 .7055 3.1925 .3483 2.8352 

.lil81 -.18981 -.945411 -.15701 -.91221 

ux1l1ary Matr1x 

1.0169 -0.3801 0.2690 -0.2258 0 0.6631 0 0.6631 ~.05241 .05311 -.06331 .06611 .05581 .05581 

.9401 .2221 -1.7415 2.0302 1. 7304 3.0192 0.6922 1.9810 

-.05571 -.04491 .16581 - .29791 .34981 .21821 .13991 .00031 

.6672 .6713 .9398 -1. 5922 - .4285 -1.0207 .1278 -.4643 

-.06841 -.18521 -.34871 .08331 -.06791 .01511 .08391 .16681 

.2966 .5880 1.6566 2.4870 .2809 1.2808 .1458 1.1456 

.09721 -. 02931 -.34881 -.75561 .00901 .00901 -.01891 -.01881 

K' ; 0.5264 la 1.5264 K'lb ; 0.3377 1.3378 

.04851 .04841 .03201 .03191 

K' = 1.1786 2.1785 K' 3h z 1.0329 2.0330 3a 

.27791 .27791 .23491 .23471 

K' 5CL = 0.0195 1.0193 K'5b = 0.3584 1.3582 

-.07701 -.07731 .04171 .04141 

NATIONAL ADVISORY , = 0.2809 K' 

COMMITTEE FOR AERONAUTICS. 

K 1.2808 = 0.1458 1.1456 7a 7h 

.00901 .00901 -.01891 - .01881 
90 NACA TN No. 1195 

TABLE XII.- VALUES OF w AT VARIOUS STATIONS ALONG THE SPAN 

(EXAMPLE I) 

y sin n~ - n 

bos n 

r 1 0.9165 

:3 -0.1099 

0.4 < 

5 -0.0956 

.... 7 0.1580 

-

w; 

F 1 0.800 

3 0.1173 

0.6 ~ 

5 -0.1994 

.... 7 0.0295 

-

W· , 

r 1 0.600 

3 0.3120 

0.8 < 

5 -0.0151 

'- 7 -0.1598 

-

W· , 

,- 1 0.4359 

5 0.3255 

0.9 < 5 0.1550 

.... 7 -0.0022 

-

)1; 

Knh sin n~ 

Kna. 

sin n~ 

n n 

0.5095 + 0.02931 0.4824 + 0.04451 

rO.1155 - 0.02581 -0.1295 - 0.03051 

r-0.03:55 - 0.00"391 -0.0018 + 0.00721 

0.0201 - 0.00261 0.0588 + 0.00121 

0.1826 - 0.00301 0.5899 + 0.02241 

0.2702 + 0.02561 0.4211 + 0.05881 

0.1212 + 0.02761 0.1382 + 0.05261 

~.0715 - 0.0083i -0.0039 + 0.0154i 

0.0045 - 0.0006i 0.0085 + 0.0003i 

0.3242 + 0.04431 0.5637 + 0.08711 

0.2026 + 0.0192i 0.5158 + 0.02911 

0.3223 + 0.07531 0.3677 + 0.0867i 

-0.0054 - 0.0006i -0.0003 + 0.00121 

-0.0204 + 0.0026i -0.0393 - 0.0013i 

0.4991 + 0.09451 0.6459 + 0.11571 

0.1472 + 0.01391 0.2295 + 0.0211i 

0.3362 + 0.07651 0.5856 + 0.09051 

0.0556 + 0.0065i 0.0030 - 0.0119i 

0.0003 + O.OOOOi -0.0006 - O.OOOOi 

0. 5387 + 0.0969i 0.6155 + 0.0997i 

NATIONAL ADVISORY 

COMMITTEE FOA AERONAUTICS. 
BAOA TN No. 1195 91 

TABLE XIII.- CALCULATION OF OJ (j (4, h) F'IlliCTIONS 

(EXAMPLE I) 

PARABOLIC BENDING 

Y Wj 1J 

- Wj f j(Y) ---1 C +-- OJ bOB fj{Y) J O-1J 

0 --- 0 00 - 001 0.5850 + 0.03091 00 -CI01 

0.4 0.1326 - 0.0030i 0.16 0.1413 - 0.0188i 0.5850 + 0.0309i 0.0832 - 0.0066i 

0.6 0.3242 + 0.0445i 0.36 -0.0994 + 0.123li 0 .5850 + 0.0309i -0.0620 + 0.0689' 

0.8 0.4991 + 0 .0945i 0.64 -0.2202 + 0.1477i 0.5850 + 0.0309i -0.1334 + 0.0796i 

0.9 0.5387 + 0.0969i 0.81 -0.3349 + 0.1196i 0.5800 + 0.0309i -0.1996 + 0.0596i 

1.0 0 1.0 - 1 0.5850 + 0.0309i -0.5850 - 0.0309 

LINEAR TORSION 

-

0 0 00 -05 i J 0.5850 + 0.03091 00-001 

0.4 0.3899 + 0.0224i 0.4 -0.0252 + 0.0560i 0.5850 + 0.0309i -0.0165 + 0.0320i 

0.6 0.5637 + 0.0871i 0.6 -0.0605 + 0.1452i 0.5850 + 0.0309i -0.0399 + 0 .0831i 

0.8 0.6439 + 0.1157i 0.8 -0.1951 + 0.1446i 0.5850 + 0.0309i -0.1186 + 0 . 0786i 

0.9 0.6155 + 0.0997i 0.9 -0.3161 + 0.1108i 0.5850 + 0.0309i -0.1883 + 0.05511 

1.0 0 1.0 - 1 0.5850 + 0.0309i -0.5850 - 0.0309' 

TABLE XIV.- VALUES' 'OF INTEGRANDS FOB. USE IN' GRAPHICAL INTEGRATION 

(EXAMPLE I) 

Z3 z· • °hZ 3 Z °h °hz 

0.4 0.064 0.0256 0.0832 - 0.0066i 0.0021 - 0.0002i 0. 0053 - 0.0004i 

0.6 0.216 0.1296 -0.0620 + 0.0689i -0.0080 + 0.0103i -0.0134 + 0.0149i 

0.8 0.512 0.4096 -0.1334 + 0.0796i -0.0546 + 0.0326i -0.0683 + 0.0408i 

0.9 0.729 0.6561 -0.1996 + 0 .0596i -0.1310 + 0.0391i -0.1455 - 0.0309i 

1.0 1.0 1. 0 -0.5850 - 0.0309i -0.5850 - 0.0309i -0.5850 + 0.0309i 

a 3 3 a Z Z Z ° ° Z ° Z 0:. 0:. 0:. 

0.4 0.16 0.054 -0.0165 + 0.0320i -o.DOll + 0.0020i -0.0026 + 0.0051i 

0 . 6 0 .36 0.216 -0.0399 + 0 .0831i -0.0086 + 0 .0179i -0.0144 + 0.0299i 

0.8 0 . 64 0.512 -0.0016 + 0 .0786i - 0 .0607 + 0.0402i -0.0759 + 0.0503i 

0;9 0 . 81 0 .729 -0.1883 + G.0551i -0.1373 + 0.0402i -0.1525 + 0.0446i 

1.0 1.0 1.0 -0 .5850 - 0.0309i -0. 5850 - 0.0309i - 0 .5850 - 0.0309i 

NAT IONAL ADvISORY COMMITTl E FOR ~ER "'toU T\C<; 
92 

Term 

k E 0.333 

A 

-

B 

-

D 

-

~ 

k . 0.40 

A 

-

B 

-

D 

-

! 

k = 0.50 

A 

-

~ 

D 

-

! 

NACA TN No . 1195 

TAIJU XV 

AERODYNAMIC CORRE=ONS AND CALCULATED JlETERMINANT TERMS CORRECTED FOR FINITE-SPAN EFFECTS (EXAMPLE I) 

2 r 11 2 ( : 0. ) !1 + 19) where g = ga. • ~ 1 

Structural part or term or term 

(tw<Hlim. ) 

1.3331 ~ _ 0.1920 ( :0. )2 (1 + 18j,] --{).0Q96 - 0 .77991 

0.3154 -3.0592 - 0 .74371 

0.3154 0.1274 + 0.19501 

0.8"789 E - ( ~ ) 2 (1 + 8a.~ 0 .9075 - 0.80171 

1.3331 ~ - 0.1920 ( w: y (1 + ~J 0.0350 - 0 .62501 

0 .3154 - 2.0431 - 0.73441 

0.3154 0 .1163 + 0.15631 

0.8789G -( ~ )2(1 +18a.J 0.6365 - 0 .63751 

1.3331 [ - 0.1920 ( ~ y (1 + 19J 0 .0794 - 0 . 47831 

0.3154 - 1.2414 - 0.67691 

0.3154 0 .1051 + 0.11961 

0.8789 -( ~ ) (1 + 18a.J 0.4227 - 0 . 48621 

- - - - - -. - ------

Detenoinant term corrected 

F1n1~pan 

ror the 

corrections 

efrect or a fin1 te span 

0.0876 + 0.26281 1.4111 - 0 .51711 - 0.25600 

0.9096 - 0 .10981 - 1.8180 - 0.85351 

--{).0210 - 0.05711 0.4218 + 0.13791 

--{). 2127 0.03431 1. 5737 - 0 .76741 - 0.8789 0 

0.0530 + 0 .19651 1. 4211 - 0 . 42851 - 0.25600 

0.5752 - 0.01031 - 1.1525 - 0.74471 

--{).0129 - 0.04271 0.4188 + 0.11361 

--{) .1324 + 0.00811 1.3830 - 0 .62941 - 0.87890 

0.0208 + 0.13121 1.4333 - 0.42851 - 0.25600 

0 . 3014 + 0 .04451 --{).6246 - 0.63241 

--{).OO54 - 0 .02821 0 .4151 + 0.09141 

--{).0692 - 0 .00651 1.2324 - 0. 49271 - 0 .8789 0 

NATIONAL ADVISORY 

COMMITTEE FOR AERONAUTICS 
RACA T}r N". 119 5 '3 3 

TAllIE AlI 

CHARACTERISTICS ALONG SPAN OF TAIL SURFACES USED I N EXAMPLE II 

I I Horizontal i Vertical tail . tail ! I 

'---·-,------j----,--- - I --r I 

b ft b/bo i b f t I b/bo i a I c e f 

r-----+--+-----, ' +-j·-+-I--..f.---+------i, 

o 4 1 I 64 11 _ I -1. ~09 I -0 . 032 0 . 246 I 1 I 

.3 1- i 3.95 .851 i - 2. 413 I .8281 

I I I ! 

.4 3. 44 1 0.850 ' 3.69 .797 I - 2.634 -. 230 .098 I .835 i 

.513 I 3. 33 .718 I - 2.970 I .877 I 

I .770 3.00 .648 I - 3.300 -. 340 .060 I 1 I .655 2 .18 .411 ' -4. 5B1 - -.404 .Oj1 I 1 I 

o 0 0 1 I 

I I .6 

I .8 

/1.$ 

3.oB 

2, 62 

o 

TlLBLE XVII 

GENERAL CHARACTERISTICS OF TAIL SURFACES USED I N E.XAMPLE II 

~rizontal tai l Vertical tail 

s 2.525 1 .800 

b o 4 ft 4.64 ft ,.. 

br 3.33 ft 3.33 ft 

ab -10 .08 ft 

n - -.40(lag) 
94 

:'3 terminant 

term 

A 

-

B 

-

C 

-

D 

-

E 

-

F 

G 

H 

I 

A 

B 

C 

-

D 

E 

F 

G 

-

H 

I 

-

A 

B 

-

C 

-

D 

E 

F 

-

G 

H 

I ,. 

NACA TN No. 1195 

TABLE XVIII.- EVAllJATIOIY OF INTIDRAlS FOR EXAMPLE II 

2 

In = ( :0.) (1 + 19) where g c ga. '" Bt l 

Two-dimens1onal Structural part aerodynamic part 

(ft) (ft) 

(kOH '" 0.686 koV = 0.795) 

67.27 Cl - 1.988 nJ 2.1228 - 14.66481 

100.1 -9 .0606 - 24.93341 

1.731 -11.2670 - 3.69931 

100 .1 4.4524 - 17.83191 

1891 (1 - n J -27.7895 - 167.0931 

23.862 -67.1873 - 38.63701 

1.737 0.3202 - 0.17281 

23.96 0.2137 - 4.7161 

5.99~ -0.1613 - 3.2510t 

(koH = 0.600 kov = 0.696) 

67.27 [1 - 1.988 n J 1.5917 - 17.39401 

100.1 - 13.9834 - 28.86861 

1.731 -14.8607 - 3.57031 

100.1 3.7185 - 20 .12901 

1891 'tl - n J -54.9543 - 184. 6361 

23 .862 -88 .6384 - 39.56451 

1.137 0.3108 - 0.20441 

23.916 2.0841 .:.. 5.31281 

5.992 -0.4069 ~ 3.50801 

(koH = 0.436 kov '" 0.506) 

67 .27 [1 - 1.988 n J -0.6677 - 25.85681 

100.1 -34.9401 - 39.19341 

1.731 - 30 .2362 - 2.9~31 

100,1 0. 7482 - 29 .57011 

1891 fl - n J -166 .9951 - 255 .16971 

23. 862 -177. 2376 - 43 .31121 . , 

1.137 0.2795 - 0. 28971 

23.916 0.7526 - 7.241h 

5,99-2 ~i,1038 - 4.72781 

'. 

Determimmt term 

two-dimensional 

aerod,yll8l!l1c part 

(ft) 

69.393 - 14.66481 - 133.73 n 

91.039 - 24.93341 

-9;5360 - 3.69931 

104.552 - 17.83191 

1863.2 - 167.0931 - 1891 n 

-93.3253 - 38.63701 

2,0572 - 0.17281 

24.1297 - 4.7161 

5.8307 - .3.25101 

68.862 - 17.39401 - 133 .73 n 

86.117 - 28.86861 

-13.1297 - 3.57031 

103.8185 - 20.12901 

1836 - 184.631 - 1891 n 

-64.7764 - 39.56451 

2.0478 - 0.20441 

26 .0000 - 5.31281 

5.5851 - 3.50801 

., 

66.6023 - 25.8568 - 133.73 n 

65.1599 - 39. 19341 

-28.5052 - 2.94831 

100.8482 - 29.57011 

1124 - 255.16971 - 1891 n 

-153 .3756 - 43.31121 

2.0165 - 0.28911 

.6686 - 24 i 

4.8882 - 4.72781 

NATIONAL ADVISORY 

COMMITTEE FOR AERONAUTICS. 

... 
RACA TN No. 1195 95 

TABLE XIX.- SOLUTION OF SET OF EQUATIONS ~Knh Anrn = Ohm (2) BY 

THE CROUT METHOD (EXAMPLE II) 

(2) 

G1ven Matrix Oh Check 

0 .4704- -0.4459 0.2507 0.0591 0.,348 

-0.119 1 0.15661 -0.09)21 0·55l81 ·0. 9581 

o 6~" !..0.2670 -0.2849 0.0768 0.1808 

- 0 :1 1 0.09781 0.11111 0.77091 0.79541 

0.6,t7 0.2k~9 -0.2657 0.09K6 0.7165 

-0.17 01 -0.0 21 0.11271 0 . 92 51 0.77301 

Supplementary Matrix 

1.9953 -0.4459 0.2507 0.0591 0.,348 

0·50591 0.15661 -0.09321 0·5518i 0. 9581 

2.45991 -0.6330 -0.0260 -0. 306

0.96231 0.2508i -0.00061 0.112 4 1 

0.9673 0.0649 0.9843 

0.34161 0 . 18091 -0.14391 

Aux1liary Matr1x 

0. 470

4 -0.9689 0·5474 -0 . 1612 0.41~2 - 0 .119 1 0.08691 -0.05911 1.13091 1.15 61 

6~~ 0.3525 -1.798~ -0.0'634 -0.86211 

-0 .1 1 -0.13791 0.007 i -0 . 02651 -0.01851 

63~7 0.8476 0.91~2 0.0010 1.0013 

-0.17bO -0.31391 -0. 32 ~61 0.19721 0.19701 

k2h -0.2031 

1.34621 0.7~70 

1.3 601 

k4n -0.0601 0.9403 

0.32821 0.32801 

NATIONAL ADVISORY 

COMMITTEE FOR AERONAUTICS. k6h 0.0010 1.0013 

0.19721 0.19701 
96 NAOA TN No. 1195 

TABLE XX.- VALUES OF 0 n FOR EXAMPLE II 

y sin n~ sin n~ 

bOB 

n n ~ n 

oo4{ 2 0.3667 -0.0745+0.49371 

4 -0.2493 0.0150-0.0818i 

6 +0.1132 +0.0001+0.022~1 

0::: h -0.0594+0.43421 

Oo6{ 

2 0.4800 .!0.0975+0. 64621 

4 -0.1344 0.0081-0.04411 

6 -0.1098 -0.0001-0.02171 

0'" h -0.0895+0.58041 

Oos{ 

2 0.4800 -0.0975+0.6Lt621 

4 0.1344 -0.0081+0.04411 

6 -0. 1098 -0.0001-0.0217i 

n= h -0.1057+0.6686i 

TABLE XXI.- AERODYNAMIC CORRECTIONS AND CALCULATED DETERMINANT TERM3 

CORRECTED FOR FINlTE-BPAN EFFECTS (EXAMPLE II) 

Term 

A 

-

B 

C 

-

D 

-

E 

F 

G 

H 

I 

koH ;:: 0.436 

Det. term 'With twodim. aero. :part. 

66.6023-25.85681 

-133·73 n 

65·1599-39.19341 

-28·5052-2.9483i 

100.8482-29·5701i 

1724-2~5~16971 -1 91 n 

-153.3756-43.31121 

2.0165-0.28971 

24.6686-7. 24111 

4. 8882-)-\-.72781 

kov = 0.506 

Finite-s:pan Oet. term corrected for 

corrections finite-s:pan effects 

5.0885+6.98841 71.6908-18.86841 

-133.73 n 

11.5520+2.86801 76.7119-36.3254i 

6.5593-3. 23601 -21.9459- 6.1843 i 

6.3861+9.04921 107.2343-20 .52091 

65·4194+16.1656i 1789.4-23§.004i 

-1 91 n 

34.6350-17.11131 -118.7415-60.42251 

0.°572+0.09461 2.0737- 0.19511 

-4.7027+1.7546i 19.9659-5.4865i 

0.3885-0.20121 5.2767-4.92901 

NATIONAL ADVISORY 

COMMITTEE FOR AERONAUTICS. 

~~ - ----
NACA TN No. 1195 97 

TABLE XXI.- Concluded 

koH = 0.600 

A 68.8617-17.39401 2.8478+3.98221 71.7095-13.41181 

-133.73 n -133·73 n 

B 86.117-28.86861 5.8026+2.49831 91.9196-26 .37°31 

c 

D 

E 

F 

-13.1297-3.57031 2.8707-1.10771 -10.2590-4.67801 

1°3.819-20.129°1 3.0639+5.64231 106.8829-14.48671 

1836-184.6311 29.5923+ 15.01001 1865. 6-169. 6171 

-1891 0 -1891 0 

-64.7764-39.~51 14.5931-5. 281 51 -50.1833-44.84601 

G 2.0478-0.20441 0.0264+0.05611 2.°742-0.14831 

H 26.0001-5.31281 -1.9233+0.32501 24.0768-4.98781 

I 5.5851-3.50801 0.1624-0.04181 5.7475-3.54981 

A 

B 

C 

D 

E 

F 

kOH = 0.686 

69.3928-14. 66481 1.9813+3.06731 

-133·73 0 

91.0}94-24.93341 4.0837+2.19551 

-9.5360-3.69931 1.9336-0.59261 

104.5524-17.83191 2.2890+4.26181 

1863. 2 -167.0931 20.5217+13.12~ 

-1891 0 

-43.3253-38.63701 9.8602-2.5°801 

71.3741-11.59751 

-133·73 n 

95.1231-22.73791 

-7.6024-4.29191 

106.8414-13.57011 

1883.7-153.9731 

-1891 n 

33.4651-41.145°1 

G 2.°572-0.17281 

H 26.053-4.7161 

I 5.8307-3.25101 

0.0182+0.04161 2.°754-0.13121 

-1.2893+0.09351 24.7637-4.62251 

0.1060-0.01721 5.9367-3.26821 

NATIONAL ADVISORY 

COMMITTEE FOR AERONAUTICS. 

-

WING MIDCHORD LIN§.... 1\ --- I ----, II 

( i ) 

I .. S bo • I .. s bo p I 

( a) x 

Figure 1.- (a), Plan form of wing, showing location of elastic axis and midch

(b), Wing section with aileron and tab, showing main in 

local semichord. 

o 

-I 

(b) 

NATIONAL AD

COMMITTEE FOR A

. , I' 
Fig. 2a 

8 

7 

6 

"U) 

o 5 

..lI: 

Y:i 

-• en 

4 

:3 

2 

o 

o 

n:a9 

n-8 

n=6 

n=5 

n=2 

2 

cos. -I.n: 

:3 4 

kos 

NACA TN No. 1195 

5 6 

NATIONAL ADVISORY 

COMMITTEE FOR AERONAUTICS 

(a) Cos ¢ ; 1; n = 1, 2, 3, 4, 5, 6, 7, 8, 9. 

Figure Z (a to h).- Function Sn(¢, kos) against kos for various 

values of n and of cos ¢. (Dash lines are for 

imaginary parts; solid lines, for real parts.) 
NACA TN No. 1195 Fig. 2b 

1.6 

1.5 cos ~ =0 .8 

1.4 

1.3 

L2 

1.1 n=3 

1.0 

0 .9 

0.8 

0.7 

n=1 

0,6 

0.5 

0.4 n=7 

0 .3 

0.2 n-5 

<i)' 0.1 0 n=9 -"C 

-&. 0 -c CJ) n=5 - 0.1 

-02 n=1 

- 0.3 

-0.4 

-0 .5 n= 3 

-0.6 

-0.7 n=9 

-0.8 

-0.9 

-1.0 

-1.1 

- 1.2 

-1.3 n=7 

-1:4 

-1.5 NATIONAL ADVISORY 

-1.6 CO ""'THE fOR AERONAUTICS. 

- 1.7

0 I 2 3 Kos 4 5 6 

(b) C08 ¢ = 0.8; n = 1, 3, 5, 7, 9. 

Figure 2.- Cont inued. 
Fig. 2c 

1. 2 

.I 

I. 0 

O. 9 

O. 8 

O. 7 

0. 6 

O. 5 

O. 4 

O. 

0 .2 

-en 0 0 .1 

.x: 

-& 0 

-• (J) -0 .1 

-0 .Co 

-0. ~ 

-0. 

.It 

'T 

-0 

-0. \"J 

-0. 7 

-0. B 

-0 .'" 

-I. n 

0 

n=8 

n=2 

n= 

n=4 

n= 

n=2 

0=4 

n=6 

2 

NACA TN No. 1195 

cos _ -0.6 

NATIONAL ADVISORY 

COMMITTEE FOR AERONAUTICS. 

3 4 5 6 

kos 

(c) Cos ~ = 0.6; n = 2, 4, 6, 8. 

Figure 2.- Continued. 

, 

J 
. . .-- ~-- -~------~ --------

NACA TN No. 1195 

n=9 

nlll :~- , 

, -, 

..j.. 

+--

0.3 rt- i-+--q::: , -: ~-..,.-l. "1- 02 H- + f.+/+- , , ; , 

. i .. Ljjc:l-:· !:~- n = 7 

A 1 ~rt~·~ ,,: +++ 'Cii' . IT. . +-, + -rl''''+- . o 'Ll.. _~ 1 

~ a .' ~. !.. ' n -7 

~ 

.. ~ ... 

!''''' T·:lf.& I ~L~' I ' , ._ I 

l!~1 I en -0.1 ' ; . .--r.' n - 3 

-0.2 ~:; .. ~ \i_~ .+-++ n =9 

-0.4 . , 

, " -+--

-0.5 

-0.6 ~ L , . 

-0.7 

-0.8 

-0.9 n=5 

-1.0 

+, 

-1.1 .. :::r -

-1.2 ~ 

-1.30 2 

F ig. 2d 

cos ~ -0.6 

3 

koS 

NATIONAL ADVISORY 

COMMITTEE fOR AERONAUTICS . 

4 5 6 

(d) C08 ~ = 0.6; n ~ 1, 3, 5, 7, 9. 

Figure S.- Continued. 
Fig. 2e 

1.0 

0.9 

0.8 

NACA TN No. 1195 

cos ~·0.4 

.... 0.7... s n=2 

0.6 

0.5 

0.4 

0.3 

0.2 

'Ci) 0.1 0 

~ 

-& 0 ---c 

(/). -0.1 

-0.2 

-0.3 

-0.4 

-0.5 

-0.6 

-0.7 

-O.S 

-0.9 

-1.0 

-1.1 

~ 

0 

n=6 

n=4 

n=8 

n=E 

n=6 

n=2 

n=4 

2 3 

kos 

NATIONAL ADVISORY 

COMMITTEE FOA AERONAUTICS. 

4 5 6 

(e) Cos ¢ ~ 0.4; n = 2, 4, 6, 8. 

Figure 2.- Continued. 

- --.-------
NACA TN No. 1195 Fig. 2f 

1.1 

1.0 

0.9 

0.8 

0.7 

0.6 

0.5 

0.4 

0.3 

0.2 

~ 0 0.1 

.:It:. 

'& 0 -II: 

en -0.1 

-0.2 

-0.3 

-0.4 

-0.5 

-0.6 

-0.7 

-0.8 

-0.9 

-\.O 

-1.1

0 

n=7 

n=1 

n=9 n=5 

n=3 

n=7 :'1' 

n=1 

n=5 

n=9 

n=3 

2 

cos ¢",O.4 

3 

kos 

NATIONAL ADVISORY 

COMMITTEE FOR AERONAUTICS. 

4 5 6 

(f) Cos ~ = 0.4, n = 1, 3, 5, 7, 9. 

Figure 2.- Continued. 
Fig. 2g 

IJ 

1.0 

0.9 

0.8 

0.7 

0.6 

0.5 

0.4 

0.3 

-en 

0.2 

0 

..x: 0.1 

-s. 

-.... 0 '" CJ) 

-0./ 

-0.2 

-0.3 

-0.4 

-0.5 

-0.6 

-0.7 

-0. 8 

-0.9 

-1.0 

-/./ 

0 

n=1 

n=5 

n=3 

n= 

n=9 

n=7 

n=1 

n=3 

2 

cos _ =0.2 

n=9 

n=7 

3 

ko 5 

NACA TN No. 1195 

NATIONAL ADVISORY · 

COMMITTEE FOR AERONAUTICS. 

4 5 6 

(g) Cos ¢ = 0.2; n = 1, 3, 5, 7, 9. 

Figure 2.- Continued. 

) 
NACA TN No. 1195 Fig. 2h 

1.0 

0.9 

0.8 

0.7 

0.6 

0.5 

0.4 

0.3 

0.2 

~ 0.1 

...x: 

-e. 0 -c: 

en -0.1 

-02 

-0.3 

-0.4 

-0.5 

-0.6 

-0.7 

-0.8 

-0.9 

-to 

-1.1 

o 

n=9 

ni:7 

n=9 

n·, 

2 

I I111 LO cos _. 

3 

kos 

NATIONAL ADVISORY 

COMMITTEE fOR AERONAUTICS 

4 5 6 

(h) Cos ¢ = 0; n = 1, 3, 5, 7, 9. 

Figure 2.- Concluded. 
Fig. 3 

0.4 

0 .2 

o r-----

.tR=3 

k 0 =0167 

-- ------f----

I 

r-.... 

-~ 

R----:-

NACA TN No. 1195 

.tR=6 

ko=0.167 

[ 0.2 

2 pU2 bo 0 

0.4 

--~ 1------- - f- -- \ 

o 0 .2 0.4 0.6 0 .8 1.0 

R_ o 0.2 0.4 0.6 0.8 LO 

0 .8 

0 .6 

0.4 

0 .2 

o 

f----r----~---~---

kol=O.313--

---

I 

r---.., 

\ 

R 

---

0 .8 

0.6 

[ 0.4 

2pU2bo 0 .2 

o 

-0.2 

--1--- ~--- -+- K- I , I 

ko=0.333 

\ 

R - -0.2 0 0.2 0.4 0.6 0 .8 1.0 o 0.2 0.4 0.6 0.8 1.0 

1.2 

1.0 

0 .8 

0.6 

0.4 

0 .2 

o 

-0.2-

-0.4 fo 

. 

t---r-.. I 

ko=0.667 --........ '" \ \ 

[ 

2p U2bo 

J RL I- -- :.:::::-' ,--

0 .2 0.4 0.6 0 .8 

Y 

1.0 

1.2 

1.0 

0 .8 

0.6 

0.4 

0.2 

0 

-0.2 

-0.4 

o 0.2 

5 bo NATIONAL ADVISORY 

COMMITTEE FOR AERONAUTICS. 

- t---~ ko=0.667 ~ \ 

I RJ 1-. ---'::>"'--

0.4 0.6 0.8 

Y 

5 bo 

1.0 

Figure 3.- Lift distributions for uniform translati on of rectangul ar 

winss of aspect ratios 3 and 6 for ko ; 0. 167, 0. 333 , 0.667. 
NACA TN No. 1195 

~=3 

-----~-- I -- I ----- R"=-=- ko=O 

-r--- L 

~ 2 P U2

bo 

\ 

o 0.2 0.4 0.6 0 .8 1.0 

I I 

ko=Q.167 f----- - ---- ---- ---- ----

- R [ 

-----~ 2p U2

bo 

1 \ 

o 0 .2 0.4 0.6 0 .8 1.0 

ko=0.333 ----- ---- - ---

I----. R [ 

t---+--f----+--+ ..-l 2 P U2 bo 

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

o 0.2 0.4 0.6 0 .8 1.0 

ko=0.667 

R 

f-

-- --= 

-~-

1 \ 

o 0.2 0.4 0.6 0.8 1.0 

Y 

sbo 

3 

2 

1----- ----

~=6 

--__ 1 ____ 

ko=O - 1------

Fig. 4 

~---- R-

~ \ 

o 

o 0.2 0 .4 0 .6 0.8 1.0 

3 

2 

o 

-----

1-----

I I 

ko=0.167 

--- -- ---- - 1-- ---

---- - --

--- -

~ 1 \ 

o 0 .2 0.4 0.6 0 .8 1.0 

. 

3 ko=0.333 R -- 2 - ~ I '\ 

o 

F ~ 

o 0.2 0.4 0.6 0 .8 1.0 

-

3 ko=0.667 

R 

-

~ o . -

o 0.2 0.4 0.6 0:8 1.0 

Y 

sbo 

Figure 4.- Lift distributions for uniform pitching of rectangular wings of aspect ratios 3 and 6 for ko = 

0, 0.167, 0.333, 0.667. 

NATIONAL ADVISORY 

COMMITTEE FOR AERONAUTICS. 
Fig. 5 

ko·O /' 

..- .-' 

..---- ..- R [ 

.-' .-' 2pUab .-' o .--.-' 

r- .-' ~ - R\ ---.-' 

o 0.2 0.4 0.6 0.8 1.0 

ko=O.l67 

---- [ 

--- ..- 2PUa _.-' R --- ~ :.---

-

-

-_ ...... ~ 

o 0.2 0.4 0.6 0.8 1.0 

ko=0.33:3 

...:- [ 

--- 2pUa 

----- R --- ~ ;..::::...-- , ----

.-;; 

= "". ~ 

o 0.2 0.4 0.6 0.8 1.0 

ko-0.667 

R _-- [ 

2 P U2bo ----- -- - ~ -

~ ~ ~ =-- , ~ 

o 1.0 

3 

2 

o 

3 

2 

o 

:3 

2 

o 

:3 

2 

o 

NACA TN No. 1195 

ko=O .-' .-' ---

.... ---- R - ........ - / ::---- ~ 

~ 

-- ~ R \ 

o 0.2 0.4 0.6 0.8 1.0 

ko=0.167 

---- -- R ---- ~~ ~ 

~ ~ I \ 

.-

o 0.2 0.4 0.6 0.8 1.0 

ko=0.33:3 

~ -ri- -- ,~ 

L------~ 

~ 

I \ .- ~ 

o 0 .2 0.4 0.6 0.8 1.0 

ko=0.667 

~ - -

,..-:-- -::::..c-

~ ~ r:o----

~ ---- I~ 

0 .2 0.4 0.6 0.8 o 1.0 

-L NATIONAL ADVISORY 

COMMITTEE FOR AERON AUTICS. sbo 

0.2 0.4 0.6 0.8 

- ysbo 

Figure 5.- Lift distributions for linear symmetrical torsion 

of rectangular wings of aspect ratios 3 and 6 for 

ko = 0, 0.167, 0.333, 0.667. 

- - - - - -
NACA TN No. 1195 

ko·O.l67 

C -.-- 2pU2~ -"---1 

- -- -- - r-- B r-~ 

D C .2 0.4 0.6 0.8 1.0 

ko=0.3331 

./ .,..." ..... ' ././ I C ./ 

2 P U2bo " ..... ~ ~ 

;?' ~ \ .......... R >-- ..--

o 0 .2 0 .4 0.6 0.8 1.0 

ko=0.667 / / / 

/ / / 1- / 

/ / 

--- ....... /y \ [ 

/' 2pU2 b /J 

...--- /' 

/ 

// 

----~- / 

---=:::.: -=:---~ 

o 0 .2 0.4 0.6 

y 

sbo 

'-

C .8 1.0 

o 

Fig. 6 

0.8 r---.------;-k-o--, .0 ..... 16=-----r------, 

0.6 

0.4 t----j---+---/----+--=-I_ 

--- 1-- 0.2 I------+--+-=,.::.:r==~~___j 

0~~-==--+==4;,;;;.=,;;~~ 

o 0.2 0.4 0.6 0.8 1.0 

0 .8 ko=0.333 

0.6 f----+---j---+---b.L---j 

...-" ..... 

0.4 f----+---j-..... ./"7"""'f'=-.....::::::F--=~ 

..... 

0.2 t----l---:;;;;,;=-+---j---+----\j 

O~~=-~=_~--~R~~ 

1.2 

1.0 

0 .8 

0.6 

0.4 

0.2 

o 

-0.2 

-0.4 

o 0.2 0.4 0.6 0 .8 1.0 

NATIONAL ADVISORY 

COMMITTEE FOR AERONAUTICS 

ko=0.667 / / / 

/ 

/ /' I 

// V \ 

// \ 

V 

w 

~ 

---:::::-

~ / 

"-:::.: ~-U 

o 0.2 0.4 0.6 0.8 1.0 

Y 

sbo 

Figure 6.- Lift distri butions for linear symmetrical bendi ng 

of rectangular wings of aspect ratios 3 and 6 fo-" 

ko ; 0.167, 0.333, 0.667. 
Fig. 7 

ko·O.l67 [ 

;;-- .... /., 

~ -- 1 - .R---~ 

0 0.2 0.4 0.6 0.8 I. o 

/ 

ko=0.333 / / 

" 

/ [ 

/,,/ I 2pU2~ / 

-;::::::. -- ..,-?- R '\ 

7' 

o 0.2 0.4 0.6 0.8 1.0 

/ 

/ 

ko=0.667 / 

/ 

/ 

/ 

/~ ~ [ 

V v I; 2pU2 b o 

-~/ 

- --==, 

~" R / 

-"""":::',",,-, r.-:- 7 '-'-, 

. o 0.2 0.4 0.6 0.8 1.0 

0.8 

0.6 

0.4 

0.2 

o 

0.8 

0.6 

0.4 

0.2 

o 

1.2 

1.0 

0.8 

0.6 

0.4 

0.2 

o 

-0.2 

-0.4 

NACA TN No. 1195 

ko=0.167 

... // " ... ,- ~ ::::.--

-- -~~ 

o 0.2 0.4 0.6 0.8 1.0 

/ 

ko=0.333 / / 

//1 

~ ~ ~ ~ R 

o 0.2 0.4 0.6 0 .8 1.0 

I 

/ 

/ 

ko=0.667 / 

/ 

I 

1- / 

/ 

/ ~ 

V \ / 

k:::::;::: ~/ 

---

---..... -

~ ~ '-.. 

o 0.2 0.4 0.6 0.8 1.0 

y 

s bo 

NATIONAL ADVISORY 

COMMITTEE FOR AERONAUTICS. 

y 

5 bo 

Figure 7.- Lift distributions for parabolic symmetrical b'ending of rectangular wings of aspect ratios J and 6 

for ko = 0.167, 0.333, 0.667. 
NACA TN No. 1195 

~=3 

[ 

3 .0 

2.5 

2.0 

ko=O 

'" " 

/' 

"," 

",/ 

~/ R 

2pU2bo 1.5 

, 1.0 

" '" ~.- R 

F ig. 8 

~=3 

'" '" /' 

2 P S'b. ~~ II---=~I =_:::;It;;=0~16;;;~:~--;:;t;; o 0.2 0.4 0.6 0.8 1.0 

0 .5 

o 

" ----- "," 

~ ./' 

o 0.2 0.4 0.6 0.8 

~ \. 

_ 101~333 ~ ~ I02P~'b.O~: ;¢--t- ~r= 3.0 

2.5 

2 .0 

[ 

2pU2bo 1.5 

1.0 

0 .5 

o 

2.0 

1.5 

ko=0.167 

./ 

, / 

, 

,...' " ,... 

./ 

,... 

R ,... 

'" // 

~ >- '\ ./ 

f---. ./ ---- ~ V I -...... '- ------ o 0.2 0.4 0.6 0 .8 1.0 

./ 

ko=0.333 ./ ././ 

./ 

./ 

"..,,~ R 

" /" / 7 

,..-/ --' I \ 

# ~ ~ ..--- --- -~ 

1.5 

1.0 

C 2p U2 bo 0.5 

o 

o 0.2 0.4 0.6 0.8 1.0 

ko=0.667 1 ---- ---- --::... ....-- ~ 

~ ;;- \ 

---- j 

--=., -=.:: -- R d 

-0.5 0 0 .2 0.4 0.6 0.8 1.0 

2.0 

1.5 

1.0 

NATIONAL ADVISORY 

COMMITTEE FOR AERONAUTICS . 

I , / 

ko=I.333 ,..-

././ ::------- ~ Vi ,./ 

7' 

0.2 0.4 0.6 0.8 1.0 C 0.5 

2 P U2 bo 0 

.7 ~ 

V 

2.0 

1.5 

2P~'1bo /.0 

0.5 

o ~ 

ko=0.667 

/;/R / 

.//. ~ ~ 

././ 

./ ,..-

.- ' 

~ \ 

a 0 .2 0.4 0.6 0.8 1.0 

~ 

5 bo 

-0.5 ~ 

-1.0 

-1.5 

-2.0 

-2.50 

~ 

~ ~ I 

J '...: .... .... .... .... 

0.2 0.4 0.6 0.8 1.0 

Y 

s bo 

Figure 8.~ Lift distributions for linear antisymmetrical torsion of a 

rectangular wing of aspect ratio 3 with ko = 0, 0.167, 0.333, 

0.667 lleft col.umnJ, and for rigid rolling 'right column) of a rectangular 

wing of aspect ratio 3 with ko = 0.167, 0.333. 0.667, 1.333,' 
Fig. 9 NACA TN No. 1195 

J:R= 3 ~=6 

ko=O - - ./ 

./ 

R " ./ 

.// , 

, 1.5 ko=O -- ./ 

./ 

R , ./ , ./ \ 

'" \ 

./ \ 

'" 

[ /.0 // I , / VLV I , /"'~ ~ 

R , 

"'" , 

, ~ 

/ 

/ / 

~ 

o 0 .2 0.4 0.6 0 .8 1.0 

ko=0.212 

------ " /// n , 

\ ./ 

.--- -.... \ 

<-/ ~ \ 

.".-

/ '" '" '" ~ -~ I 

2 P U2

bo 0 .5 

o 

1. 5 

[ 1.0 

2p U

2

bo 0 .5 

o 

o 0.2 0.4 0.6 0.8 1.0 

ko=0.424 

---- -- R // 

/y -- -.. 

~ ~ 

£= 1::::"- -- - 1-- - _ _ 

I 

, , 

, ~ \ 

\ 

~ 

1.5 

[ 1.0 

2p U2 bo 

0 .5 

o 

R \ 

\ 

~--";?' 

I \ / \ / 

/ 

o 0 .2 0.4 0 .6 0 .8 1.0 

ko=0.212 

./-----R -, , 

/ ::-- ~\ ~ ~ 

'" / 

- - -T 

o 0 .2 0.4 0 .6 0 .8 1.0 

ko=0.424 

---R ./ 

./ ..... 

./ 

-;::::-- ~\J 

~ ~ -:::""--;;..--~ ;...... . 

----t:-----.. 

I 

o 0.2 0.4 0.6 0 .8 1.0 0 0 .2 0.4 O. 6 0 .8 1.0 

ko=0.847 1.5 

.... , [ 1.0 ,....//'- R " ~ ~- -.... "",, \ 2p U2 bo 

0 .5 

~ ~ .... ':l 

I ~ .......... \ 

" 

o 0.2 0 .4 0.6 0 .8 1.0 

-L 

sbo 

o 

ko=0.B47 _ r- -

--- I"-, 

~ ~ ~, 

/ ~ -, 

N 

I 

o 0 .2 0.4 0.6 0 .8 1.0 

--L 

s bo 

Figure 9.- Lift distributions for linear symmetrical torsion 

of elliptical wings of aspect ratios 3 and 6 with 

ko ; 0, 0.212, 0.424, 0.847. 

--

NATIONAL ADVISORY 

COMMITTEE fOR AERONAUTICS. 
NACA TN No. 1195 

f-ko·q.212 

-- ---r -...... [ 

--- ~ 2PU:i - ~. _I! 

o 0.2 0.4 0.6 0.8 1.0 

I-ko=0.424 - --- - .......... - I "-

..... ;::::. 

, [ ,.... r--... \ 

-/ 

.- ~ 2pU:ib -~ 1---_ o 

o 0.2 0.4 0.6 0.8 1.0 

ko=0.S47 ---- " .......... I \ ..-

/ t::-- --~\ 

A V \\ C 

,......// , 2p U:ibo / 

~ R / ,..,...,... ,... 

~ " ~- ~ ..... ~ 

o 0.8 1.0 

0.4 

0.2 

o 

0.4 

0.2 

o 

O.S 

0.6 

0.4 

0.2 

o 

-0.2 

Fig. 10 

-ko=0.212 I -------"-1---

--::::::-~ - - _R ~ 

o 0.2 0.4 0.6 0.8 1.0 

I-ko=0.424 ....- ----I - "-, 

/ ~ ~\ 

V R ~ 

- - - >-

o 0.2 0.4 0.6 0.8 1.0 

ko=0.S47 ..... '" /---- 1 ...... , 

........ " 

/ I;:::=--

~\ 

~ V ~ 

V/ 

/ 

~ 

R "/ """,-t..... ,/" 

---- _// 

-0.4 0 0.2 0.4 0.6 0.8 1.0 

Y 0.2 0.4 0.6 

Y 

S1); NATIONAL ADVISORY 

COMMITTEE FOR AERONAUTICS. 

sbo 

Figure 10.- Lift distributions for linear symmetrical bending 

of elliptical wings of aspect ratios 3 and 6 with 

ko = 0.212, 0.424, 0.847. 
Fig. 11 

.Afl= 3 

ko=0.212 [ 

-- I .... , 2pUzb - --- ~ o 

IR 

o 0 .2 0.4 0.6 0 .8 1.0 

ko=0.424 -- -...... ..... - I '\ L - ?-r- ~ 2 P U2

bo - R - - - I 

o 0 .2 0.4 0.6 0 .8 1.0 

o 

ko=0.847 

0 .2 0 .4 0 .6 0 .8 1.0 

Y 

sbo 

NACA TN No. 1195 

.Afl= 6 

0.4 I---+----I---+---+----l 

ko=0.212 

0 .2 I---+----I---+--=="'-+- --l 

o 

o 0 .2 0.4 0.6 0.8 1.0 

0.4 1---f--+---+---I---4 

0 .2 I---t---+--:: "F---t-~\-j 

o fiAii~~r-===f=~~~ 

o 0 .2 0.4 0.6 0 .8 1.0 

NATIONAL ADVISORY 

COMMITTEE FOR AERONAUTICS 

0.6 I---+---+---I--..,..-+-r-"'c-----J 

0.4 t-------i--t- ~-+~M 

0 .2 I---t---=:::'r---+---+---\\I 

o 

-0.2 l,----,-,l. U.:C ..,-- U.4 0~=::; .6 0;t::: .8 ~ 1.0 

-ysbo 

Fi gure 11.- Lift distri but ions f or pa raboli c symmet rical bending of el l i pti cal wi ngs of a sp ect rat ios 3 and 6 

with ko = 0.212, 0.424, 0.847. 
NACA TN No. 1195 

2.0 

/.5 

ko=O 

... / / 

..-R=3 

/ ... / ... 

----, R , 

\ 

\ 

\ 

Fig. 12 

~=3 

1.0 1 ko=0.212 I 

I ---- ----- -- - ---- ::::::..... 

\ R \ R / / \ 

-- r-.. I o 0 .2 0.4 0.6 0.8 1.0 

1.5 

1.0 

c 

2p U2 bo 0.5 

o 

// v--- ~ "\ 

o 0.2 0.4 0.6 0.8 1.0 1.0 

k

1

o=0.2112 ----- "-

-- " ... - IR. \ ,/ 

... ~\ / 

---->- / 

... V \1 ~ I 

- 1.0 

- -- ---- o 0 .2 0.4 0.6 0 .8 1.0 0.5 

[ 

2 pU2 bo 0 

. 1 

ko=0.424 

,- ---- --..... 

~ ,....... 

~ ~ 

IR 

o 0.2 0.4 0.6 0.8 1.0 

. 1 I ko=0.847 ---- -.... --:.:::: " 

~ ~ ~ = 

-----1--- = ---.::::.. ~ 1.5 

1.0 

k~=0.424 R 

-0.5 0 0 .2 0.4 0.6 0.8 1.0 

----r-' 

---- R', ...... --1---- , 

-- L----- ~\ 

~ 

... V I ~ !=-==- ---- ---- r=------

o 0 .2 0.4 0.6 0.8 1.0 

1.5 

1.0 

0 .5 

[ 

I.5.---ko,---=-0-.a-4,---7-----r------r---, 2 P U2 bo 0 

-0.5 

-/.O 

- - -----b-----ol, [ 

2 P U2bo

O.

5 

o ~-+--~-~---r--~ -1.5 

J -- -- ko=1.695 100-" 6.? I ~, 

~ \ 

L \ 

,~ /) '\~ ~ // " 

'-'" ~ ~ 

o 0.2 0.4 0.6 0 .8 1.0 o 0.2 0.4 0.6 0.8 1.0 

-L -'1-

s bo sbo 

Figure 12.- Lift distributions for linear antisymmetrical torsion 

of an elliptical wing of aspect ratio 3 with ko ~ 0, 

0.212 , 0.424, 0.847 (left column) and for rigid rolling of an 

elliptical wing of aspect ratio 3 with ko ; 0.212, 0.424, 0.847, 

1.695 (right column). 

NATIONAL ADVISORY 

COMMITTEE FOR AERONAUTICS. 
Fig. 13 NACA TN No. 1195 

c 

o 0 .2 0.4 0.6 0 .8 1.0 

o 0.2 0.4 0.6 0 .8 1.0 

2 

2 

.5 

.0 

[ I. 5 

2pUz bo I. 0 

r::. 

---- ----

ko=0.424 O.4I---I-----+---'------L----I 

ko=0.424 

O. f---f-----+--+---+-~ 

------ -_ R ........... ' .... 

......... 

~\ 

M,(e) 0 .2 ~~l=---l---j.----+--~ 

2puz boz 

O. -_. - --- .:::::::::- I 

0 r-::::--: -- r--~ 

0.1 r===j=====f~~~ 

o ~ 

2.5 

2.0 

C 1.5 

2 P UZbo 1.0 

0.5 

o 

-- - () 0.2 0.4 0.6 0.8 I. 

I 

ko·0.847 

1----- ---- ---- R ~ --f":.-_ 

------ ~ ~-........ 1', 

",",', 

.... \~ 

o 0 0.2 0.4 0.6 0.8 1.0 

NATIONAL ADVISORY 

COMMITTEE fOR AERONAUTICS. 

0 .4 - -

0.3 

M,(e) 0.2 ~--If...----+-~o-----+-- 2pU2 boz 

O.I I---------+---+----+-~-.l-~ 

o b~---1.-..-=-=f-:::::::=::::~ 

o 0 .2 0.4 0.6 0 .8 1.0 o 0 .2 0.4 0.6 0 .8 1.0 

-y- ....L 

sbo sbo 

Figure 13.- Lift distributions and hinge-moment distributions 

for full-span aileron with uniform spanwise deflection for an elliptical wing of aspect ratio 3 (e = C, ~ = 

0.2b/oo) for ko = 0, 0.424, 0.847. 
NACA TN No. 1195 

[ 

1.5 

1.0 

ko=O 

~=6 

r-, 

I 

I 

I 

I 

----" 'R- ~ \ 

2PUa boO.5 

o ------

V 

Vi I 

I 

N 

" \ 

[ 

o 0 .2 0.4 0.6 0.8 1.0 

1.5r---'---ko '----. t.r>:J.2"4,.----.----.-----, 

r-_ 

I -_ I. 0 I---+---+--+---+----'-..=--+-R- I " 

l/~~. 

~ Vi 1 \ I \ 

L __ ~____ ---

o 0.2 0.4 0.6 0.8 /.0 

1.5 

1.0 

ko=0.847 

r-

: ---.. R-

[ -- 2p Uabo 0 .5 

o .....-

V 

Vi 

~-

~, 

1 \ 

_ .... 

o 0.2 0.4 0.6 0.8 La 

-y5 bo 

Fig. 14 

ko=O ~-

I 

~ ;>,~ : 

I ~\ , I 

i ~ 

.03 

M.(e) .02 

2puz boz .01 

o 

o 0.2 0.4 0.6 0.8 LO 

.03 I-------+,ko = 0.424 -t---+-- -I 

,- _ . 0 1---+---+-~-+-=~ -~ 

Me(e) 2 p UZ bel .01 1---+---+-____ --1--- -~---1 

O~-+_--+----L-_+_----+=~::.........j 

a 0.2 0.4 1.0 

.03 ko=0.847 _ ~-

r-

_ .02 

Me(e) 

2p U2 bo2 .01 

o 

1 __ 

--~ I- -

~ ~ .. I '~ 

: I k 

o 0.2 0.4 0.6 0.8 1.0 

Y 

5 bo 

NATIONAL ADVISORY 

COMMITTEE FOR AERONAUTICS 

Figure 14.- Lift and hinge-moment distributions for an elliptical wing of aspect ratio 6 with half-span ailerons 

of approximately one-quarter the wing chord and with ko = 0, 

0.424, 0.847 (t = O.lb/bo ). 
Fig. 15 

LINEAR ANTI SYMMETRICAL 

TORSION 

CR 

a 

1.00 I----II-----+-t.--I---I----j 

0.75 1--.--1---+-I'r--+---t---I 

, 

, 

0 .50 ~--l----l---T' -L------::-->-:=-=-i \ ko=0.667 

\ I I 

0.25 I---+-+-+-=~\ ko=.Oj333 

o I----+-Q=:::::+k;;o~= 0~1..:..6 7.:.....j 

o 1.0 

1.00 \ 

, 

, R 0.75 , 

..... - 0.50 

0 .25 

0 

0 0.5 \.0 1.5 2.0 

ko 

'OR 

-==- h 

NACA TN No. 1195 

ROLLING OF RIGID WING 

1. 0 

--

/ ko=0.667 

f----.i!, ~'k 0 = 0.3 '3 3-+----i 

ko='O.167 o ~~~--~--~~ 

o -\.O 

R 

NATIONAL ADVISORY 

COMMITTEE FOA AERONAUTICS. 

0.75 

0.50 

0 .25 

0 

-0.25 

-0.50 

-0.75 

" 

/ V 

L "Z 

L v.:L 

" ), 

~ ~ 

~ .~ 

~ 

'\ 

\\ \ 

\ 

o 0.5 \.0 1.5 2 .0 

ko 

Fi":'.lTe 15. - Rcll i ng- mon.ent coe.!'ficients for a rect&~gular wing of aspect 

ratio ;" in l·lbl..i I'011ing a:ld f o r <l rectiJ.~gular wing of aspect 

ra.::io " in linea..:.' antis/mtnct.ri~al tn~nion pl:>tted a3 ve~tor diagrams and A functio~ s cf ko_ 

- I 
NACA TN No. 1195 

LINEAR ANTI SYMMETRICAL 

TORSION 

0.8 

ko=2.542 i= 

0.6 

I 

I 

ko=1.695 ::: I 

I 

OA I 

I 

I 

ko=0.847 I 

I I ..... N I l \ ko=p.4~ \ 

0.2 

ko7o.2;12) r-.: 

t--.... 

(ko=O o ~ _9 

o 0.2 0.4 0 .6 

R 

0.6 .1 

\ If 

\ V , 

, ..... _- R -~ ::;..=::: I-=--

0.4 

/' 7 

/ 

kll ~I / 

.... _/ 

0.2 

o 

o 1.0 2.0 

ko 

Fig. 16 

ROLLING OF RIGID WING 

1.0 

ko=2.542 

O. 5 1---1I-----t-~~--+-__+-__+_____I 

o 0 -0.5 -1.0 

R 

NATIONAL ADVISORY 

COMMITTEE FOR AERONAUTICS. 

0.5 

o 

VCR 

4h 

-0.5 

...,"" ~ 

~ ..... "-

~ V "" ..... 

~~ ';." R f:::,., 

" 

1.0 

ko 

..-,2 ~ 

v 

r'\. 

'\ ~ 

~ 

~ 

2.0 

l igure 16 . - Rolling-moment coefficients for an elliptical wing of aspect 

ratio 3 in rigid rolling and for an el liptical wing of aspect I~t o 3 i n linear antisymmetrical torsion plotted as v ector diagrams and 

as functions of ko . 
Fig. 17 

lTCl 

Ba 

UNIFORM PITCHING 

3 

--'" 

ko=1.695 I 

2 I 

I 

I 

ko=?817,t'i 

I .!. 

ko=O.4f~ \\ \ 

o 

o 

ko~O.2\2~ 1 1 r\, 

k: = 6"-:;::? 

2 

R 

I 

2.5 

2.0 

1.5 

tY 

\ I \ 

\ / , 

\ 

"-

~~ , 

11'-- ::.::&-

1.0 

0.5 

0 

I 

/;' V 

-

3 

o 0.5 l.0 1.5 2.0 2.5 

ko 

2 

~ I 

I 

NACA TN No. 1195 

UNIFORM VERTICAL 

TRANSLATION 

;o-bi 

~ ~ 1 

,,'.-" ko=1.695 

//'V /~ ~o=0 847 

~~o~O"l24 ko=0.212 

lo=P I 

o 0 -I -2 -3 

TIel 

Btl 

. 

3 

2 

0 

-I 

-2 

-3 

R 

NATIONAL ADVISORY 

COMMITTEE fOR AERONAUTICS . 

./ 

.V Y 

L ~ 

-"':::". "-

R\ 

1\ \ 

o 0.5 1.0 1.5 2.0 2.5 

ko 

Figure 17.- Lift coefficients for an elliptical wing of acpect ratio 3 

in uniform pitching and for an elliptical wing in uniform 

vertical translation plotted as vector diagr ams and as functions of ko . 
NACA TN No. 1195 

o:~ j 

o 0.2 

1.5 

1.0 

~"O'2Im 2P~'b. O~5 0.4 0.6 0.8 1.0 -0.5 

-/.o 

Fig. 18 

-- ---I-- I 

---r---::::.: ....................... 

ko=0.847 ~ ..... , 

~, 

l ,-, 

R ~~ " ~ 

-~ ~ I 

o 0.2 0.4 0.6 0.8 1.0 

1.0 

0.5 

o 

-0.5 

NATIONAL ADVISORY 

COMMITTEE FOR AERONAUTICS. 

k 0= 0424 

1----- ---r __ 

r---r r:::..-....... 

......... 

f- -- - ~ R 

o 0.2 0.4 0.6 0.8 1.0 

y 

s bo 

[ 

2p U2bo 

:3 

2 

0 

-\ 

-2 

-3 

-4 

I -~ "-

ko=I.695 ~ ~ , '" 

t 

// 

.? . /V 

~ V 

Lf? 

-;::P '7 

,......7 

p:"--

o 0.2 0.4 0.6 0.8 1.0 

.-L 

s bo 

Figure 18.- Lift distri butions for uniform translation of an elli~tical 

wing of aspect r atio 3 fo r ko = 0 .212, 0.424, 0.847, 1 . 695. 
Fig. 19 

3.0 

2.5 

2.0 

r---_ 

-

--. 

ko=O 

1", 

, 

R 

-Z 

....... 

"-"-

\ 

~ \ 

\ 

'\\ ~ 

2.0 

1.5 

[ 1.0 

2p U2bo 

0 .5 

o 

NACA TN No. 1195 

ko"'O.424 

--. t--_ ..... 

----r- ....... , 

~ ....... 

" r-- I ~\ --- ~ t---~ -- o 0 .2 0.4 0.6 0 .8 1.0· 

1.5 

1.0 

0 .5 

00 0 .2 0.4 0.6 0 .8 1.0 

2 .5 

2.0 

1.5 

1.0 

0 .5 

---__ ko=0.212 2. Ol---+---- ko =0.847 --+----i 

F==-~-~-= - -_ R 

o 

o 

-r-_ , 

-t::::::", " ~ .... I. 5 1-----.::p!II"-=::::+==""""-o;;::-'f"'...-::::--+---t 

~ "-

\ 

"\\ 

[ /.0 1------I1----+--1'-..:::--~~__1 2 pU2 bo 

51---+--+---+--~~tt1 

I , -- ---r- - 0 0 0 .2 0.4 0.6 0.8 

0.2 0.4 0.6 0.8 -L 

Y 

sbo 

1.0 

s bo 

NATIONAL ADVISORY 

COMMITTEE FOR AERONAUTICS. 

4.0.---,-----,,---.c-:-----,-----, 

3. Ol----I-----'f' -t----t-----j 

[ 0~~::::::::L ~ 2pU2 bo 

I. 0 1--+----t---+----;:;~~--1 

00 0 .2 0.4 0.6 0 .8 1.0 

Y 

s bo 

Figu're 19.- Lift distributions for uniform pitching of an elliptical 

wing of a spect ratio 3 for ko = 0, 0.212, 0.424, 0 . 847, 1_695. 
Fig. 22a,b 

[l 

\ 

\ 

" I'-. ....... -- --f-- ----

1.0 

0.9 

0.8 

0.7 

0.6 

0.5 

0.4 

0.3 

0.2 

0.1 

o 

-0.1 

""""'t.::::::. ::.::::.:: r.:::J P) . ....::--

~(C) 

R(C+~ 

.- ~--+-- ~ 

-- R(~) -. 

I(Q) --

-

-0.2 1~9-V --

I(C+CT) 

-0.3 

-0.4 

1.0 

0.9 

0.8 

0.7 

0.6 

0.5 

0.4 

0.3 

0.2 

0.1 

o 

-0.1 

-0.2 

-0.3 

-0.4 

1.0 

(a) AR; 1.5 ko 

\ 

\ 

\ 

"'-I'-. ....... ~) 

R(C+CT) r---R(Qf -----

--t-- t--.L_ 

R(P) -- --

I(Q) 

~ :::::: --f- .-r--

-..:..: "- '- -!F-) 

"- -- k- - I(C +CT) 

--I(C) 

1.0 

(b) AR == 2.0 ko 

NACA TN No. 1195 

R(C(ool) 

... - --I---f---

1-- ----I---

_. .-

-

2.0 3.0 

NATIONAL ADVISORY 

COMMITTEE FOR AERONAUTICS. 

R(C(oo)) 

-. f- .-f---

-- -- ----

-

2.0 3.0 

Figure 22 (a to d). - Uniform pitc)ung and tl'anslat1on of rigid elliptical wing (one.::point 

collocation) . Comparison of functions C, C + cr with functions F and 

Q of refe~ence 5. 

o 

C\J 

C\J 

W 

cr 

=> 

(9 

l.L.. 
NACA TN No. 1195 

1.0 

DEFLECTION MODES 

FIRST TORSION 

O. 51---+---,I+--+---+-\----+--~ 

-0.51---+---+--~--+--+-~ 

-\.0 '----'-------'-_--'-------'--------'6 

o 0.2 0.4 0.6 0.8 1.0 

o 

C -0 .1 

2 P U2bo-0. 

-0. 

2 

7, 

FIRST BENDING 

~ B .~ 

r---. 

~ --J 

I ............ 

o 0 .2 0.4 0 .6 0 .8 

Y 

1.0 

5 bo 

0 .3 

0.2 

0.1 

L 

2plfbo 0 

-0.1 

-0.2 

-0.3 

Figs. 20,21 

SECOND BENDING 

I ,- ~-...... 

.-Y ........ 

~ [:!' " .- R ~~ ~-\ 

\ 

\ 

o 0.2 0.4 0.6 0 .8 1.0 

3 

2. 

[ 

-:;:;'l 

/ 

o 

o 

FIRST TORSION 

--- // 

~. -=--~ y \ / 

./ 

17 

- I 

----- ---

0.2 0.4 0.6 0.8 1.0 

Y 

5 bo 

Figure 20.- First-torsion., first-bending, and second-bending modes for a 

rectangular wing of aspect ratio 13.5 and lift distributions 

for the~e modes for ko = 0.0965. 

1.5 I--------l_---,:k.c... o ..:.:. 1.6.c...9~5=----+_---I /.--

L \'0 r-I-fI-=:t:::~~1 

2 p U2bo 

0.5 

o 0 0.2 0.4 0.6 0.8 

Y 

2.4r--.------~-~-~ 

0·~-+-- -J~~~-~ 

1.6 I---J---...~ 

[ 1.2 f------,II---l----l---- ---l 

2 pU2 bo 

O . 8f---+--l----h..--~t:=_--.::!~\..----.. 

0.2 0.4 0.6 0 .8 1.0 

Y 

sbo 

Figure 21.- L1ft distributions for linear ant1symmetrical torsion of an 

elliptical wing of aspect ratio 3 for ko = 1.695 and 2.542. 

----------. 

NATIONAL ADVISORY 

COMMITTEE FOA AERONAUTICS. 
NACA TN No. 1195 Fig. 22c ,d 

1.0~ 

0.91-+\-+---+--+--+--+----+--+--+--+-+-+--+--+--t---t--t---r---1 

0.8 \ 

0.7 ,~ 

6~~~_ 

0.5 R(P" -- __ _ _ - R (01 ---. tR(Cloo» 

I '. r---- _:..;,)~ ____ ~_ 

0.4~4-~~--+-~-+--~4-~~--+-~-+--r-1--r~~ 

0.3~4_~-4--+-~-+--~4-~-4--+-~-+--r_~_r_1~ 

0.2~4-~~--+-~-+--~+-~~--+-~-+--r-1--r~~ 

0.1 ~4_~_4--+-~-+--~+-~_4--+-~-+--r-~_r~~ 

o r-:::...'-r-. !.1.Q. .__ --f---i-- ---- I ........ ~ -,[P)"'-'-'- -- -0.1 ~(C+(7"V--~ 

-0.2 ~C)-+--+--+-t--+--+--+---+---1f---+--+--+--+--+-t--1 

-0.3~~-+~--+-~-+--~1--+~;-+--r-+--~~-+~~ 

- O. 4,L...---L----'-----' __ ....L.--:-'=-..L __ -'--....L..----'---='.....-....L.----'---..L __ .I.....-...;;=-'"=---'-----'--' 

1.0 2.0 3.0 

1.0 

0 .9 

0.8 

0.7 

0.6 

0.5 

0.4 

0.3 

0.2 

0.1 

o 

-0.1 

-0.2 

-0.3 

-0.4 

(c) AR >= 3.0 

~ 

\ 

~\ ~ 

~R(Q) 

R(P0 

R(C+cr)' 

~]P)\ I(Q} 

~ V ~ 

r--I(cr 

(d) AR = 6.0 

ko 

NATIONAL ADVISORY 

COMMITTEE FOR AERONAUTICS 

R(Cloo}' 

.. 

~ 

1.0 2.0 3.0 

ko 

Figure 22.- Concluded. 
Fig. 23 NACA TN No. 1195 

0.3.---.----.----.---.----.----.---~--~----~--~--~--~ 

0.2~--4----+----~--4----+----~--~--~----~--~--~--~ 

?8.6 mph f' ! --34.2 mpn 

-ko a:: 0.4 O.l~--4----+----~--4----+--~j~~~ -+~--~--~--~--~ 

~s~d damping P- -' 1 1--- .--r--Tr--- - --- +-- --+---- ~ +---l 

Using two-~imenFional-- -I, ~:--k a:: 0.333 

aero termS I I / I / i ° 

M 0r-~--~--_+---~1~~/r41~/-+~1+---+---~~~-4--~ 

~ ,0 = F' 5 -- :> 

~ I I / / --ko = 0.4 

~ k o ,= O.6jP 

'HoG) -0. ). I----+-----l-----+I----/-f-:L---+---I j ---+----+----f---+------t----+-----I 

P- ---ko c 0.5 

() Flutter mode 1 / 

M ~--~--_4----4----+--~+'~-Using three-dimensional aero ~--~ 

~ terms ·rl 

j:). 

~-0.2 f---~---1--~---+---+--~--~--~--~--~--4-~ o 

NATIONAL ADVISORY 

COMMITTEE FOR AERONAUTICS. 

\ -0.3~-+---rr1\~-~~_--L---~-~--1--+--4---+--~-~ 

ko 0: O.fJ~ \ -k

o =1

0

•

5 

\ ,' 'Using two -dimenai \ onal aero terms 

\ 1\ r Flutte~ mode 2 

-0.4 ~--+---~-+~~-+1--~,----+---4---~---+--~L---~~ 

\ _ --ko ;c p. 4 

ko = O.5 ---~ , ---.----+--~~~~--~----4----+----+_--~----~--~--~ 

\ \- uSini thrjee-di,enai1onal laero I erms 

_0.5~ __ J-__ -L ____ ~ __ ~ __ -L ____ L-__ ~ __ ~ ____ L-__ -L __ ~ __ ~ 

o 10 20 30 40 50 60 

True airspeed in mph 

Figure 23.- Damping coefficient a.gainst true airsp eed for a 

rectangular wing of aspect ratio 6. bo ; 10 

inches; g = 0.070. (Analysis by method of reference 4.) 
NACA TN No. 1195 

- 0 .8 

-0.6 

- 0 .4 

.cI 

~ 

- 0. 2 

o ~ 

0.4 0 . 6 

z = y/ebo 

Fig. 24a,b 

Rl 

/ 

V 

/ 

!----V 

LV 

± 

0.8 1.0 

11.0 4 

Figure 24 (a to d).- Graphical 1ntegration. (a) For AA: dh Z dz = 

-0.040 + 0.0101; ko = 0.4. 0 

- 0 .8 

- 0. 6 

-0.4 

- 0 .2 

o 

0 . 2 

o 

NATIONAL ADVISORY 

COMMITTEE FOR AERONAUTICS. 

---

0 . 2 0.4 0. 6 z = y/sbo 

R--

1 

/ 

/ 

L ~ V 

L 

V 

i 

0.8 1.0 

Jnl.O 

Figur e 24.- Continu ed . ( t. ) For ~: de(,;r3 d z ~ - 0 . 042 + 0.0141; k.o : 0 .4 . - 0 
F ig. 24c ,d 

-0.8 

-0. 6 

-0.4 

-0 . 2 

o 

0.2 

o 

-

O..cl 0 . 4 0.6 z = y/sbo 

NACA TN No. 1195 

R--1 

/ 

V 

/ 

f-.-----' ~ 

~ 

../' 

± 

0 .8 1.0 

J l.O 3 

Figur e 24 .- Oontinu ed. (c) For 6 D: d Z dz = - 0.043 + O. Olai; ko = 0 . 4 . = h 

-O.B 

-0. 6 

-0 .4 

d ~ 

-0.2 

o 

0 .2 

o 

o 

NATIONAL ADVISORY 

COMMITTEE FOR AERONAUTICS. 

0 .2 0.4 0 . 6 Z = y/sbo 

( a

' ) /,1. 0 2 Fi gure 24 .- Conc lud ed . For L'>!: J" d a:, z dz 

o 

R---j 

I 

/ 

V 

/ 

.------' V 

~ 

../' ~ 

i 

O.B 1. 0 

- 0 _018 + 0 . 017i ; ko c 0 . 1. 
-03.0 

--r--... 

---

-5.0 

r--... r--.. dorizontal tail I-.. 

r--r-- -4.0 

........ r--r---- -3. 0 

Stabilizer 

-2.0 I 

>- - Y-Axis - 0-::::::: 1 -

- 1---t- __ ·Semichord line r-j_ - - 1---

-1. 0 

ft 

o 

I Hinge line 

1.0 

" Fus el ag e torsion 

axis I I ilevator 

2.0 I--- NATIONAL ADVISORY 

I COMMITTEE FOR AERONAUTICS 

I--- l---

J..-l---l---

3.0 

4.0 

o 0 .2 0.4 

10.1 ft 

0 .6 

y/sb o 

---

-

t-- - r--

__ V 

0.8 

Fl gur e 25. - Horlzontal tall used in example II. 

-6.0 I 

-5.0 I 

I """'"' 

I 

) 

Fin 

-4.0 -3.0 

'\ 

\\ -2. 

.r 

) 

I -IJ 

- r--, 

Y-Axls 

I- r-...::::::: I ) 

........ 1----- Fin-rudder I . 

line 

----1--

o RUdier hing_e line · --- - I

-1. 

it 

I I I 

"Fuselage t orsion axl B :>:l 

, 

/ 

(]) 0 

I 

I Rudder .. 

(]) :> 

1. 

/V 

V 

I 0 <1> z4 = 4.25 it 

0 

t:>" 

0 0 .. 

Po 2. 

z3 2 . 5 it - .I 

........ 

0 f-"' 3. 

l---r--

, 

-

- 8.35 

1.0 4. 0.2 0.4 0 .

y/sb o 

Figure 26.- Fln-rudder combination 
Fig. 27 

+" 

~ 

Q) 

·rl 

o 

·rl 

'H 

0.06 

0.04 

0 . 02 

o 

'Q;-0 . 02 

o 

o 

t:D 

~ 

·rl 

P. 

~ -0.04 

-0.06 

-0.08 

-0.10 

o 

NACA TN No. 11£3 

1 I I 743 mph--l I 

I I 

1/299 mph-; 

Measured 1/ damping Flutjer m~de 1 

-t--t--- ~ 

~t ------ - r--[:/i kov = 0.506 - I -- ./ 

U.in~ twoldime~.ion~l ae o teJmS-j 

V I V Lf' 

Using three-dimensional aero terms---- --// i /kov ~ 0.696-_;p V 1 

ko .. v o.~ V I ,,- I 

NATIONAL ADVISORY - COMMITTEE FOR AERONAUTICS 

100 200 300 

True airspeed in mph 

Figure 27.- Damping coefficient g against true airspeed for 

the tail-flutter problem of example II; g = 0.038. 

Notel Flutter determinants solved by method of reference 4.