NO. 68-73 -

A DOUBLET LATTICE METHOD FOR CALCULATING LIFT DISTRIBUTIONS ON OSCILLATING SURFACES IN SUBSONIC FLOWS

b Y

E. ALBANO and W. P. RODDEN Northrop/Noroir Hawthorne, California

AIAA Paper

No. 68-73 ...,

Meeting NEW YORK, NEW YORK/JANUARY 22-24, 1968

F l i r t wblicalion rights reserved by American Inflitule of Aeron3uiic1 and Ailronoulics. I290 Avenue of the America$. New Yo&, N. Y . 10019. Ab5lmct1 moy be published without permiision i f credit is airen to author and lo AIAA. [Pr ice -A IM Member $1.00, Nonmambw $1.501

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A DOUBLET LATTICE METHOD FOR CALCULATING LIFT DISTRIBUTIONS ON OSCILLATING SURFACES IN SUBSONIC FLOWS

E. Albana, Senior Engineer W. P . Rodden, Consulting Bngineer,

Northrop Corporation, Norair Division Hawthorne, California

i

Summary

Approximate solutions from the lineariied for- mulation are obtained by simulating the surface by a set of lifting elements which are short line- segments of acceleration potential doublets. normal velocity induced by an elanent of unit strength is given by an integral of the subsonic kernel function. The load on each element is determined by satisfying normal velocity boundary conditions at a set of points on the surface. It is seen 5 posteriori chat the lifting elements and collocation stations can be located such that the Kutta condition is satisfied approximately. method obviates the prescription of singularities in lift distribution along lines where normal velocity is discontinuous, and is readily adapted for problems of complex geometries. Results com- pare closely with those from methods which pre- scribe lift-mode series, and from pressure measure- ments. The technique constitutes an extension of a method developed by S. 6. Hedroan for steady flow,*

The

The

Symbols

m b c

K k M NC NS

P.(k P U 2 )

wu

t

X.5

X,Y,Z

Aspect ratio

Semichord Chord Kernel function

Reduced frequency k =Wb/U Free stream Mach number Number of boxes on a chord Number of boxes on senispan

Complex amplitude of lifting pressure Complex amplitude of normal velocity at surface

Time

Coordinates on the surface Cartesian coordinates

w Frequency of oscillation P Free stream density

to the pressure difference across the surface

P(x,s,t) = Re p(w,s) e 1- by a singular integral equation and the Kutta con- dition at the trailing edge (TE):

where (x,s) are orthogonal coordinates on the sur- face S such that the undisturbed stream is directed parallel to the x-axis.

Rodemich'') has given a derivation of the kernel function for a nonplanar surface, and Landahl(2) has presented a relatively concise formula:

e-ik1U11R2(1 + u1 2 % ) 1 K = -312 - iklM r

2

-iklu1 Mr u e 2 2

R R + + 4k2)312 Introduction I

The linearized formulation of the oscillatory subsonic lifting surface theory relates the normal velocity at the surface

du

c x = x - i , Yo = Y -7. 2 = z -

2 2 2 % R = (xo + p rl 1

i6Jtl

I 2 2 % r1 = CY, + z,, ) ,

*At the time of this writing, the authors learned that a similar extension had been developed fnde- u1 = (MR - x,)l@ 2 rl, kl =wrl/U, @= (1 - n 2 % ) - pendently by Stark.(19) ( 4 )

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system is i l l u s t r a t e d i n Fig. 1. The t h a t t h e i n t e g r a l i n Eq. (1) is

t h e " f i n i t e p a r t "

The t r a d i t i o n a l m e t h o d - f o r o b t a i n i n g approxi- mate s o l u t i o n s f o r p when w i s g iven is t o a s s m e a series approximation

and t o determine t h e c o e f f i c i e n t s aij by s a t i s f y i n g no rma l -ve loc i ty c o n d i t i o n s on t h e surf_ace. p r o p e r t i e s o f t h e f o r c e d i s t r i b u t i o n p such as t h e behav io r n e a r s u r f a c e edges are b u i l t i n t o t h e approximation by a p p r o p r i a t e cho ice o f t h e func- t i o n s i n t h e s e r i e s . T h i s t e c h n i q u e has been used s u c c e s s f u l l y f o r a t l e a s t t e n y e a r s , a l t hough most a p p l i c a t i o n s have been f o r p l a n a r wings wi thou t c o n t r o l s u r f a c e s .

Known

C o n s i d e r a t i o n o f t h e e f f o r t s needed t o develop a single computer program t o hand le a f a i r l y general c l a s s o f nonplanar problems h a s l e d us t o s e e k a t echn ique which would o b v i a t e t h e p r e s c r i p - t i o n of t h e behav io r o f along edges and c o m e r s and i n f a c t , would remove a p r i o r i r e s t r i c t i o n s on t h e g l o b a l behav io r o f t h e f o r c e d i s t r i b u t i o n . The "box methods" f o r t h e s u p e r s o n i c problem p rov ide examples of t h e methods sought .

Doub le t -La t t i ce Method

We d e s c r i b e a method which i s an e x t e n s i o n of t h e one developed f o r s t e a d y subson ic f l a w by H e & ~ a n ( ~ ) i n 1965. The elements o f t h e t echn ique a re t o be found i n t h e v o r t e x - l a t t i c e method o f F a l k ~ r ( ~ ) . Since i t appea r s a t t h e p r e s e n t t ime t h a t a r i g o r o u s a n a l y t i c a l b a s i s f o r t h e method i s no t a v a i l a b l e , we p r e s e n t an o p e r a t i o n a l d e s c r i p - t i o n :

It i s assumed t h a t t h e s u r f a c e can b e approx i - mated by segments of p l anes . The s u r f a c e is d i v i d - ed i n t o small t r a p e z o i d a l p a n e l s ("boxes") i n a manner such t h a t t h e boxes a r e a r r anged i n columns p a r a l l e l t o t h e f r e e s t r eam (Fig. 2), and so t h a t s u r f a c e edges, f o l d l i n e s and h inge l i n e s l i e on or near box boundaries . The 114-chord l i n e o f each box is t a k e n t o c o n t a i n a d i s t r i b u t i o n o f accelera- t i o n p o t e n t i a l d o u b l e t s o f uniform b u t unknown s t r e n g t h . In s t e a d y f low, each doub le t l i ne - seg - ment w i l l b e e q u i v a l e n t t o a v e l o c i t y - p o t e n t i a l ho r seshoe v o r t e x whose "bound" p o r t i o n c o i n c i d e s w i t h t h e doub le t line..

Let n b e t h e n m b e r o f boxes and l e t t h e doub le t s t r e n g t h i n t h e j t h l i n e segment be

z j d/l 14ZP

where ? j is t h e complex ampl i tude of t h e f o r c e pe r u n i t l e n g t h a long t h e l i ne , and dfl t h e inc remen ta l length. The normal v e l o c i t y (downwash) induced a t a p o i n t ( x i , s i ) on t h e s u r f a c e is

where each i n t e g r a t i o n runs along t h e l i n e segment whose l e n g t h i s C j . downwash p o i n t s on t h e s u r f a c e , t h e f j a r e d e t e r - mined. The f o r c e on t h e doublet l i n e is t aken as t h e f o r c e on t h e box and t h e p r e s s u r e d i f f e r e n c e across t h e s u r f a c e i s approximated by

If Eq. (5 ) is a p p l i e d a t n

= fo rce / (box a r e a ) = P.L. l (box a rea) J I

where i s t h e box average chord and A. t h e sweep a n g l e 0 2 t h e doublet line. J

We n o t e t h a t t h e induced downwash c a l c u l a t e d by Eq. ( 5 ) w i l l be i n f i n i t e i f t h e downwash p o i n t l i e s on a doub le t l i n e segment or downstream from i ts end p o i n t s r Furthermore, t h e Ku t t a c o n d i t i o n has n o t been imposed.

From numerical expe r imen ta t ion w i t h t h i s t ech - n ique , i t has become appa ren t t h a t t h e Ku t t a condi- t i o n w i l l be s a t i s f i e d approximately if each down- wash p o i n t is t h e p o i n t a t midspan and 314-chord o f a box. _ _

Eq. (5 ) i s w r i t t e n

(7)

where

1 2 P = i I T P u

I f A a re t h e elements of t h e m a t r i x whose i n v e r s e i s t k a m a t r i x o f D t hen

i j '

" Pi = EAij w j

j=l (9)

prov ides t h e approximate s o l u t i o n f o r t h e f o r c e d i s t r i b u t i o n .

Gene ra l i zed f o r c e c o e f f i c i e n t s a re computed approximately by

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where

S k = a r e a of box k IiJ 4 [(qo2 - t o 2 ) ~ + V ~ B + c]. J t o / - l t a n

i $

pkJ = f o r c e d i s t r i b u t i o n in mode j a t box k

= d e f l e c t i o n i n mode i of 114 chord, mid- - span p o i n t of box k

I

r - 2T10 e > e2 + 2 e A

1 1

1

+(? B + VoA) l og r + 2q0 e + e

b = r e f e r e n c e l e n g t h (15)

where Working F o m s

z L z Approximate e v a l u a t i o n of t h e i n t e g r a l i n Eq. rl = Vo + to

(8 ) is achieved by approximating t h e i n t e g r a n d by a simple func t ion . We cons ide r t h e downwash induced a t a r e c e i v i n g p o i n t R = (XR, y ~ , ZR) by a doub le t l i n e segment whose end p o i n t s are S- and S+ and whose midpoint i s So. Let

For t h e p l a n a r case (t0-0)

(11) 2 f = r K 1

Denote

Define t h e c o o r d i n a t e System (Fig. 3 )

V = y c o s y . - z s i n % L7

( 1 2 ) t= -y s i n 3 + z COSY

L7

Since t h e l e n g t h of t h e doub le t segment is small, i t may be expected t h a t a p a r a b o l i c approximation f o r a long t h e segment would b e s u f f i c i e n t i n e v a l u a t i n g the integral .

+ (+ B + Vo A) log(=Y+ 2 eA ‘1, + e

In o r d e r t o converge t o Hedman‘s v o r t e x l a t t i c e r e s u l t s f o r s t e a d y f low and t o improve t h e approx i - ma t ion o f Eq. (15), we have found i t conven ien t t o s u b t r a c t t h e s t e a d y p a r t (W= 0) from 17 b e f o r e app ly ing t h e above formulas , and t h e n t o add t h e e f f e c t o f a ho r seshoe vortex.

Approximate e v a l u a t i o n of t h e i n t e g r a l s ‘$,and 12 i n t h e k e r n e l f u n c t i o n may b e accomplished in many ways. f o r 11 i n t e r n s o f i n f i n i t e s e r i e s . However, we choose t o approximate t h e i n t e g r a n d s by s imple f u n c t i o n s . It is s u f f i c i e n t t o c o n s i d e r nonnega- t i v e a r g m e n t s because of symmetry p r o p e r t i e s of t h e in t eg rand . I n t e g r a t i o n by p a r t s gives

L, Schwarz(6) h a s g i v e n an expres s ion

-iklU

Il(ul; k l ) = f ( l + u ) e 2 312 d‘

U.

where ~ a t k i n s ( ~ ) h a s g iven t h e formula 1

J Vo = (y, - yso) cosy i (z, - z

to = - (yR - yso) s i n Y + ( 2

t / m m 1 - 0.101 e x p ( 4 . 3 2 9 t ) e = - & c o s A

2 j ) s in%

0 so

- 2 c o s % - 0.899 exp(-1,4067t) 0 R So

2 - 0.09480933 exp(-2.90t) sinnt, (t>G) A = (f - 2ko -I t - ) l 2 e +

which when s u b s t i t u t e d i n t h e above y i e l d s a s i m p l e expres s ion f o r 11’ The i n t e g r a l ‘I2 i s approximated in s i m i l a r f a sh ion .

I n view o f t h e lack of a r i g o r o u s b a s i s f o r t h e fo rego ing a s s m p t i o n s , it i s n e c e s s a r y t o demon- - The r e s u l t of t h e i n t e g r a t i o n is

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s t r a t e t h e adequacy of t h e doub le t l a t t i c e method by comparison of r e s u l t s w i t h s o l u t i o n s o b t a i n e d by o t h e r means, and w i t h experimental da t a .

R e s u l t s for Two-Dimensional Flow

I- For p l a n a r f low t h e i n t e g r a l equa t ion (1) be-

comes t h e one-dimensional i n t e g r a l equa t ion of Possio ( s e e , e.g., s e c t i o n 6-4 o f Ref. 8 ) and t h e doub le t l i n e segments become d o u b l e t s on t h e chard. The r e s u l t s o f t h e doub le t l a t t i ce method p resen ted h e r e have been o b t a i n e d by d i v i d i n g t h e a i r f o i l chord i n t o equa l i n t e r v a l s and t a k i n g a send ing and r e c e i v i n g p o i n t a t t h e 114-chord and 3/4-chord, r e s p e c t i v e l y , of each i n t e r v a l .

Exact s o l u t i o n s a r e a v a i l a b l e f o r incompres- s i b l e flow. S o l u t i o n s for an a i r f o i l w i t h o s c i l - l a t i n g f l a p are compared i n Fig. 4, where measure- ments r e p o r t e d by Bergh(9) a r e included.

The doub le t l a t t i c e method may be thought o f as producing s t e p - f u n c t i o n approximations t o t h e p r e s - Sure d i s t r i b u t i o n - Ra the r t h a n p r e s e n t t h e s e i n g raphs , we have chosen t o draw cu rves through t h e v a l u e s f o r p a t sending p o i n t s . r e s u l t s , we use t h e term "boxes" t o mean i n t e r v a l s ( i n 2-D f low) or small p a n e l s ( i n 3-0 f low), and

NC = Number o f boxes p e r wing chord

NS = Number o f boxes p e r semi-span ( p l a n a r

Also i n d e s c r i b i n g

wings)

The t a b l e below is a comparison o f g e n e r a l i z e d f o r c e s from t h e p r e s e n t method and from t h e t a b l e s of Ref. 10. The c o e f f i c i e n t s are de f ined by

30 "boxes"

-0.0197 -1.119 i

-1,574 -0.07951

-0,4824 +0.08231

0 3 3 0 3 +O.O924i

-0.0572 -0.8623i

-0.4105 +0.00581

0.0591 -0.0638i

-0.0762 -0.14741

-0.0919 -0.07121

L =7rpU2 b eikt(Aka + Bkb + Ckc)

M = TpU b e ( h a + Bmb + Cmc) 2 i k t

Re fe rence 10

-0.0444 -1.12011

-1.5755 -0.02671

-0.4803 +O0O868i

0.3309 +0.07601

-0.0946 -0,86191

-0,4147 +O.O248i

0.06OlZ -0.066091

-0.0798 -0.15041

-0.0931 -0.07391

N = TpU2 b e i k t ( h a + Bnb t Cnc)

where b i s t h e semichard, k t h e reduced frequency based on b and

L = Force o f a i r f o i l and f l a p , p o s i t i v e down-

M = Moment o f wing and f l a p about midchord,

N = Moment o f f l a p about h i n g e a x i s , p o s i t i v e

ward

p o s i t i v e t i i l heavy v t a i l heavy

Ab = T r a n s l a t i o n ampli tude, p o s i t i v e downward

B = Amplitude o f a i r f o i l r o t a t i o n about mid-

C = h p l i t u d e of f l a p r o t a t i o n , p o s i t i v e

cha rd , p o s i t i v e t r a i l i n g edge down

t r a i l i n g edge down

R e s u l t s shown h e r e are f o r t h e case M = 0.8, k = 0.9, T = ( f l a p c h o r d ) / ( a i r f o i l chord) = 0.3.

The t a b u l a t e d r e s u l t s i n d i c a t e t h a t t h e doub le t l a t t i c e approximation would b e v a l i d f o r h i g h sub- sonic Mach numbers and reduced f r e q u e n c i e s o f o r d e r u n i t y ,

R e s u l t s f o r Three-Dimensional Flows

Fig. 5 compares c a l c u l a t e d l i f t d i s t r i b u t i o n s w i t h measurements r e p o r t e d i n NACA RM A51G31 f o r a swept, t a p e r e d wing a t i n c i d e n c e i n s t e a d y flow. For t h i s problem, t h e semi-wing was d i v i d e d i n t o 48 boxes, each c o n t a i n i n g a horseshoe v o r t e x as i n HehanIs method. R e s u l t s f o r t h e wing s t a t i o n s shown were o b t a i n e d by i n t e r p o l a t i o n ,

C a l c u l a t i o n s and measurements f o r a wing w i t h p a r t i a l - s p a n f l a p are p r e s e n t e d i n F i g , 6. The curves l a b e l e d "experiment" were o b t a i n e d by reduc- t i o n o f g r a p h i c a l d a t a o f Ref. 1 2 and f o r t H s reason, are r a t h e r imprecise . C a l c u l a t i o n s w e r e made w i t h e i g h t y ho r seshoe v o r t i c e s on t h e semi-wing e i g h t v o r t i c e s be ing on t h e f l a p , Landahl(13) has shown t h a t b o t h t h e form and s t r e n g t h of t h e p r e s - sure s i n g u l a r i t y a t t h e h i n g e line can b e determined a n a l y t i c a l l y . The expres s ion f o r t h e l i f t d i s t r i b u - t i o n f o r u n i t f l a p a n g l e is:

v

-0.0075 -1.118 i

-1..571 -0.09211

-0.4835 +O.O789i

0.3293 +0.10051

-0.0391 -0.86041

m -0.407 5 +O. 00211

n 0.0584 -0.0626i a -0.0745 -0.14561 "b -0.0912 -0.06961

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where

x = coordinate of hinge

A= = hingeline sweep angle

c = local chord

For the wing considered here, the singular part of Eq- (16) becomes

hc A 30°, M = 0.6;

be studied so that its full implications may be brought out. The second kind of approximation is concerned with the evaluation of integrals in the kernel and in the downwash-pressure influence CO-

efficients, Eq. (8). These procedures may be im- proved and optimized according to the standard techniques of nmerical quadrature.

The advantages of the doublet lattice method arise from being able to disregard the special behavior of the lift distribution where the normal- wash is discontinuous. So long as edges do not intersect boxes, a computer program based on this technique does not need to discriminate among side edges, fold lines, hinge lines, etc., and this fact is important when problms of intersecting surfaces are considered. Furthermore, since the influence coefficients Dij are independent of the properties of the normalwash distribution, the same matrix computed for a given wing will yield solutions for a large class of normalwash distributions. For example, generalized forces for many different con- trol surface configurations may be obtained from the same influence coefficient matrix.

For applications in aeroelastic analyses, aero- dynamic ipfluence coefficients that relate control point forces to deflections have been defined in Ref. 18, The doublet lattice method leads imnedi- ately to this definition of influence coefficients since the control point force is given by the product of lifting pressure and box area, and the dawnwash is the substantial derivative i n the streamwise direction of the deflection. (The sub- stantial derivative requires curve fitting "in-the- small" along the surface strip; e . g . , a parabola may be passed through the control point and the points upstream and downstream of it.) number of degrees of freedom is desired for the aeroelastic analysis over the number of boxes employed in the aerodynamic analysis, the number of control point forces and deflections may be reduced by a streamwise curve fit and the method of virtual work as discussed in Ref. 18.

If a reduced

References

1. Vivian, H. T., and Andrew, L. V., ''Unsteady aerodynamics for advanced configurations. Part I - Application of the subsonic kernel function to nanplanar lifting surfaces," Air Force Flight Dynamics Lab. Report FDL-TDR-64- 152, Part I (May 1965).

2. Landahl, M. T., IlKernel function for nonplanar oscillating surfaces in a subsonic flow," A I A A J r 2, 1045-1046 (May 1967).

theoretical aerodynamics," British A.R.C. RM 2424 (1951).

4. Hedman, S o G., IWortex lattice method for cal- culation of quasisteady-state loadings on thin elastic wings," Aeronautical Research Insti- tute of Sweden Re'port 105 (October 1965).

5. Falkner, V. M., "The calculation of aerodynamic loading on surfaces of any shape," British A.R.C. R&M 1910 (1943).

6. Schwarz, L., tlInvestigation of some functions related to the cylinder functions of zero

3 , Mangler, Ka W., '%nproper integrals in

The graph of this expression is labeled "local solu- tion" in Fig. 6a. Fig. 6b shows the distribution of lift on the semi-wing calculated by the horseshoe vortex lattice technique.

Both kernel function calculations and measure- ments of lift distribution reported in NASA TND-344 are compared in Fig. 7 with results from the doub- let lattice method, The R 3 rectangular wing con- sidered is oscillating in a bending mode described approximately by

6 =0.180431f( + 1.702559'- 1.136881y 3 I + 0.253~377~

yhere 6 is the nondimensional deflection amplitude, y the nondimensional spanwise coordinate based on the semispan.

Results of computations for an aspect ratio 2 rectangular wing with full span 40% chord oscil- lating flap are shorn in Fig. 8. These are compared with kernel function calculations (with built-in hingeline singularity) given in reference 15, and experimental data of reference 16. The doublet- lattice calculations were made w4th 99 boxes on the semi-wing and of these, 45 boxes were on the flap.

Some preliminary results of computations for a

.J

rectangular T-tail are presented in Fig. 9, where measurements by Clevenson and Leadbetter(17) are reproduced. TO account in part for the tunnel waL1, the image system of the fin was included in the c a l - culations; the image of the horizontal stabilizer was neglected. and forty on the horizontal stabilizer semi-wing, The discrepancy between calculations and experi- mental data for increasing reduced frequency might be due to the relatively small number of boxes used, or to the incomplete modelling of the effect of the tunnel wall. The T-tail results illustrate that nanplanar problms are easily approached by the doublet lattice technique.

Forty boxes were placed on the fin

Conclusions

Within the context of the linearized subsonic lifting surface theory, two types of approximations are involved in the doublet lattice method. The assmption that for purposes of calculating lift distributions the surface can be represented by a system of line segments of acceleration potential doublets is seen to be a valid approximation in view of the results obtained. As far as the authors are aware, an analytical basis for this approximation has not been established and certainly deserves to - order," Luftfahrtforshung, V a l r 20, No. 12,

341-372 (1944),

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7. Watkins, C. E., Runyan, H. Lo, and Cunningham, H. J., "A systematic kernel function procedure for determining aerodynamic farces on oscil- lating or steady finite wings at subsonic speeds," NASA Tech. Report R-48 (1959).

8. Bisplinghoff, R. L,, Ashley, H., and Halfman, R. L., Aeroelasticite (Addison Wesley, New York, 1957) Chap. VI, p. 294,

9. Bergh, H e , "A new method for measuring the pressure distribution on harmonically oscillating wings of arbitrary planform," National Aeronautical and Astronautical Research Institute, Amsterdam, Report MP.224 (1964).

10. Anon., "Tables of aerodynamic coefficients f o r an oscillating wing-flap System in a subsonic compressible flow,,! National Aeronautical and Astronautical Research Institute, Amsterdan, Report F,151 (May 1954).

11. Kolbe, C, D., and Boltz, F. W., "The forces and pressure distribution at subsonic speeds on a plane wing having 40° of sweepback, an aspect ratio of 3, and a taper ratio of 0.5," NACA p\M A51G31 (19511,

12. H m o n d , A. Do, and Keffer, Bo M,, "The effect at high subsonic speeds of a flap-type aileron on the chordwise pressure distribution near midsemi-span of a tapered 35O sweptback wing of aspect ratio 4 having NACA 65A006 section," NACA RM L53C23 (1953).

13. Landahl, M. T., "On the pressure loading func- tions for oscillating wings with control sur- faces," Proceedings of the AIAAIASME 8th Structures, Structural Dynamics and Materials Conference, 142-147 (March 1967). i/

14. Lessing, H. C., Troutman, So L., and Menees, 6 . P., Experimental determination of the pres- sure distribution on 2 rectangular wing oscil- lating in the first bending mode for Mach num- bers from 0.24 to 1,30," NASA TN 0-344 (1960),

15. Curtis, A. R., Gikas, X. A., and Hassig, H. J o , "Oscillatory flap aerodynamics - comparison between theory and experiment," Paper presented at the Aerospace Flutter and Dynamics Council Meeting, Cocoa Beach, Florida (Novo 1967).

16. Beals, V o and Targoff, W. Po, llcontrol surface oscillatory coefficients measured on low-aspect ratio wings," Wright Air Development Center Technical Report No, 53-64 (April 1953).

17. Clevenson, S a A., 2nd Leadbetter, S. A", "Measurements of aerodynamic forces and moments at subsonic speeds on a simplified T-tail oscillating in yaw about the fin mid-chard." NACA Tech, Note 4402 (1958).

18. Itodden, Wo P., and Revell. J. D o , "The statu of unsteady aerodynamic influence coefficients:' I& Fairchild Fund Paper No. FF-33 (1962).

19. Stark, V. J. Eo, Private comunication to W. P. Rodden, 11 Oct. 1967.

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I

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LINE OF \ DOUBLETS

DOWNWASH COLLOCATION POINT

FIGURE 1. COORDINATE SYSTEM, FIGURE 2. WING AND PANEL GEOMETRY

L

SUR FACE 7

v

FIGURE 3. COORDINATE SYSTEM OF EQUATION (12).

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FIGURE 4. LIFTING PRESSURE DISTRIBUTION ON WING WITH OSCILLATING FLAP IN 2D FLOW. M = O K = 1.0

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d

- 7.0

6.0

5.0

4. (

3. c

2. c

1.C

C

L

la PRESENT METHOD; NC = 6, NS = 8 1 0 EXPT, NACA RMA51G31 I

,x FIGURE 5. LIFT DISTRIBUTION ON SWEPT WING IN STEADY FLOW

0.2 0.4 1 0.6 0.8 (a) COMPARISON WITH tXPERlMENT

.o 75

x

fb) CALCULATED LIFT DISTRIBUTION ON SEMIWING FIGURE 6. LIFT DISTRIBUTION DUE TO DEFLECTED PARTIAL-SPAN FLAP

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0.2 0.4 0.6 0.8 1.0 X/C

FIGURE 7. LIFT DISTRIBUTION ONLR3 RECTANGULAR WING OSCILLATING IN BENDING MODE ( 0 ) $= 0

~~

0.2 0.4 0.6 0.8 X/C

FIGURE 7. CONTINUED. (b) = 0.9

0

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KERNEL FUNCTION (M = 0.2) REF. 15

0.5

0.4

A

-

I

A

- 80

L;i -120 A W IY A

- 0 w n

-200 ~

I I I 0.5 1.0 1.5

k-

FIGURE 8. FLAP HINGE MOMENT DUE TO FLAP OSCILLATION, FOR A 40% CHORD, FULL SPAN FLAP ON m 2 RECTANGULAR WING

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”# $’

DEGREES

EXPT, NACA TN 4402

k

FIGURE 9 SIDE FORCE COEFFICIENT FOR T-TAIL OSCILLATING IN Y A W ABOUT THE FIN MID-CHORD. M = 0.

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