The aerodynamic derivates of an aerofoil oscillating in an infinite staggered cascade

Report No. 125

3 HS DELFT

' THE COLLEGE OF AERONAUTICS

CRANFIELD

THE AERODYNAMIC DERIVATIVES OF AN

AEROFOIL OSCILLATING IN AN INFINITE

STAGGERED CASCADE

by

COPPIGENDUM

Professor Sisto has pointed out that equation (19)

y ^ ( ê ) -y^.(g)e

is valid only if m = 0 or i , and that arbitrary phase

difference can be included only if different complex

operators are used for harmonic time dependence and

for the complex velocities.

The appropriate sentences of the summary,

introduction and conclusion should be amended to;

"the theory allows for arbitrary stagger angle and

for phase differences of 0 and ir between adjacent

blades"

T H E C O L L E G E O F _A S R O N A U T I O S G R A H F I E L D

The aerodynaüiic derivatives of an aerofoil oscillating in an infinite staggered cascade

b y

-A, H, Graven, M.Sc,, Ph.D,, D,G,Ae.

SUIvlIARY

Thin aerofoil theory is used to obtain, in integral form the

aerodjToamic derivatives of an aerofoil oscillating in an infinite cascade. The theory allows for arbitEary stagger angle and phase differerice bet\7een adjacent blades of the cascade. The expressions obtained reduce, for zero stagger and for in-phase and antiphase oscillations, to knovm results,

CONTEDITS

^ae,e

Summary

List of Symbols

1. Introduction 1

2. The lift and moment equations 1

3. The vorticity distribution in the cascade 2).

4. The velocity of flcTW over the reference blade 9

5. The aerodynamic derivatives 10

3.1, The lift derivatives 10

5.2, The moment derivatives 12

5.5>

Comparison vd-th previous results 14

6. Conclusion 15

7. References 16

F i g u r e s : 1 . Cascade geometry

2 , The t r a n s f o r m e d a e r o f o i l

The a u t h o r vd-shes t o acknowledge t h e p e r m i s s i o n g i v e n by t h e Commandant of t h e Royal A i r F o r c e T e c h n i c a l C o l l e g e t o u n d e r t a k e t h e stuc3y hei'eln d e s c r i b e d .

LI^_OF_SÏ]fflgLS

a e l a s t i c a x i s p o s i t i o n measvired from midchord

c chord l e n g t l i F , F , F f u n c t i o n s of 77 and n d e f i n e d by e q u a t i o n s (54) G , G , G i n t e g r a l s i n v o l v i n g t h e F - f u n c t i o n s d e f i n e d by e q u a t i o n s ( 4 2 ) G a f u n c t i o n of G and H d e f i n e d i n e q u a t i o n (5"!) G a f u n c t i o n of G , H^, I ^ and J ^ d e f i n e d i n e q u a t i o n (57) H , H , H i n t e g r a l s i n v o l v i n g the Q - f u n c t i o n s d e f i n e d by eqioations (^^8) h b e n d i n g d i s p l a c e m e n t of t h e a e r o f o i l s 1 , 1 , 1 i n t e g r a l s i n v o l v i n g the F - f u n c t i o n s d e f i n e d by e q u r . t i o n s (54) J , J > J _ i n t e g r a l s i n v o l v i n g t h e G - f u n c t i o n s d e f i n e d by e q u a t i o n s ( 5 5 ) k r e d u c e d frequency i-^^ L l i f t p e r u n i t span M moment about t h e e l a s t i c a x i s i k

p Tx + "^

q u - i v s s p a c i n g beti'reen a d j a c e n t b l a d e s U i m d i s t u r b e d f r e e s t r e a m v e l o c i i y u l o c a l v e l o c i t y r e s o l v e d i n f r e e s t r e a m d i r e c t i o n V l o c a l v e l o c i t y r e s o l v e d noirual t o f r e e s t r e a m d i r e c t i o n X, Ê. c o - o r d i n a t e s i n s t r e a m d i r e c t i o n ; o r i g i n a t mid-chord 6 a n g i i l a r d i s p l a c e m e n t of a e r o f o i l /? s t a g g e r a n g l e y v o r t i c i i y d i s t r i b u t i o n r t o t a l c i r c u l a t i o n » TTC i/3

^

2 i ^

p d e n s i t y 0 v e l o c i t y p o t e n t i a l e l / t a r i k "K n e t a n k 7^ /i e t a n k Xx

1 • jjitrqduction

The flexure-torsion flutter of aerofoils in unstaggered cascade has

been the subject of theoretical studies by Lilley (1) and Mendelson and

Carroll (2), These authors use thin aerofoil theory to derive the lift and

moment equations for an aerofoil moving in phase or in antiphase vd.th its

neighbour, Lilley includes structioral stiffness terms and determines the

conditions for flutter to occur. Sisto (3) finds a general expression for

the vorticity at any point on the oscillating aerofoil in the fori'i of an

integral equation v^iich is solved approximately for the case of zero stagger

angle. The numerical results for the derivatives agree T/ith, the exact

calculations of Mendelson and Carroll and the approximate values found by

Lilley.

Legendre (4)» using a confomial transfoimation method, lias considered

the general case of flutter in a cascade with stagger. This is an extension

of the work of Timman (5) for zero stagger. Expressions are given for

the velocity potential and circulation from which the pressure distribution

can be calculated. Eichelbrenner (6) gives details of calculations based

on Legendre's method for one gap/chord ratio and one stagger angle. lie

simplifies Legendre' s integral expressions by the extended use of theta and

zeta functions.

The present paper uses thin aerofoil theory to extend the v/ork of

Mendelson and Carroll to include arbitrary stagger angle and phase difference

betv/een adjacent blades. An integral equation relating local velocity and

vorticity is solved and the aerodynamic derivatives are found in integral

form,

2 • The Lift and Moment Equations

Oansider an infinite cascade of oscillating aerofoils of unit

^ond-chord at zero incidence, set at a stagger angle /? and having a gap s ^^Fig. 1 ) .

The uniform velocity far upstream of the cascade is U. We shall assume

that "üie oscillations are of small an^ilitude so that velocity perturbations

are small conipared with the free stream velocity.

The eq\:iations of motion for the pertiarbed motion reduce to

9t ^ ^ Ox P ax

^^^

at ^ ^ 3 x p 9y

^^^

2

-equations (1) and (2) become respectively

axat - 2

ox

. 1 ^

*" "p ax

(4)

ayat

u ^-^

9x9y

.. 1

P ay JE

(5)

Adding (4) and (5) we have, if the operator d = -^ dx + g^ dy, dp = - p d

at * ^ ax

(6)

The difference in velocity above (u ) and belov7 (u^) an aerofoil is

• -u-^=(i^4-(-i), «

and, from thin aerofoil theory, tliis velocity difference can be repnresented by a distribution of vorticiiy along the chordline of tlie aerofoil and its wake,

u^ - u-^ = y (x,t) Thus substituting (7) and (8) in (6)

^ P = P^ - P i = -P ( u y +

(8)

f y(x,t)eoc') (9) -1 ^

and since there can be no pressure difference across the wake

u y j x , t ) + ^ / yj^»"*^) ' ^ + at / y(^»^^) <ax = 0 (10)

-1

where ^ ( x , t ) i s the v o r t i c i t y i n the \7ake,

T/

If r(t) is the total circulation about the aerofoil

'1

r ( t ) = / y ( x , t ) dx -1

and (10) becomes

3

-I f the oscillr^tion i s simple harmonic a l l q u a n t i t i e s have a time v a r i a t i o n

propoi'tional t o e and vie can escpross our equations i n t e r n s of the

^ c / o

reduced frequency k defined by k = -^i (v/hcre -,• i s the semi-chord and i s

2U

talcen as u n i t y ) . Equation (11) becomes

y (x) + ik / y (x) dx + i k f = 0

'vr ' / 'w "^

(12)

where y (x) and T are now the amplitudes of the yreke voi'ticity and circulation respooti'vely and are thus conplex quontities independent of tine,

Equation (12) can be solved for the vorticity in the wcJce in terr.is of the circulation round the wing in the form

•w f \ 'T V llc(l-x)

y (x) = -ik-L e ^ '

and equation (9) becomes

A p = - p U y ( x ) + i k y(x) dx

-1

(13)

(14)

Integrating (14), the lift on the aerofoil ia

1 1 X

L = - P U

j y(x) dx + ik /

y(C)

dC

dx

-1 -1 -1

and, if the moment is measured about the elastic axis x = a,

(15)

- P U (x-a) y(x)dx + ik I (x-a) / y(S)dg dx -1 -1 -1

z,.

-3. The vorticity distribution^ in the cascade

For the infinite cascade the induced complex velocity da = du - i d V at ary point x on the reference blade

(zeroth blade) due to the vorticity y (g) at the point z = ^ + i n s e"" ' on the n aerofoil and its wake is given by

"^V^) = -

2^x

-

z T

±13

or /.Titing \ = —^—— (with — = 1) where A i s coniplex,

r IT')

I I t i s r e a l for ^ = 0 and imaginary for ^ =- — ,;

'^y J S ) dê

d ^ ( x ) = " (17)

L ^

The coirplex v e l o c i t y induced by the complete cascade i s

\ r CO y (H)d£

-1 " - •|-'^5-x)

I f tlie phase difference betvreen t'.TO adjacent blades i s 6 = 2 Trm

(O < m < l ) then the phase difference 6 bet\7een the n blade and the

reference (n=0) blade i s

6 = 2'7r m n

n

y^(S) = y ^ ( S ) e 2 i " ^ (^5)

and

where y (S) = y ( 0 is the vorticity distribution on the reference blade,

-1

5

-Thus from (18) and (19)

^ /• '^ CO , 2±7mn

l^ n - ~ ( C.-X)

2imiin . X l - 2 m ) ( ê ~ x )

But ? ^ .^ = -^-yTc—T (Re f .7) • - n - # ( ? - x ) sanhT^TS-xl

and tlias (20) becomes

•^ / " " V?> K l - 2 m ) ( S - x )

~ ' 2ir J s i n h A^ê-x) ^ ^ -1

We can nov/ use (13) to express q(x) in terms of the local vorticity and total circulation on the aerofoil, Yfe obtain

, , Mcr .CO e^(^-^) e'^(^-2m)(C-x) 1

i X(l-2m)(?-x)

iX ƒ J < i ) „ e _ _ _ _ _ ^ ^ ^22) 2^^ j , sinhX(5-x)

a s t h e i n t e g i a l e q u a t i o n which must be s o l v e d f o r the l o c a l v o r t i c i i y y(x) on t h e a e r o f o i l i n terms of t h e p e r t u r b a t i o n v e l o c i t y q ( x ) , I f we p u t t a n k ?^x = u/e t a n k ?^ = ri/e t a n k \ = l / e •vrfiere /J, 77, <; a r e complex, t h e n (22) becomes (23) C / \mf /2X

c

1

6

-where the contour C i s the path of 77 = •r' •,• 'v'^ i n the range -1 < ? < 1

and G i s the path of 77 i n the irange 1 < g < co , P i g . 2 sho\7s tlie contour

C^ f o r the case X = e ' .

Equation (24) can be simplified by v^Titing

, ik

,m-1 , r i k r r.

^N

" ^

and the i n t e g r a l equation reduces to

i k /iJ7

QW = ^ / ^^^^^^4^7-?^-^ (25)

2 ^ / (e+r])"^-1(e^„77^) '^"^

1

77 = e corresponds to the point g = «> and hence does not l i e on the

contour C . Thus the k e r n e l of equation 25

K(„) „ ^")(^-")°'^ ^ (26)

has no s i n g u l a r i t i e s - en 0 , ,

Thus we have to i n v e r t the equ^.tion

Q(.) = ^ ƒ K(.) ^

AIL

'/J

(27)

Ci

a ^ ( t ) + - i T f f f ^ d s = f ( t ) (28)

which i s a s p e c i a l case of the general Cauchy equation

0

with a = 0 and b = 1 an.d C an unclosed continuous contour, The solution of this equation is (Ref. 8)

0

7

-. ,, - 1 T a + b

wrch.

p = u-• .' • In — — - r

^

2 T 1 a - b

and

o:

./? the first and last points of the contour C.

In the special case ocnaidered here

and /S = ^ g _ ^ = 1

thus the solution of equation (27) is

c

or, substituting back the actual vorticity and velocity distributions

k r e ^ / i£:£il '- dr^^ l /i^d-Y d^L /..N

ijk

r '

m-1

ik

Now

r = [ y(C)d5

-1

thus, using the substitution of equation (23), the circulation on the reference

blade can be expressed as

!• = i ƒ 4^. •ï" (52)

0

8 -_ 2 ^ r - Tf±k (, e+7]) /77f 1 (e-r,) m T7- 1 L C

[i^^ M

m-1 j^j,^ilc m - " 2 7 ^ .

C, (e + njP

A2i f Ü j r J .

di" ACT? dT]

which can be manipulated to give the c i r c u l a t i o n i n terms of the v e l o c i t y

over the blade and the cascade geometry i n the form

r =

v\

2e / i^^f"^ h^<^ "

1 +

(e-r?)

m y\-r\ q ( / i )

^j^"^ /jj-iNP d/L

( Ê + / ^ ) ' m VA' + 1/ A'-'? d77 i k _{(efr?)"^-^ /1 + 77N2} C

/ErlY - f ('^-^i)'' AIL

m-1 / sm-I f nov7 we w r i t e F^('7) = > -^

(e-n)

^ 1 -T] 0 ^"'"^^ z i J Ai"77

(33)

dr? 1 .

_ , , ^ ( e - r t ' ° ' ^ / (J - : ^

^ - 77 (34)

'•™

= / e

then (33) becomes

r =

( e - r , ;

Cg ( e + n ^ ^ p-1, d77 ^ -r? 1 _d77_ /J -77

H / P,(^) / q ( / i ) F (Ai,77) dAi dT7

C

1 +

e k e i k P i ( ^ ) P (^) cl^ (35)

and s u b s t i t u t i n g i n t o (31) we have the v o r t i c i t y d i s t r i b u t i o n on tlie

reference blade given i n terms of the v e l o c i t y d i s t r i b u t i o n q (A*) by

xiJlL = I p (,)

e^ n^ V 1 q(Ai) P ( M , n ) - ^ " ^ P f r ? )

-C, ~ ' '^'^ S . - ^ ^

F ( T J ) P (T?) CJ) ''••• 0

(36)

9

-4.

t i s

_The_yelocity of flovj- over t h e r e f e r e n c e b l a d e

The v e r t i c a l d i s p l a c e m e n t of a p o i n t x on t h e a e r o f o i l a t time

y ( x , t ) = h + ( x - a ) Ö (37)

^ere h and Ö a r e f u n c t i o n s of t and a i s t h e p o s i t i o n of t h e e l a s t i c a x i s

measured from mid c h o r d ,

iTe assume t h a t t h e i n d u c e d p e r t u r b a t i o n v e l o c i t y i n t h e s t r e a m d i r e c t i o n i s s m a l l enough, compared w i t h the f r e e s t r e a m v e l o c i t y , t o b e n e g l e c t e d . The v e l o c i t y n o n u a l t o t h e s u r f a c e must be z e r o ( r e l a t i v e t o t h e s u r f a c e ) a t a l l p o i n t s of t h e s u r f a c e , Thus u ( x , t ) = U v^x t') - U ^ + ^ (38)

or

v{/i,t) = n + u6 -

+ | ^ l o £ e +1-1 e -/J and

q(/j, t) = U - i [ h + u e - a ê + l^log ^gg ] (39)

i w t

Now <^f h and Ö a l l have a time v a r i a t i o n p r o p o r t i o n a l t o e c o n s i d e r i n g t h e a m p l i t u d e of t h e time dependent t e n n s , vre have

T h e r e f o r e ,

ci(Ai) h - 0 ^ a + i - 1 ^ l o g - ^ (40) and s u b s t i t u t i n g f o r q(Ai) i n (36) from (40) we o b t a i n t h e v o r t i c i t y

d i s t r i b u t i o n on t h e r e f e r e n c e b l a d e i n terms of t h e b l a d e motion i n t h e form

2 ^ TT F (r?) (h - e |a + V k ] ) J P2(Ai,T7)dM i k 2\ f , . n ^+^ ^„ k e e „ / V r (h -el a + V k ] ) G + 7 ^ G 1

] F,(A^,n) l°g — . ^ - - ^ ^ ?(^) L I — l - . ^ i - L ^ 2 . ^

G, 1 + - ^ G (41.)

10

-where

G = 1 G = 2

j Ï ; ( ^ ) J \it^,v) d/id77

ƒ F^(r?) ƒ P2(/^,n) log 1 ^ d/idrj

(42) G = 3 / P (Tj) F (77) d 77

5 , The -Ar:-r'0'5.vnar'ic D e r i i s t i v e s

5 . 1 . The_ l"..ft ^derivatives

From ('15) the l i f t per unit span i s given by

L = - pU

/ y(x)dx + i k / / y ( ? ) d ^ d x

-1 -1 '-A

or using the transformations defined in (23)

L = - PU

L I Y

(^)

C.

2 2 e - 77"^ dT7 + i k e ' .2 ^2 c;(77)' 2 _ r 2

dg dn

e - ^ nV Ne"-&'

(43)

where C'

(v)

is the part of the contour G between -1 and ^

From (41)> using the G functions defined by (42)

J±l

, 20)

_{Tv, A^ i/l^l^ ® n k e ^ _ r(h-ea- e^A)G + ^ G l '}

h - e(a+ /k) G + -Tï- G - • — G_

'

^ ' 1 2X z

1^ ^ ^ ^j 1 2X 2 ^ ^ 3 - - — - • . ^ - ^ ^ — - J

1 + ^ ^ ^ G

T^X

'^

. (44)

Similarly

2 2 ^ / e - 77''

„ 1 . ik ((h-ea . J | ) G 4|HG

h-e(a. A)l G ; . - G ^ - - ^ , ^ ^ ^ ^ ^ ^ ^ ^

(45)

1 1 -Yihere G ' (n) = G ; ( 7 7 ) / F^(S) f P^(/i,?)dAJdS

c;(^) c^

F (5) f F^(Ai,S)dAid? 0;(77)

(46)

G ; ( 7 ] ) _{P ^ ) F ^ ( ? ) d?}

c;(^)

and f u r t h e r m o r e 2 2 e - 77*^ C. c;(77)

^^^ d^n = f

e^- e

h

-e(a

+

Vk)

where H^ = G ; ( 7 7 ) 2 2 k e e ^ •• .7-X ""^ dT? ; H = 2 H +-rT H 1 2A. 2 (h-6a. - i 2 ) G + e ^ 2X 1 i k

1 +^J|-G

TT'X 3

(47)

G'(77) f Qr'iv) . ^ a n 5 H =. / ^ ^ ^ ( i ^ )

c^ ^-^ o / - ^

S u b s t i t u t i n g from (44) and (47) i n t o (43) -irpu 2__^ TT^Xk ^ , ike ^ i ö Y „ i k e ^^ kee „ / 3 A. 3 i k n r M 1 i k G + i k e H ^X|^ 2 /. 2 ^ ^ 2

(49)

The t\TC) d i m e n s i o n a l l i f t d e r i v a t i v e s a r e found by c o l l e c t i n g t h e c o e f f i c i e n t s 1 i

of - — , r and 1 in (49) . We Viorite z for h to conform vd.th the usual notation for such derivatives,

Bius, remembering that the chord length is 2

\ = ^

V P U ^ = ° ' ^5 = ^ ^ 2 p U

ie

12

-1..

z

= L

_{fe-A^ 2^X^}

ilc vAiere G. = (51) and l o =

1..

6

^V;

2pU^ L „ /o G / ' ^ P

^ V " ^^ ' h = ^ê/4pU = i ' " -

2 a G . - ~ - ( H J ^ G G ) - X 1 X e Bvr^X^

m

i k G - 2 i a H + - 2 ^ ( 2aG - - ^ ) G A 1 _ 2 ^ 1 A.' 4 7 ^

(52)

5 . 2 . The morient d e r i v a t i v e s

Prom (16) the moment is given by ^ ,1 M = - P U / ( x - a ) y ( x ) d x + i k / ( i & . a ) |

-1 -1 -^

y(5)d5 dx

o r , u s i n g t h e t r a n s f o r m a t i o n s of (23) M = - pU e X C. -1 e+77 l o g ——i - a

Yinl .^ . ike^ f ^°S T3? -^

d77 + -Y2 e - 77 '^ e - 77 J^/ e - 4 J .2 „ 2 dSdT?! ' 1

(53)

S u b s t i t u t i n g f o r t h e v o r t i c i t y from (2+4) and (45) and u s i n g t h e f u n c t i o n s I^ , Ig , I3 and J^ , Jg , J3 d e f i n e d by

h

=

ƒ P,(7?) l o g | i 2 ƒ p^(/j,T?)d^dr7

= j P^(7]) log | i 2 ƒ P^(A/,r)) iog|±gdAJd77 (54)

1

e+7)

I 3 = / P,(77) F^irj) l o g - g ^ d T ?

13 -J = n „ If s. ^ e + 7] 2 2 e^ - 77^ n = 1 , 2 , 3

(55)

t h e moment e q u a t i o n (53) becomes -TTPU M^ 2 2 6 TT^Xk

( h - 6 a - i 2 ) [ i ^ -aG^ + ^

i ( j - aH ) - - ^ G ^ G ^ i k 1 1 ^ 1 5 k - ^ 1

_ ^ > ^ - ^. + 4^ [" i ( j - aH ) - - ^ G G 1 }

2 X ) 2 2 X L ^ 2 2^ ^ 2 5 J J Ö T T P k e + TT; ) I_ - aG

(56)

where

(57)

1 i C o l l e c t i n g t h e c o e f f i c i e n t s of - — , -^ , 1 vre o b t a i n t h e moment d e r i v a t i v e s k2 ^ m. = My2PU^ = 0

_{' ^z = V40U}

i e 2 7r2X ( I , - a G j i k m„ = M., / z z / o p

f^X^ [ i ( j . -aH,) «J-GiG,

(58)

and i e

"% = V4pl? = " t ^ X ^^1 "''^^^

m, = M, _:/8pu m„ = M„

e

e

/ I 6 f 4 w^x e" STT^X^

i al

x(l^ - aGj- i ^ (I^ - Gj J- I [_i(j^ - aH^ )

i k ^ G G If" ' = L ^ i k Gg

-+ V G^(afi^ - ^ ^ )

V7^

(59)

14

-5 . 3 . Cqmpojrison vd.th p r e v i o u s r e s u l t s

The b a s i c e q u a t i o n s of t h i s pamper ( 1 5 , l 6 and 24) a r e i n a.greement, f o r t h e s p e c i a l c a s e of z e r o s t a g g e r and antiphs.se o s c i l l a t i o n , T/ith t h o s e of L i l l e y (Ref, 1 , e q n s . 2 , 1 0 , 2,11 a>-.d 2,27) and the s o l u t i o n of t h e

i n t e g r a l equa.tion a l s o a g r e e s , L i l l . y e x p r e s s e s t h e aerodynaoaic d e r i v a t i v e s d i r e c t l y i n e l l i p t i c f u n c t i o n s and f u r t h e r compai'ison of t h e tv70 p a p e r s i s n o t p o s s i b l e e x c e p t t h a t t h e p r e s e n t a u t h o r a l s o f i n d s t h a t 1 = m = 0 and 1 . = 1„

2 0 and m, =-- mo, 2

Mendelson and Carroll (Ref. 2) present their results for the unstaggered cascade oscillating in phase or in antiphase in the form of functions L, , L^, M, , M^^, v/hich shov; the dependence of the lift L and moment M on flexural displacement h and angular displacement Oj such that

L = TTpo) ( L, h +

_{Lj^h + I Let "(i + ^ ) \ I '^ J}

M = w-pcj" _{M^ - ( ^ + a)Lj^ h +}

_{M„ -(? + a)(ltt + \Xh^f\W}

(Ref, 2 eqn,B.37)

In corresponding form the results of the present paper for the special cases are „ ^ , G ieH ^ ik TT^X TT^X

k " X"

1 4 ik „. /G id-I ik \ ^2 / G .^„ ik T - ± T a. 21 e / _ j . . _L ee \ _£_ f „2. . leH £e__

TT^X ' V ^ ^

_TT^X 2 4 " ^ = - ; ; ^

( I -aG

2 e 1 1 1

i ( j - a H ) - ~ - G G

i 1 W^ 1 Ï ik e_

(60)

k [\-'%*r [i(J. - aHj

- ^ G , G J j

15

-with m = O or •§-, X r e a l and the i n t e g r a l s along C becoming i n t e g r a l s ' along the 7?-axis bet\,VBen -1 and 1 , the i n t e g r a l s along G^ becoming i n t e g r a l s along the 77_axis between 1 and ^ and C (7?) becoming t h a t paxt of the r7-axis betv/een -1 and log -^-^

e-77

I f we s u b s t i t u t e fojV the G, H, I and J i n t e g r a l s i n (6o) we o b t a i n resvilts which sho\7 s u b s t a n t i a l agreement with equations B,38, 39, 40 and 41 of

reference 2 . However Mendelson and C a r r o l l have been able to siiMolify the i n t e g r a l s

1 X j x V(?) dë dx and / (x - a) f y(5) & dx

-1 -1 -1 -1

further than the present author and hence the H and J integrals of this paper are more complicated than the corresponding integrals of refe2?ence 2.

6, Conclusion

Thin aerofoil theory can be used to find the aerodynamic derivatives of an aerofoil oscillating in an infinite ca^scade. The theory taJces account of stagger angle and phase difference betvTcen adjacent blades of the cascade, The derivatives are expressed in terms of complex integrals (except foi

the degenerate case of zero stagger and antiphase oscillation \7hcn the integrals are real) T/hich ha.ve to be evaluated along the aerofoil and its wake,

— 1 D <-7 , Rqüferences 1 , L i l l e y ; G,M, 2 , Mendelson, A, and C a r r o l l , R.W. 3 . S i s t o , F . 4 . L e g e n d r e , R. 5 . Timman, R. 6 . E i c h e l b r e n n e r , E A. 7 . Bromwich, T . J , 8 , M i k h l i n , S,G. An iiv.vestigation of t h e f l e x u r e -t o r s i o n f l u -t -t e r c h a r a - c -t e r i s -t i c s of a e r o f o i l s i n c a s c a d e . C o l l e g e of Aei'onautics R e p o r t 6 0 , 1 9 5 2 . L i f t and moment e q u a t i o n s f o r o s c i l l a t i n g a i r f o i l s i n an i n f i n i t e iJnsta-ggered c a s c a d e . NAGA TN.3263 1954 Unsteady aerodynamic r e a c t i o n s on a i r f o i l s i n c a s c a d e . J n l . A e r o . S c i e n c e s , May 1 9 5 5 . P r e m i e r s e l e m e n t s d ' u n c a l c u l de 1' a m o r t i s s e m e n t aerodynamJ.qje des v i b r a t i o n s d ' a u b e s de compr-esseurs, La Recherche Aeronautique No.37 1954.

The aerodynamic f o r c e s on an o s c i l l a t i n g a e r o f o i l betv/een tv7o p a r a l l e l v / a l l s , App, S c i , R e s . Vol,A3 No. 1 1 9 5 1 .

A p p l i c a t i o n numerique d' un c a l c u l d ' a m o r t i s s e m e n t aerodynaniique des v i b r a . t i o n s d ' a u b e s de c o m p r e s s e u r s , La Recherche Aeronautique No. k-6 1955

An i n t r o d u c t i o n t o the t h e o i y of i n f i n i t e s e r i e s .

( M a c h i a i l a n ) 192)2.

I n t e g r a l e q u a t i o n s . (Pergamon P r e s s ) 1957.

fy

/

J",

\—*^/ •*- X - I /

FIG.I. CASCADE GEOMETRY

I - o

- 0 - 5

FIG. 2. THE TRANSFORMED AEROFOIL

tanhX