An investigation of the flexure-torsion flutter characteristics of aerofoils in cascade

TECHNISCHE HOGESCHOOL VUEGTUIGBOUWKUNDE Kcmaalniaat 10 - DMiT REPORT No. 60

2 7 JUN11952

THE COLLEGE OF A E R O N A U T I C S

CRANFIELD

INVESTIGATION OF THE FLEXURE-TORSION FLUTTER

CHARACTERISTICS OF AEROFOILS IN CASCADE

by

G. M. LILLEY, M.Sc. D.I.C., A.F.R.Ae.S. of the Department of Aerodynamics.

This Report must not be reproduced without the permission of the Principal of the College of Aeronautics.

TECHNISCHE HOGESCHOOL

VUEGTU1G80UWKUNDE KcmaalstKiat 10 - DELFT

MAY, 1952 2 7 JUN11952

T H E C O L L E G B O F A B R O N /l TT T I C S C_R A H F T B L D An i n v e s t i g a t i o n 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 caEca^le -hy-G.H. L i l l e y , M . S c , D . I . C . , A . F . R . A e . S . of t h e Department of Aerodynaniics S U MM A R Y

Part 1 of this report describes the results obtained from a series of tests on the flezur-e-torsion flutter character-istics of cascades of similar aerofoils having symmetrical sections. The critical flutter speeds and frequencies of the aerofoils in cascade have been compared with their isolated

values. The investigation has included the effects of gap-chord ratio and of stagger. The Reynolds number, based on wing chord, was about 0.15 x 10 .

It vras found that the critical flutter speed decreased as the ga.p-chord ratio was reduced. The variation of critical flutter speed with stagger angle vras relatively small. In both cases the critical flutter frequency was greater t|;ian that f or the isolated aetxfoils. It wa.s noted that din-ing flutter adjacent aerofoils were oscillating approximately 180 out of php.se, a.nd hence alternate blades were in phase. The mode of oscillation was of the flexure-torsion type.

The accuracy of the experiments was limited by slight variations in the structure of the models and in the case of

the wooden aerofoils, by noticeable changes in their torsional and flex-oral stiffnesses with humidity and temperature.

MEP

-2-Part 2 of this report is a reviev/ of the theoretical studies on oscillating aerofoils in cascade. Since, as is noted in Part 1, adjacent aerofoils vibrated 180 out of phase, the problem is analoguous to that of a single oscillating aero-foil placed bet^veen parallel walls. The air forces have been calciilated approximately enabling the flutter characteristics of the aerofoils, described in Part 1, to be computed and a com-parison made v/ith the experimental results. Pair agreement has been obtained, and such differences as there are, it is suggested, are due to the neglect of the effects of finite aspect ratio and thickness of the aerofoils and the rigid body movements. The latter problem together vdth an accoiant of simplified flutter calculations are discussed in appendices.

The major part of the experimental v/ork discussed in Part 1 was reported by K. Aiming, G.E. Gadd and W.P. Yfiles in an unpublished note,

-3-I N D B X

Page

Notation 1 5

Part 1 S1 Introduction

3

§2 Apparatus 10

S3 Test Procedure i ' 11

§4 Experimental results 12

§5 Discussion

^6

§é Acknowledgements j 16

§7 Conclusions 18

Part 2 S1 Introduction

₂₀

!2 General theory 22

'3 Flutter with a single degree of freedom 36

References 38

Appendices

1. The calculation of the critical flutter

speed and frequency 40

2. Finite amplitude effects 46

Figures 1, View of the tunnel showing the unstaggered

cascade in position

2. Back of the turntable showing blocks for

clamping the aerofoils

3. Cascade of aerofoils m t h stops in position

4-. General view of the metal blade

5. View of the aerofoil moimted on the

vibration table

6. Decline in flutter speed with age

7. Critical flutter variation v/ith gap-chord

ratio

8. Flutter speed variation vdth stagger angle

/9. ...

-4-9. V a r i a t i o n of c r i t i c a l f l u t t e r frequency w i t h g a p - c h o r d r a t i o 10. V a r i a t i o n of c r i t i c a l f r e q u e n c y v d t h g a p - c h o r d r a t i o and s t a g g e r a n g l e 1 1 . V a r i a t i o n of the c r i t i c a l f l u t t e r speed w i t h g a p c h o r d r a t i o f o r the u n -s t a g g e r e d ca-sc-s.de 12. V a r i a t i o n of t h e r e d u c e d frequency w i t h g a p - c h o r d r a t i o f o r t h e u n s t a g g e r e d cascade s o 13a A e r o f o i l s i n cascade — = 0.^ a" = 1 5

c 13b • • • • 0- = 0 ° 13c I s o l a t e d a e r o f o i l a = 0 14a Mode i n f l e x u r e 14b Mode i n t o r s i o n N o t a t i o n 15b

16. V a r i a t i o n of A and B v/ith gap-chord r a t i o 17. V a r i a t i o n of t h e p a r a m e t e r q v d t h g a p

-c h o r d r a t i o

18. Variation of the functions Q., Q„, Q,, Q, 19. Typical values of the classical

aero-dynamic derivatives

20. Notation for flutter calculations

21. Typical variation of the coefficient c. with a and k.

1

-5-NOTATION A r e a l p a r t of C (see below) A. f l e x u r a l moment of i n e r t i a 1 a. n o n - d i m e n s i o n a l form of A. 1 1 A f l e x i H ' a l - t o r s i o n a l p r o d u c t of i n e r t i a a , n o n - d i m e n s i o n a l form of A_ 3 3 b = 'cos/27tc

B imaginary part of C (see below) B, direct flexural damping coefficient b, non-dimensional form of B.

1 1 B , compound torsional damping coefficient

b , non-dimensional fonn of B_ 3 3 C. flexural stiffness 1 c. non-dimensional form of C c chord

C (= A-iB) generalised Theodorsen function C , torsional cross-stiffness

c-, non-dimensional form of C ,

3 3 E (k) complete elliptic integral of the second kind

f frequency f (ri) f l e x u r a l mode

F(TI) t o r s i o n a l mode

• • c

F(a,b;o,z) hypergeometric function

G. f l e x u r a l - t o r s i o n a l p r o d u c t of i n e r t i a g. non-dimensional form of G

G , torsional moment of inertia 3

g, non-dimensional forra of G,

he distance of flex\xral axis from leading edge Pj(2) jj(2) Hankel functions

1 ' o

J. compound flexural damping coefficient J> non-dir;".ensional form of J^

1 1

-é-J, direct torsional dsjnping coefficient

3

j , non-dimensional form of J,

3 3

k 3 tanh 'A

k' =

J^ -

k^

K(k),K'(k) complete elliptic integrals of the first kind

K. flexural cross-stiffness

k. non-dimensional form of K.

1 1

K_ direct torsional stiffness

k, non-dimensional forra of K ,

3 3

L f l e x u r a l moraent

h

span

flex\;iral stiffness

2 ^' ty

aerodynamic derivatives

9 I

h

0 S •'''é aerodynamic derivatives

M pitching moment; torsional momentj Mach No. in

free stream

m mass/unit length of span

mQ torsional stiffness

m m. m» aerodjTiaraic d e r i v a t i v e s

Z 2Ï 2 m„ m* m%' aerodynamic derivatives p pressure Q Q„Q,Q, functions of q q 5 exp(-7iK'/K) q q.qpq,q, dimensional coefficients q q.q^q^q. non-dimensional coefficients r distance (see eqn. 2.22)

n

r.-CrPiV, coefficients in flutter equations

1 2 3 4

R RJl, coefficients in flutter equations

s gap

t aerofoil thickness*, time

u perturbation velocity conaponent in direction OX

V freestream velocity - isolated aerofoil

V* freestream velocity - aerofoil in cascade

w perturbation velocity canponent in direction OZ

-7-X coordinate in chordwise direction

x„ distance of reference section from origin

y coordinate in spanwise direction

Z force in direction OZ

z coordinate normal to chord

TT.

aerofoil displacement in direction OZ

a = T- tanh M X , angle of incidence', aerodynamic stiffness parameter

P = exp(7':c/s)* elastic stiffness parameter

Y bound vorticity; phase angle

Y coefficient in series Q,Q.

I" circulation

e free vorticity

y\

=

y/i

e aerofoil rotation', torsional coordinate

1 ^ H

Qc/l

(A)

radius of gyration

2s

7\^)s^/Su/ aerodynamic derivatives

7^n7^^/^y aerodynamic derivatives

O D D

|i^ |iA/ |i*A aerodynamic d e r i v a t i v e s

M-o IJ-Ó l-^Q aerodjTiamic d e r i v a t i v e s

V kinematic viscosity ? S x/c

eg distance of the centre of gravity from the leading edge

eg distance of the flexural axis from the

leading edge

coc/V frequency parameter (reduced frequency)

p air density

cr stagger angle

%

= Y + e

0

flexural coordinate' velocity potential

^ acceleration potential

0) circular freqioency

CO-, natiiral frequency in flexure

8

-w, n a t u r a l frequency i n torsion

Suffix c denotes the value of a quantity a t v/hich f l u t t e r i s

j u s t maintained,

Suffix 0 denotes the free stream value.

TECHNISCHE HOGESCHOOL

VLIEGTU1G80UWKUNDE

Kcmaalstiaat 10 - DELFT

-9-5l. Introduction

The effect of the interference of adjacent blades in a cascade of aerofoils in modifying the isolated critical flutter characteristics of the aerofoils has received little attention. A recent paper by Bellenot and Lalive d'Epinay

(reference l) describes some tests on cascade flutter made at one gap-chord ratio over a range of stagger angles. They found that the modes of vibration during flutter were either pure torsion or pure flexure and these are therefore different from the type of flutter investigated in this report. In Part 2 of the present paper the problem of flutter with one degree of free-dom is discussed.

An experimental investigation of the flutter characteristics of aerofoils in cascade has been conducted in the Aerodynamics Laboratory of the College of Aeronautics

between 1948 and 1950. Two types of model aerofoils have been losed in these experir.ients

(a) Aerofoils manufactured from a light wooden framework covered with doped silk

(b) Rigid metal aerofoils supported from combined flexure and torsion springs at the root.

The two types of aerofoils had approximately the same chord but the spans v/ere different. It was found, however, that there was in the main qualitative agreement between the two sets of results. Therefore, in order to avoid confusion and \mdue repetition, and noting that the aerofoils of type (a) are more allied to practical aerofoils, only the results obtained from

type (a) aerofoils will be presented here. The small differ-ences in the flutter characteristics obtained between aerofoils of types (a) and (b) have not been caapletely explained but it is considered that these differences are probably due to the variations in the end fixing, the modes of vibration and the aspect ratio.

The accuracy of these experiments was limited for reasons which will be discussed.

The major part of the experimental work, relating to the wooden aerofoils, described in this report was reported by K. Aiming, G.E. Gadd and ''.''.P. Wiles in an unpublished note^ The experiments on the metal aerofoils were completed by E.S. Farris, E.T.B. Smith and C.G. Hughes.

• ^ £ ^ • • t

critical value. The frequency could not, however, be conven-iently obtained corresponding to the critical wind speed. Con-sistent readings of the critical frequency were, however,

obtained by measuring the frequency at each steady vdnd speed above the critical wind speed. Since the latter was

obtained as stated above the critical frequenqy could easily be obtained by extrapolation (see figure 9 ) .

The aerofoils v/ere then selected so that a cascade of blades could be fotmd such that the isolated characteristics

of the aerofoils differed by less than + 5 per cent. The aerofoils were arranged in cascade so that the weakest aerofoils were near the centre. The critical flutter speed and

-10-§2. Apparatus

The experir.ients were conducted i n a blower type vdnd

tunnel, \7hose vrorking s e c t i o n dimensions were l 8 . 7 5 i n . x 8.75in.

and the speed range was zero to 170 feet per second. The d i s

-t r i b u -t i o n of ve].oci-ty across -the working s e c -t i o n ou-tside -the

boimdary l a y e r v/as uniform t o v d t h i n +_ 0.5 per cent.

The a e r o f o i l s -nere cantilevered from a turntable i n a

side wall extension to the vdnd tunnel contraction (see figures

1 and 2 ) . The wooden a e r o f o i l s were of rectangular planfom,

3in. chord and 8in. span, and \rere of NACA 0010 section. Each

a e r o f o i l had a mahogany spar 0.15in. square and eight mahogany

r i b s each 0.1 Oin. t h i c k . The framevrork was covered '.idth s i l k

which v/as doped vdth a mixture of vaseline and chloroform. The

blades vrere provided with stops (see figure 3) 'in order to l i m i t

the amplitude of the blades during v i b r a t i o n .

The metal a e r o f o i l s were f i r s t made of s o l i d l i g h t

a l l o y . They had a 14 per cent thick symmetrical section, a

chord of 2.9in. and a span of 2.9in. The blades were fixed t o

various forms of spring hinges connected to the working section

t u r n t a b l e . The f l e x u r e - t o r s i o n springs were designed so t h a t

the natural frequencies of the blades i n flexure and t o r s i o n

were nearly the same as those of the vrooden a e r o f o i l s .

Although many d i f f e r e n t types of springs were t e s t e d they a l l

-12-were in all cases set at zero incidence relative to the upstream direction. A few measurements were, hoY/ever, made at 5 inci-dence and since no change in the flutter characteristics could be detected it v/as assunied that the blade incidence v/as not a very important parameter at least in the range + 5 .

It was noted that when fluttering, adjacent aerofoils v/ere approximately 180 out of phase, and hence ^Adth zero stagger, they might be regarded as images of each other in a rigid plane boundary midv/ay between them.

The above measurements v/ere, therefore, repeated for the case of a single aerofoil placed midway betv/een tv/o parallel plates. The gap between the plates was varied; the distance apart of the plates being assumed to correspond to the gap between adjacent aerofoils when in cascade, but with zero

stagger.

§4. Experir.:ental results^

The fall of the critical flutter speed with age is shown in figure 6. The temperature and humidity variations have also been plotted on figure 6 and it is seen that little

correlation was obtained with the changes in the flutter speed. This does not necessarily indicate that temperatiore and hmaidity

do not affect the flutter characteristics but rather that fatigue of the flexible v/ooden structure was probably predaninant.

Further tests did in fact show that the elastic stiffnesses, especially the torsional stiffness, v/ere altered by changes in temperature and huraidity. It was also found that the porosity of the silk covering v/as not always uniform and extreme care was required in applying the chloroform-vaseline dope. It was

-13-compared vdth the mean speed in the gaps beti.veen adjacent aero-foils. It is clearly seen that the very large decrease in the critical flutter speed with gap-chord ratio cannot be entirely due to a blockage effect.

The resiolts of the two tests can be expressed by the folloTrdng empirical law

V' c

V 1

-c (l+s/-c)'' where

n = 4.24 for the aerofoil in cascade

= 3.15 for the single aerofoil between two parallel plates

V' = the critical flutter speed of an aerofoil in cascade V = the critical flutter speed of the isolated aerofoil

c

s/c = gap-chord ratio.

The effect of the cascade stagger angle on the critical flutter speed of the aerofoils is shov/n in figure 8. It is seen that the critical flutter speed is nearly independent of stagger angle,

The frequency-speed curves for different gap-chord ratios of the aerofoils are shown in figoores 9a and 9b; the former shows the res-ults obtained from a single aerofoil placed between two parallel plates whilst the latter shows the results for an unstaggered cascade of five aerofoils. The dotted lines show the variation of frequency with vdnd speed v/hen the aero-foils are fluttering above their critical m n d speeds. The blacked in points correspond to the extrapolated critical frequencies (see paragraph 3 above) at each gap-chord ratio. It is seen that again the results are similar in the tv/o cases and that the critical frequency of the aerofoil in cascade is only slightly greater than that of the single aerofoil between plates. The corresponding effects vdth cascade stagger angle are shov/n in figure 10. The critical flutter frequency increases with stagger angle althoiigh the increase is not

large for stagger angles below 20 . It v/as observed that for small gap-chord ratios the frequency increased rapidly for small increases above the critical flutter speed. This fact probably accounts for the scatter of the observed points for s/c equal to 0,25 in figure 10.

-14-The experimental results for the unstaggered cascade are compared vdth the theoretical values, obtained from Part 2, in figures 11 and 12. In figure 11 the square of the critical speed ratio has been plotted against gap-chord ratio. Good agreement betv/een theory and experiment is obtained except at the smallest value of gap-chord ratio. This is not surprising since the finite amplitude and the effect of the aerofoil thick-ness, i=Mch have both been neglected in the theory, vdll increase in importance as the gap-chord ratio decreases. In figure 12 the sqimre of the reduct-d freq\K:ncy ratio has been plotted

against s/c. Good agreement betiveen the observed and theoretical values is again obtained except at the lowest value of gap-chord ratio. This agreement betv/een theory and experiment is encour-aging but not conclusive, since the theoretical values do not agree so well with th3 results obtained from the tests on the single aerofoil betv/een parallel plates. Nevertheless, the theory shoiild apply equally well to both systems, provided that the aerofoils in cascade are vibrating exactly in antiphase, and hence further investigation of these differences is desirable,

Figure 13 shows a series of photographs taken in

stroboscopic light for a cascade having a gap-chord ratio of 0.5 and tv/o stagger angles cr = 0 and 15 . The corresponding pictures taken for an isolated aerofoil are also given. The antiphase motion betv/een adjacent aerofoils is clearly indicated. The motion, in detail, can be described as

follows.-(i) An aerofoil in its mid-position and moving upvards say, has a positive t-m.st i.e. its leading edge is above its trailing edge.

(ii) As the motion progresses the twist is reduced until

/at the ,,.

+ These differences may be accounted for as follows,-(i) The aerofoils in a cascade had isolated flutter character-istics which differed by about + 5 per cent, and experimental inaccuracies of at least + 5 per cent will therefore exist.

(ii) In the case of tloe single aerofoil oscillating betaveen parallel plates the boundary'' layers adjacent these plates may produce important changes in the flow even though a rough calculation has shown that the equivalent blockage effect is small.

-15-at the point of maximum flexure the tv.dst is approx-imately aero or slightly negative.

(iii) As the aerofoil starts its do--.'nv/ard path its tv/ist becomes more negative reaching a maximvim at about the

mid-position of the flexural displacement.

(iv) As the flexui'al motion proceeds dov/nv/ards the tvdst of the aerofoil is redixsed and reaches zero or a slightly positive value at the position of majcimum negative displacement.

(v) Whilst the motion described in (i) is taking place the aerofoil above is moving dov*nv/ards vdth negative tvdst which is decreasing as the flexural motion progresses.

(vi) Similarly the aerofoil belov/ is also moving dov/nwards v/ith negative tvdst which is decreasing as the flexural motion progresses.

Hence alternate aerofoils in a cascade, both unstaggered, have similar motions. Their motion corresponds to the classical flexure-torsion vibrations in v/hich the torsional motion lags behind the flexural motion (see reference 2),

The type of motion discussed above was present for all arrangements of aerofoils except that at angles of stagger above 35 the flutter amplitude did not remain constant. In this case a pulsation of the aerofoil was superimposed on the steady

oscillations. The reasons for this require further investigation,

The measured elastic stiffnesses and the natural frequencies of \mcoupled flexural and torsional vibrations in still air varied for each aerofoil in the cascade. Typical values for the central aerofoil in the axscadc together vdth its isolated flutter characteristics are given below in Table 1.

/Table 1 ...

+ 'Pulsating flutter' of the type encountered at large angles of stagger is probably due to the disturbances created by the oscillatory wakes affecting the motions of adjacent

aerofoils. In the case of small angles of stagger the fluttering aerofoils are moving tov/ards the surfaces of adjacent aerofoils, but at large a.ngles of stagger the fluttering aerofoils are moving during one half of their motion towards relatively undisturbed air, v/hilst on the other half thejr are moving towards the dis-turbed v/akes of adjacent aerofoils.

-16-TABLE 1

Quantity Flexural stiffness

Natural flexin-al circular frequency

Distance of flexural axis from the leading edge Torsional stiffness

Natural torsional circular frequency

Critical flutter speed (isolated aerofoil) Critical flutter circiilar

frequency Reduced frequency Reynolds number Symbol

I

'0

tOr,

he

m

_e

w.

Measured Value

2.49 l b . f t . / r a d i a n

152 r a d . / s e c .

0.25 c

0.15 lb,ft./radian 326 rad./sec. ^ c Ü)

c

c

V

87,0 f . p . s .

220 r a d . / s e c .

1.0

1.37 X 10^

From the res\iLts quoted in Table 1 above and the results plotted in figure 9 it can be seen that the critical flutter frequency increases towards the natural frequency in torsion as the gap-chord ratio is reduced.

The modes in flexure and torsion, obtained from static tests, are shov/n in figvires 14a and 14b respectively.

§5. Discussion

The main reason for the decrease in critical flutter speed with gap-chord ratio arises from the increased negative value of the aerodynamic torsional-stiffness derivative, (see Part 2) even though the corresponding variations of the aero-dynamic torsional and flexural clamping derivatives are many times greater. It vms first thought that the reduction in the critical flutter speed was due to the aerodynamic forces and moments which arise when the displacements of adjacent aerofoils

in the cascade are not infinitesimal. It is shov/n in Appendix 2 that these forces and moments arising from rigid body movements £ire inversely proportional to the gap-chord ratio. The

-17-numerical value of the results given in Appendix 2 are unlikely to be correct owing to the drastic nature of the assumptions used. However, even v/ith these values of the aerodynamic derivatives added algebraically to those calculated from the

'classical theory' of Part 2, the critical fLvitter speeds and frequencies v/ere changed but slightly from the values calculated using only the derivatives of the 'classical theory'.

The aerodynamic forces v/hich arise due to the rigid body movements (even though these may be relatively small

com-pared vdth, say, the gaps between adjacent aerofoils) are important hov/ever in controlling the type of antiphase flutter occurring betv/een adjacent blades in a cascade. The aerodynamic forces and ensidng motions probably arise as follows. ^'Jhen a given aerofoil in a cascade is vibrating v/ith a harmonic motion of small amplitude in an othervdse steady airstream, the air vel-ocity over its upper and lo^.ver surfaces vdll respectively in-crease and dein-crease as it pursues say the upv/ard part of its motion, since if vre assume that initially the adjacent aerofoils, above and below it, are at rest the effective upper and lower gaps vdll respectively decrease and increase due to the aerofoil motion. But the increase in velocity over the top surface of the given aerofoil v/ill also exist over the lower surface of the adjacent upper aerofoil on v/hich in consequence a normal force in the dov/nward direction v/ill be induced. Similarly a dov/n-v/ards induced fcrce v/ill be exerted on the adjacent lov/er aero-foil. These induced forces Vvdll be sinusoidal and vibration of these aerofoils vdll therefore be excited by the oscillations of the parent aerofoil and the motions of these aerofoils vdll be in the opposite phase to that of the parent aerofoil. It appears therefore that the structurally weakest aerofoil in a cascade of aerofoils vdll commence fluttering at a critical speed determined by its elastic stiffnesses, inertias and.the aerodynamic

deriva-tives calculated from the 'classical theory' (see Part 2). This aerofoil will in turn excite the adjacent aerofoils.

The corollary to be piained from this explanation is that the critical flutter speed of a cascade of aerofoils is that corresponding to the structurally weakest member in the cascade provided that the variations in the elastic stiffness are not very large,

The variation of the critical flutter speed v/ith gap-chord ratio, as determined in this paper, is not a \miversal

-18-curve for all cases of aerofoil geometiy, elastic stiffness, and inertia. The results, as quoted, apply only to the particular type of aerofoil tested and each particular arrangement requires a separate investigation.

The good agreement between theory and experiment for gap-chord ratios greater than 0.5 suggests that future v/ork on the compressible flov/ problem at high subsonic Mach numbers, follov/ing on similar lines to that suggested in Part 2 of this paper, is v/orthy of consideration.

s6. Acknov/1 edgements

This v/ork v/as initiated by Professor W.J. Duncan who directed it in the important early stages. Diiring the course of the investigation stimulating discussions were held vdth Professor A.D. Young and other members of the staff of the

Department of Aerodynamics. i

Thanks are due to Dr. S. Kirkby v/ho checked certain sections of Part 2 of this paper and to Hr. S.W.Ingham v/ho was responsible for sane of the numerical calculations.

The wooden aerofoils v/ere maniofactiared by Hr.C.D.Bruce and Mr. S.H.Lilley constructed the metal aerofoils and spring arrangements. Some of the later experimental data on the

wooden aerofoils v/ere obtained by Messrs. A.R.McLean and J.Bowles,

§7. Conclusions

1. Experimental resiolts have shown t h a t when a e r o f o i l s a r e placed i n cascade, a t small incidence, t h e i r c r i t i c a l f l u t t e r speeds are reduced compared with t h e i r i s o l a t e d v a l u e s . Similarly the frequency of the f l e x u r e - t o r s i o n v i b r a t i o n s , a t the c r i t i c a l f l u t t e r speed, i n c r e a s e s as the gap-chord r a t i o of the a e r o f o i l s i n the cascade arrangement i s reduced.

2. Adjacent a e r o f o i l s i n the cascade v i b r a t e i n a n t i p h a s e . Hence a l t e r n a t e a e r o f o i l s have s i m i l a r motions. 3. The frequency of the f l u t t e r i s i n general nearer to the

t o r s i o n a l n a t u r a l frequency than to the f l e x u r a l n a t u r a l frequency.

1 9

-The v a r i a t i o n s i n the c r i t i c a l f l u t t e r speed and frequency

\ d t h stagger angle (at small angles of incidence) are

r e l a t i v e l y unimportant f o r stagger angles belov/ 30 ,

At stagger angles g r e a t e r than 35 constant amplitude

f l u t t e r could not be maintained and the steady o s c i l l a

-t o r y mo-tion v/as dis-turbed by p u l s a -t i o n s probably

o r i g i n a t i n g from adjacent v/akes.

P a i r agreement betv/een theory and experiment has been

obtained. In viev/ of the p r a c t i c a l iiaporta.nce of t h i s

work, as for example i n connection vdth the design of

blading i n cjcial compressors, i t appears desirable t o

extend both the range of the experiments and the theory

to high subsonic Ilach numbers.

20-PART 2

SI. Introduction

As far as is known to the author no theoretical pampers (apart from reference 1) have been published on the flutter of aerofoils in cascade. It appears, however, in the light of the experimental results reported in Part 1, that the general theory of the flutter of aerofoils in cascade can be simplified, in a restricted sense, in viev/ of the antiphase motions of adjacent aerofoils. The flutter characteristics can, hov/ever, only be calculated v/hen the aerodynamic forces and moments on the oscillating aerofoil are known.

If v/e consider the air flow past a cascade of oscilla-ting aerofoils in antiphase motion (see figure 15b) it can be seen that the flow about the mid lines, A'A' and B'B', betv/een adjacent aerofoils, vdll be symmetrical for all positions of the aerofoils. The lines such as A'A' and B'B' are therefore streamlines of the motion and can thus be replaced by solid boundaries.

The flow around oscillating aerofoils in cascade at zero stagger is therefore equivalent to the tunnel v/all inter-ference on a single oscillating aerofoil, provided that adjacent aerofoils in the cascade have antiphase motions.

The three-diniensional problem of tunnel v/all inter-ference on an oscillating aerofoil has been investigated by YiT.P. Jones (reference 3 ) . This theory is based on the vortex

sheet method v/hich replaces the aerofoil and its wake by suit-able distributions of doublets, satisfying the follovdng boundary conditions.

(a) The velocity at the trailing edge is finite, (b) The normal induced velocity at the aerofoil, due

to the doublet distributions, is equal to the normal components of the velocity of the aerofoil. ,

(c) The normal velocity at the walls is aero. The calculation of the airloads, using this method, is very lengthy and unfortunately numerical values are quoted only for one height-chord ratio, which is considerably greater than the val lies of the gap-chord ratio of interest in this investigation.

2 1

-The corresponding problem i n tv/o-dimensions has been

i n v e s t i g a t e d by Reissner (reference 4) and Timman (reference 5 ) ,

for incompressible flov/ and by Riinyan and Watkins (reference 6)

for compressible flov/,

The e s s e n t i a l d e t a i l s of the 1r.70-dimensional

incan-p r e s s i b l e theory i s incan-presented below and the a i r l o a d c o e f f i c i e n t s

are given i n a form such t h a t rapid c a l c u l a t i o n i s p o s s i b l e .

Only a b r i e f review of the e s s e n t i a l r e s u l t s , quoted i n the main

i n reference 4 and 5 i s given. The method of presentation,

hov/ever, has the advantage t h a t a. c l e a r physical pict\ire i s

obtained of the e s s e n t i a l feature of the theory and simple r e s u l t s

can be obtained for the values of c e r t a i n a i r l o a d c o e f f i c i e n t s

as the gap-chord r a t i o approaches zero. As v d l l be shov/n i n

Appendix 1 i t i s s u f f i c i e n t t o c a l c u l a t e the v a r i a t i o n of the

s t i f f n e s s d e r i v a t i v e s vdth gap-chord r a t i o , i f approximate

values only of the f l u t t e r c h a r a c t e r i s t i c s are required.

The extension of the incompressible theory t o subsonic

compressible flov/ i s not considered i n t h i s paper. The theory

developed i n reference 6 i s not i n a form s u i t a b l e for the

evalviation of the a i r l o a d c o e f f i c i e n t s . An important r e s u l t

obtained, however, i n reference é i s t h a t the normal induced

v e l o c i t y a t the a e r o f o i l becomes i n f i n i t e fcr c e r t a i n values of

w s / c . This resonant condition corresponds t o values of the

frequency parameter given by

^ 9^f^-M^

CO =

^ M c

where M is the freestream Ifeich number. According to this criterion the circular freqijency o) is infinite for an incom-pressible fl\dd but it has finite values in a comincom-pressible fluid when M equals zero. These results are mentioned here since

they may have an important bearing on the theory of cascade flutter a.pplied to subsonic and supersonic compressible flov/.

The theories outlined above only apply to the case of a cascade of aerofoils at zero stagger and zero incidence. The extension of the theory to other cases is being considered. The resiilts disc\:issed in Part 1 give the order of the variations involved, at least, for the case of stagger.

The calculation of the critical flutter speed and frequency is straightforward once the aerodynamic derivatives

-22-and the stmactixral coefficients have been evaluated. The class-ical treatment of this problem in the case of flexure-torsion flutter is given in Appendix 1. It is important, however, not to overlook the fact that in reference 1 flutter vdth a single degree of freedom ^was experienced. This problem is also dis-cussed belov/.

S2. The aerodynamic forces on an oscillating tv/o-dimensional aerofoil in cascade in incompressible flow,

2.1. General theory

The axes of the fixed coordinates OX and OZ are taken as sho'vn in figure 15- Thé origin of coordinates is at the midchord of aerofoil (o). The aerofoils, which are assumed to be infinitely thin, are oscillating vdth constant infinitesimal amplitude,

Let the components of velocity at (x, z) in the directions OX and OZ respectively be

V + u, w

where the perturbation velocities u,v/ are small compared vdth the freestream velocity V. From the equations of continuity and motion for an inviscid and incompressible fluid, v/hen second order terms are neglected, it can be shown that

V^0 = 0

2,1

v^l

= °

2.2

v/here 0 is the perturbation velocity potential, 0 = is the acceleration potential, p and o denote the pressure and density respectively and suffix o denotes the free stream value.

Bernoulli's equation for the unsteady motion of an incompressible fl\dd when second order velocity components are neglected becomes

d0

„ p ^o

dt 0 0

2.3

which can be written

i-[Tt*^h)>i ^•'^

/if the

TECHNISCHE HOGESCHOOL

VUEGTUIGBOUWKUNDE KaaaalatKiat 10 - DHJT 2 3

-If the suffices + and - refer to the top and bottom siurfaces of the aerofoils and their associated wakes then,

(It - ^ s ) k* - O ' ^ - Ï- " ^^^^-^ ••••'•'

=

0 1^.x,s:oo 2.6

since the pressure is continuous across the wake,

It is convenient in the further development of the theory to define Y (x,t) as that part of the vorticity associated vdth the pressure loading and (Y+S) as the total vorticity,

associated -vdth the velocity difference, across the aerofoils. We vdll refer to Y (X, t) as the boiind vorticity and s (x, t) as the free vorticity. Using the above definitions it follows that

J^ - J_ = V Y 2.7

and u_^ - u_ = -^ (0^ - 0_) = Y + S 2.8

The condition for finite loading at the trailing edges is satisfied, by Y ( ° / 2 ) eq\ials zero. It follows, from equation 2.8, that the condition for finite velocity at the trailing edges is that e (c/2) shall be finite. Since the aerofoils are infin-itely thin the velocities around their leading edges vdll be infinite. Hence Y ( - C / 2 ) vdll be infinite. By definition the valioes of e (-c/2) and e (oo) are zero.

It follows that the total lift force, and the total pitching moment about the reference section, x = x„, on each aerofoil at time t are respectively eqi;ial to

r(t)

=

nc/2

(Y+e) dx 2.9 -c/2

+ In two-dimensional steady aerofoil theory the free

vorticity, as defined above, is of course everywhere equal to zero, In two-dimensional imsteady aerofoil theory the free vorticity over the aerofoil and its wake is due to the time variation of the bound vorticity.

2 4 -Z ( t ) = pV I Y ^ 2.10

J-c/2

nc/2

- c / 2 - M(t) = pV Y(X-XP dx 2.11

I f the a e r o f o i l s a r e i n simple harmonie motion -vith circvilar frequency co and s i m i l a r motions a r e assimed for t h e i r v/akes, and i f we w r i t e a l l time v a r i a b l e q u a n t i t i e s as Y = Y e » s = ë e e t c . where Y and ë are complex q u a n t i t i e s , then i t follows from equations 2.6, 2.7 and 2.8 t h a t

0 - 0 = ^ e 2,12

Y + e = ^ 1 ^ 2,13

CÜ Ö X

and Ye = — -r" (e e ' ) . . , 2.14

Ü) d x '

If we integrate equation 2.13 vdth respect to x between the c c

limits -r and - •r then from equation 2.9

r(t) = f^e(f ,t) 2.15

Thus the instantaneous value of the circulation aroiJnd each aerofoil is proportional to the free vorticity at the trail-ing edge.

In addition the condition that the total vorticity is zero becomes

(Y+S) dx = 0 2.16

/and ...

+ It is to be understood that only the real (or imaginary) part is finally taken.

* , Or the rate of change of circulation around each aero-foil is equal to minus the product of the free stream velocity and the free vorticity at the trailing edge.

-25-and since Y equals zero in the we.kes

r\o^

nc/2

e dx = -c/2

(x+e) dx = - V (t) = - ^

e(§,t) 2.17

=/2 Ü) by virtue of equation 2.15.

Thus the instantaneous valve of the total free vorticity in the wake is equal in magnitude, but opposite in sign, to the instantaneous value of the circulation around the aerofoil.

Again if we use the condition that Y equals zero in the v/akes, equation 2.14, when integrated vdth respect to x gives s (x, t) icoxA 1Ü) V t/-c/2 icox/V, c ^ ^ c Ye ^ dx - 2 :^ x ^ ^

= e(§,t) e^"'°/2V

_{• ^ < : x < < ? o} 2.18

Alternatively from equation 2.13

and

0^

(X, t) = - :^ ( Y + E ) dx.' -c/2 nc/2

e(f,t) =

ICO V ( Y + S ) dx -c/2 2.19

The amplitudes of the lift force and pitching moment become respectively on each aerofoil

01

z

•pVc ë(x')dx' + i ^ ^ -1 U) 01 M pVc' 1_

4

(X'-Xi) S(x')dx' + i ^ ^ (I-Xl) - -^f-, 01 -1 2 Ü) f ^ 2 -of ë(x')dx'

a.i

,2.20 /where ,».

-26-where ÜT = ~ denotes the non-dimensional frequency parameter and dashes denote values of x divided by c/2,

The discontinuities in 0 across the surface of the aerofoils and their wakes can be represented by distributions

of doublets of strengths equal to the local discontinuity. Thus CxD no^ 0(x,z,t) = ^-^ ^ ^ " ^ T(X.,z ,t) 1 * n' ' T — log r dx dx. dz ° n 1 -c/2

n

'-03 2.21 v/here x = Y+e . 2 / >,2 / N2 r^ = (,x-x^) + (z-z^)

and suffix n refers to the nth aerofoil from the axis OX. •".Then adjacent aerofoils are oscillating in antiphase vdth small amplitude T (x., z , t) = (-1) T (x., z , t) and on

differentiating equation 2. 21 with respect to z and taking the lindt as z tends to zero, v/e find that the normal induced velocity adjacent to the zeroth aerofoil is

^oo w(x,0,t) = ^

x(x^,0,t) dx^

-c/2 ^^"^ Ï ^^-^l"^

,2.22

where s is the gap between adjacent blades. But v/(x,0, t) must equal the normal velocity of the oscillating aerofoil. This can be written in terms of ,6 its displacement and rotation respectively about the reference axis as

w

(-'^) = Clt^^li) (?^(--f)^3

- v [ ^

+ e ff (x'-x') + il

/Ö?i

2.23

The following relations are used

OX -r—T— log r dx = — r — ÖZ ÖZ * n 2 n r CX> n z = ns n o^ and cosech y = „ ^ ^ (-1) 2 2 2

-27-On equating equations 2.22 and 2.23 and dividing through by e we obtain finally

w

(x') = iw?-'

' iür'(x'-xp

+ 1 -\Ö^ V =

2ii

2% U T ( X ^ ' , 0 ) dx^'

_^ sinh

?V(X'-X^') ,2.24 = -r— a n d d a s h e s d e n o t e q u a n t i t i e s d i v i d e d by c / 2 . (\oc poo p 1 I f v/e f u r t h e r vnrite

U

from equation 2.18

-1

M U-1 and substitute

w

(X) =

2i

2% n1 -' "^^^1^ -'^1 >^g(1) sinh /\ (x-x.) 2-K

-1

(7»

1

expl

_{< - 1}

sinh

- ^ • • •

/ " i

x-x^)

.2.25

where, for convenience, the dashes on x have been omitted. Apart from changes in notation, equation 2. 25 agrees vdth equation 68 of reference 4- On making the substitutions

tanh A = k tanh ./Nx = ka tanh 7\x. = ka^ equation 2.25 becomes "^ T (a^ ).da^ 2.26 w(a) /^ , 2 2 N/ 1-k a 1 2 T: -1 ( a - a ^ ) N / I - k ^ a ^ 27t 1

exp f^ \1 - -^

J da^

1 ( a - a ^ ) y 1 - k a^ 2.27 The s o l u t i o n of t h e i n t e g r a l e q u a t i o n 2. 27 c a n be o b t a i n e d by t h e methods s u g g e s t e d i n r e f e r e n c e 4 . A f t e r a p a r t i a l i n t e g r a t i o n t h i s becomes

If g(x) = - i

A^

^ ^ t h e n f -1

« = K ¥ ' J . / ^ / ^

-28-H-) / m ._2 f ,/i:!i

1-k a U-1 w(a^ ) da^ 1 ^ 1 ( a - a ^ ) Y I - k a^

üil

1 „. A / . ~ ' n t a n h ka^ \ )1/k / :- ^,_. / 10) . " _{ya^+1 exp \ ^ ~ M} -V T T T

y ^1

ü^

1 _{( a - a ^ ) V l - k a^} ,2.28 The v a l u e of ê ( l ) i s o b t a i n e d by i n t e g r a t i n g T (a) v d t h r e s p e c t t o a between the l i m i t s 1 a n d -1 and s u b s t i t u t i n g from e q u a t i o n 2 . 1 9 . A f t e r some r e a r r a n g e m e n t i t can be shov/n t h a t ,

i\ r

ë(l)

= -

2k I jlZl

1+a^ w(ci^) / / l -k a^ J.

T

_{V 1+a} da U - 1 ( a - a , ) V 1 - k a'^ da. 2 i

sr

k

nVk

A/ a -1 exp

(rC-• (rC-• - ^ t a n h " ka '1

SZZ-i

/ r 12 2

A/1 - k a^ l - g l+a da -1 ( a - a , ) V I - k a da, 2.29 The a m p l i t u d e s of t h e l i f t f o r c e a n d p i t c h i n g moment can be found d i r e c t l y from e q u a t i o n s 2 . 1 9 , 2 . 2 0 , when T i s s u b s t i t u t e d from e q u a t i o n 2 . 2 8 t o g e t h e r w i t h 2 . 2 9 .

I t i s p o s s i b l e t o s i m p l i f y t h e above i n t e g r a l s f o r t h e c a s e of s m a l l g a p - c h o r d r a t i o s s i n c e when s / c s C < ' 1 , k.<^^f and A i s v e r y l a r g e . The v a l i d i t y of t h e r e s u l t i n g e x p r e s s i o n s i s l i m i t e d however t o v a l u e s of t h e g a p - c h o r d r a t i o below 0 . 5 .

More g e n e r a l s o l u t i o n s t o the above i n t e g r a l s can be o b t a i n e d by t r a n s f o m d n g them i n t e r m s of Jacobean e l l i p t i c

f u n c t i o n s . The f i n a l e x p r e s s e i o n s o b t a i n e d a r e i n agreement v d t h t h e s o l u t i o n o b t a i n e d i n a d i f f e r e n t way by Timman ( r e f e r e n c e 5 ) .

/ O n l y . . . In particular in this region the airload coefficient Zg = Z = ' ^ K2(k)

7^ A

* Timman u s e s t h e method of confoiraal t r a n s f o r m a t i o n . I t i s an e x t e n s i o n of T h e o d o r s e n ' s method f o r t h e t w o - d i m e n s i o n a l i s o l a t e d o s c i l l a t i n g a e r o f o i l .

-29-Only t h e f i n a l e x p r e s s i o n s f o r t h e a i r l o a d d e r i v a t i v e s v d l l be q u o t e d h e r e . The a i r l o a d c o e f f i c i e n t s v / i l l be e x p r e s s e d a s f o l l o w s %oóV

—2 = ^12 'i ' hk '

,2.30

^ = l l . 5 +ll„ ë

TCpC 2^2 - '"12 c " ^ 34 where Z,„ = Z. + iZ2 M 2 = M, + ii'l2 e t c

an^- -M . now r e f e r s t o t h e a m p l i t u d e of t h e p i t c h i n g moment a b o u t t h e mid-chord a x i s . Then, ~ .'V l 6 s -Z, „ = 10) Q , 1 2 71:0 1 i + (C-1) \ 2kK> 16 0) s Q,

2 T"

•3t C t • f . • <^. . 9 1 •'34 7CC 1

i + (c

_{V 2kKy_}

.-f 32s •*• ^"^ 2 2 •jt c Q + - - ! - ^ (C-1 )7c'^ _ 2kK^ 2.32 M '12 ./s.'32s „ - iü) * ~ - ^ Q, "K C

*^<=-<^-e)l ••••^•"

M 34 2 2 ^3 7C C

i +

( C - 1 ) ( 1 ^ 2kK .-..2.34 ^ 2 3 2 s ^ „ ,r^ 32s-^ „2 /_ ,>, 0) - y - j - Q, - 10) 5 — , Q (C-1) A ^ ^ T^k K^ c^ ^

where t h e g e n e r a l i s e d Theodorsen f u n c t i o n C((ï,—) S A-iB =

+ Timman ( r e f e r e n c e 5) e x p r e s s e s t h e a i r l o a d c o e f f i c i e n t s i n t e r m s of a g e n e r a l i s e d Ki-issner f u n c t i o n Tf ^ , k ) = 2C-1. E q u a t i o n s 2.31 t o 2.40 a r e n o t i d e n t i c a l v d t h t h e e x p r e s s i o n s

q u o t e d i n r e f e r e n c e 5» s i n c e a number of minor e r r o r s e x i s t i n t h e p r i n t e d p a p e r .

3 0

-ibP

( L i b - ^ ; i b , 4 ) - - ^ i ^

F(^.ib+|;ib+254)-(ib+l)p

f -

| _ . , _ i i i ^ \ F ( i , i b 4 ; i b + i j 4 )

k

.2.55

p

ibP(i,ib-i,ibj4) - /^^^^)„F(^,ib+|;ib+2,4)+

(ib+l)p'

1 / - 2 E . , , i l i l £ k ) F ( l , i b 4 , i b + 1 , 4 ) "

p V Kk'"^ k ' y (3 _

v/here F(g-,ib-g-;ibj—r) e t c . are hypergeometric functions

b =

0) s

27;c

and

3 = e

inc/s / s

I t reduces to the standard value

i H,(') (i)

C = 2.36

i H^^'^ f ) - H^2) (|)

when — tends to infinity.

In equations 2.31 to 2.35 inclusive,

" ^ ' n+5/, 2n+1v

" \ ' 2n+1 /, 2n+1 \

Q2 = ^ - . ^

^^ \

...2.38

^=^ (2n+l)(l-q2^^^^)

x " "" n/, 2nN

Q, =2:.Y, ^ - ^ ^ 2.39

n=1 (l-q )

+ The functions Q, to Q, used in this report are not

identical with the functions which are sometimes used in connection

vdth the theta function.

-31-m=n-1 m=0 /_ , \ /r, n A \ (A 2m+1 -. /, 2n-2m-1 \ (2m+1) (2n-2m-1) (1 -q ) (1 -q } 2.41 ^ - - .2m+1 - 2 n ^ - ^ "^=° (2m+1) (2n+2m+1) (l -q^^^) (l .q2n+2m+1 ^ q = e - ^ ' / ^ 0 < q < : i 2.42 k = t a n h f TTT 1 ^ k^ + k ' ^ = 1

(ff);

K(k) j K'(k) are the complete elliptic integrals of the first kind of order k, and E(k) is the complete elliptic integral of the second kind of order k.

§2.2. Approximate value of C for small values of the gap-chord ratio

The hypergeometric functions in equation 2,35 can be expanded in the follovdng power series,

•n/' -u. N A 2L.b a(a+1 )b(b+1) 2 , \ I _-.. F(a,b;c;z) = 1 ^. :J-^ z -H ^]^^'JQ^) Z when jzl^CI.

For small values of the gap-chord ratio — r <C."<Cl and then, for most practical applications, we need only the first two terms of the above series. Also in evaluating c(o)^ — ) from

P c equation 2.35 it is noted that S(k) and k tend to unity and

9 +

k' ' to zero as the gap-chord ratio tends to zero. If v/e use

g t h e s e a p p r o x i m a t i o n s we f i n d t h a t i n t h e rajige 0<— <^^ e q u a t i o n c 2 . 3 5 r e d u c e s t o p ( E - ^ ) . 8 b ^ 2 A = -^^ ^ 2 . 4 3 a j i d

( E - S C I ! ^

. I 6 b ¥

2bK B = ^: TT^ 2 , 2 ^ / F o r v e r y + F o r s m a l l v a l u e s of s / c K(k)>::<;log ^ a n d K ' ( k ) ; ~ § . „•2 O K d Hence 7tK' ~ ./TCC , , „\ q = e - l ^ ; ^ . e 2 ( 2 ^ + l o g ^ 2 )

3 2

-For very small values of the gap-chord r a t i o

r,..2 / „ \ 2

A = ^^^- ~ r 2.45

A ' " 2 1 + Ü)

l (

-, >v2 1 + to (\ + ^ lo< \ ^ •KC ^ l o g 2 ) V.C ^ e / H + ~ l o f i ; A 2

%9

2

and B = ' ' —r .« , . . 2 . 4 6

The asymptotic values of t h e i r d e r i v a t i v e s vdth respect

t o the gap-chord r a t i o are

9 9

ÈL— - > •^ g 2.47

(1 + 0) }

s/c-^O

a v ^ =4-

- - Y - 2.48

TïTo (^ * " )

The functions A and B are plotted in figure l6 for certain values of the frequency parameter 'co.

§2*,3. The evaluation of the functions Q. jQpjQ-rjQ, .

The functions Q., Q-, Q , and Q. are determined from the infinite series in the function q (see equation 2.42). Now for small values of the gap-chord ratio q varies from 0.1 to 1.0 and in this range the series are not rapidly convergent. They have been evaluated, by direct summation, to three decimal places and the results are given in table 3 and figures 17 and 18,

§2.4. The two-dimensional aerodynamic derivatives

If the lift and pitching manent are written respectively

Z = Z z + Z.z + Z.,^ + Z„e + Z-0 + Z-'O 2,45 Z Z Z Ö Ö Ö

M = M z + M. z + M..'i + 1,6 + M?,è + Mv*é z z z o ö o

-33-t h e n from e q u a -33-t i o n s 2 , 3 0 , ^12 = ^•34 = M,2 = M„ = \ TtpV^ ^6 Tr2 TCpCV z . r 2 •TipcV i c f e . TipcV - ^ 6 r.pc V icüfl. — \ * i t p c V

is/[^

r + ^ ^ Z , 2 TCpC . ^ 2 „ a ) Z g •rtpC GO M„ TipC t ,2.50 34 _{TCPC VT • K p C ' ^ TtpC} The v a l u e s of t h e aerodynainic d e r i v a t i v e s r e f e r r e d t o t h e m i d - c h o r d a x i s a r e , f r o n e q u a t i o n s 2.31 t o 2. 34 i n c l u s i v e ,

i' =-

= 0

pv

z.

z pcV c 1 • K \ " ^

t + (A-1) (1 - — I J

P _fz i6sl

' ^ ^ = " 2 = 2 pC TtC . O B 7.C / ^ ' ^ 1 \ 0) V 2kK <_

1 = - ~ S . = l 6 f Q.

® pcV^ ° ^ (A-1)

p _ . i 6 - 1 2 s !

* • ~ 2, " 2 6 p c V •rtc

(-^)1 3.

_ ^ 2 k K 2 j ° ^ 'S? V 2 k K 2 /

^•é

, 0 2 / 7 ^ Q.Q;.B m = z m. = z m„ = z m^ = pc M z pcV^ M pc TtO ^1^3 \

2kK:V/

= 0 2 ^3 •rtC ^ + (A-1) (1 M.. _z pc M 3 2 s " „ B 2 ^ 3 ^ 7tC 0)

e

32s

.V

7i:c n Q: V 2kK'^/ V 2kK'^/ • W (A1) (1 -V 2: 2^ . - , •Q 2kIC

A

_{m\ =}

i

3 4 -rai = -2

-'- ^'^ Q (A-i)+^Q./'i'---4V

V 2 W r / 'S' Ö " p c \ ~ k K V ^ ^ " . c ^ ^3 _pc % 32s^ , ^^^^^3 ^ 0 4 3 4 4 1 Tr2 3 '^'' pc^ u c ^ k K c 0)

The v a r i a t i o n s of the d e r i v a t i v e s Jf. , i^Q , m^ , m^ vdth gap-chord r a t i o a r e shov/n i n f i g u r e 19.

I f t h e reference axis i s taken he forward of the mid-chord p o s i t i o n and t h e corresponding values of the d e r i v a t i v e s a r e denoted by J' , in e t c . then the transfor.nation formulae are

•' z * z "^z = "^z ^ A '' "^z = ^z ^ ^ 4 ' ^è' = ^ - "• ^f z m^ = m^ -H h(m^ + | Q ) + h ^ l ^ . . . . 2 . 5 2 m^ =m^ + h(mj, +^.^) + h^t^ m.. = mv + h(m,. +^.) + h f , . . 0 Ö Z 0 Z

s2,5. The three-dimensional flutter derivatives

Although the theory derived above can be logically

extended to aerofoils of finite aspect ratio in cascade the added complications appear unnecessary in this prelimins-ry estimation of the magnitude of the cascade effect. It vdll be assumed therefore that the lift loading at all spanwise positions on the aerofoil vdll be similar and directly proportional to its local displacement and rotation. The aerofoils vdll be assumed to be of constant chord.

If the normal displacement of the zeroth aerofoil is given by

Z = ^0 f{r\) + (x-xpeF(Ti) 2.53

where "n = y/Z

= the distance from the root to the reference section. + Since the wake plays a minor role in the determination

of the airloads on an aerofoil in a cascade of sr.iall gap-chord ratio, it might be inferred that the effect of the tip vortex would not be very marked.

-35-f (-n) and F(TI)

0,0

= the flexural and torsional modes of vibration = the angular displacements in flexure and

torsion at the reference section relative to the root.

= the chordwise position of the reference axis in the reference section,

then the amplitudes of the flexural and torsional moments about the root section can be expressed by,

-# = =-^^v^

M - p ; ^ = " ^ ^ ^ ^ " 3 ^ ^ 7 > - - ^

Jt '

where

The aerodynamic d e r i v a t i v e s a r e d e r i v e d frcya

,2.54

where, 12 = c . + i(J5r b . - 0) a.

34

= c , + 10) b , - 0) g^ M. ₁₂ M

34

= k^ + ioT j . ^ - 'tir a , k^ + 10) j ^ - 0) g ^ . '^-' > ^.2

= I

K

n^

f^d-n

1 ^7,

ni

f^d-n

2.55

= I

= t

11

o o o \1 m j f F d-n - A l n.^ 2 f dTi f F dTi f F dn g, 1 ^ 0 f F dn o o ^1 = -^0 m.

O'

R^

^3 = ^0 f F dri

f P ^ ^ . . . 2 . 5 6

F2an

/ d-z — . . . + The d e r i v a t i v e s a r e v / r i t t e n a s ^ , in-' , . , , t o show t h a t t w o - d i m e n s i o n a l v a l u e s a r e b e i n g ^used. I t i s more u s u a l t o use t h e n o t a t i o n / ^ » . . . , HV • • • , f o r t h e s e same d e r i v a t i v e s when t h r e e - d i m e n s i o n a l v a l u e s a r e b e i n g u s e d .

^3 = ^'f

-36-^ 9

F dn

and the reference axis is taken at he forward of the mid-chord position.

The complete coefficients a. , b etc. must include both the aerodynamic and the structural components.

S3. Flutter vdth a single degree of freedan

Uncoupled flexural or torsional oscillations can occur when the coupling terms G. , J. , K and A , B, , C, (see equations A1.1) are zero.

The equations of motion for uncoupled flexural and torsional oscillations are respectively,

A ^. + B ^ + C 0 = 0

^ ^ ^ 3.1 G, 0* + J, 0 + K, 0 = 0

t> i i

where 0 is the flexural coordinate and 0 is the torsional coordinate.

Flutter can therefore occur in tlrie flexural and tor-sional modes, when respectively B and J, are zero. In the notation of §2 this would require J'. or m?, to be zero,

z o Hence for flutter idth a single degree of freedom either

g^ = 0 ; j^ = 0 ; k., = 0 ; b^ = 0

or 3.2 a^ = 0 ; b^ = 0 ; c^ = 0 ; j^ = o

It can be shov/n from an analysis of the terms given in equations 2.51 and 2.56 that these conditions cannot be satisfied. It is probable, however, that for oscillations having finite amplitude, the values of the aerodynamic deriva-tives vdll be reduced belov/ those stated in equations 2.51 (see Appendix 2) and more exact analysis may show that under certain

conditions equations 3.2 can be satisfied,

3 7 -TABLE 3 <1 0

0,05

0 , 1 0 . 2 0 , 3 0 . 4 0 . 5 0 . 6 % 0

0.272

0.465

0.953

1.702

2,939

-9.456

Q2 0

0.061

0.151

0,472

1.148

2.623

-15.209

s

0

0.061

0.151

0.472

1.148

2,623

-15.209

\ 0

0.003

0.018

0.133

0.613

2.481

-47.128

Note.

1. Tabvilated values have been obtained by term by term

^ summation.

2. Five decimal places have been used throughout.

3. The follovdng values of Y were c a l c u l a t e d and were

used i n the evaluation of Q, and Q, .

3 4

q Y^ Y2 Y3 Y^ Y5 Yg

0 1.0 1.333 1.533 1.676 1.787 1.878

0,05 1.143 1.446 1.642

0.1 1.309 1.572 1.763 1.903 2.013 2.103

0.2 1.732 1,882 2.055 2.191 2.299 2.388

0 . 3 2.338 2.307 2.44!f 2.570 2.675 2.762

0 . 4 3.262 2.927 2.990 3.094 3.191 3.275

0.5 .

-0.6 7.571 5.616 5.192 5.104 5.120 5.168

<1 fj Yg Y5 Y^o Y^1 Y^2

0 1.955 2.022 2.081 .

-0.1 2.179 2.246 2.304 _ - .

0.2 2.463 2.538 2.588 - _ .

0,3 2,837 2.903 2.961

-0 , 4 3.349 3.413 3.47-0 3.522 3.597 3.611

0 . 5

-0.6 5.223 5.278 5.330 5.379 5.422 5.464

^ ^13 ^14 "^15 ^16 "^17 ^18

0 . 4

-

-

-

-

-0.5

-

-

-

-

-0 . 6 5.5-03 5.538 5.572 5.6-03 5.633 5.661

Table 3 - Contd. -38-0 . 4 0 . 5 0 . 6 ^19 "^20 "^21 22 5.687 5.712 5.736 5.759 C5Ö - ' ^ ( l - q ^ ^ ^ ^ ) n=0

a

V

q^i

+

q^")

n=1 (.1 - q ) oo 2n+1 2n+1

"=° (2nH.l)(l - l^"-'')'

-SX3 Q

4

-«r—^ „ 2n/., 2n\ > Y 2 q (1 + q ) — - f ' n /. 2nN n=1 n(.1 - q ) - ^ Y^ = n m=ö^ _{( 2 m + l ) ( 2 n - 2 m - l ) ( l - q ' ' ° ' - ' ) ( l - q '} 2m+1 N /, 2n-2i:i-1 2m+1 ^ ^ 0 ^ (2m+1) (2n+2m+1) (l -q^^"^^) (1 -q2n+2m+1 ^ _m= REFERENCES No. Author 1. Ch. Bellenot and J. Lalive d'Epinay 2, W.J. Duncan 3. W. Pritchard Jones 4, E. Reissner Title etc.

Self induced vibrations of turbo-machine blades

Brown Boveri Review Vol. 37, pp.368-376. 1950.

The fundamentals of flutter. 1951 A.R.C. R. and M. 2417,

V^ind tunnel interference effects on the values of experimentally determined derivative coefficients for oscillating aerofoils, 1943. A.R.C. R. and M. 1912.

Boundary value problems in aero-dynamics of lifting surfaces in non-uniform motion.

Bulletin of the American Math.Soc. Vol. 55, 1949, pp.825-850.

39-No. Author 5. R. Timman

6. H.L. Runyan and

C E , Y/atkins

7« W, P r i t c h a r d Jones

8. H, Lamb

T i t l e e t c .

The aerodjTiamic forces on an

o s c i l l a t i n g a e r o f o i l between tv/o

p a r a l l e l w a l l s .

Applied S c i e n t i f i c Research Vol.A3

No. 1, 1951, pp.31-57.

Considerations of the e f f e c t of vdnd

tunnel v/alls on oscilla.ting a i r forces

f o r two-dir;iension.al subsonic

compress-i b l e flow. 1951.

N.A.C.A. T.N. 2552.

Sui-.mary of fon:iulae and notations used

i n tv/o-dimensional d e r i v a t i v e theory''.

1942. A.R.C. R. and M. 1958.

Hydrodynamics 6th e d i t i o n 1932.

pp. 190-192 (C.U.P.)

-40-APPËITOIX 1

The calculation of the critical flutter speed and

frequency for coupled flexure-torsion flutter.

A brief account of the essential features of flutter

theory are given, for completeness, below. A more detailed

account can be found, for example, in reference 2,

The equations of motion describing the vibration of a

rectangular cantilevered aerofoil in cascade are (refeirence 2)

Ajf +

BJ

+

C 0

+ G 0 + J 0 + K 0 = Ö

^ ^ ^ ^ ^ 1 A1.1

A'0

+

B,0 + C,0 + G'Ö + J,0 + K,0 = 0

3 3 3 3 3 3

where

0

is the flexural coordinate (the downward displacement

z of the extremity of the flexural axis at the tip

section divided by the span -i )

ajid 0 is the torsional coordinate (the tvdst at the tip section

- positive when the leading edge rises and the trailing

edge falls),

If f (ri) and F(ri) are the flexural and torsional

modes respectively then the downward displacement at the point

(x,y) is

z =

0lf(j])

+ 0c

(fApi-n)

... A1.2

where T) =

y / ^

? = x/c

f c is the distance of the flexural axis from the

leading edge

y ,

= span (root to tip)

c = chord

and x,y, z are rectangular cartesian coordinates having

their origin at the leading edge of the root

section (see figure 20).

The boundary conditions are that

f(o) =

F ( O )

= 0

f(l) = F(l) = 1

f'(0) = F'(1) = 0 ^^""-^

f" (l) = f " (l) = 0

:_ /If ...

+ Good approximations t o t h e f l e x u r a l and t o r s i o n a l modes

are f

(T))

= vi^

4 1

-If, f o r a harmordc motion, 0 = 0 e and

0 = 0 e''"'^ , v/here 0 and 0 a r e the ar.iplitudes and o) i s the c i r c u l a r frequency, t h e n from e q u a t i o n s A 1 . 1 , when 0 and 0 have b e e n e l i m i n a t e d , 1 T O q^ 0)^ - iq^o) - q_^^o + i q ^ + q^ = 0 . A 1 . 4 which has t h e s o l u t i o n s 2 and where 0) = q . / q ^

^

'^2

^3 " ^o S -

'^l

\

% = "^ = A, G^

A 3 G 3 I

^ '^ A 3 J 3 . + ^ = 1 B 3 G 3 = 0 .A1.5 .A1,6

'S

^ 3 "1 •^3 ^ l - ' l B 3 J 3 + " l " ^ , 0 3 = 3 .A1.7 =1 ^1 B 3 K 3 +

°1^1

C 3 J 3 \ ^ 1 ^ 1 C3 K3 I f ra

i c

2j^2 m c y \

I

₀

m 0 ~ P = V = mass p e r u n i t l e n g t h of spa.n d i s t a n c e of t h e c e n t r e of g r a v i t y from t h e l e a d i n g edge

the polar moment of inertia per unit length about the leading edge

flexural stiffness torsional stiffness d e n s i t y vdnd speed t h e c o e f f i c i e n t s A , B. e t c . can be e x p r e s s e d i n t h e f o l l o v d n g n o n - d i m e n s i o n a l f o m s ,

pc r

U •|1 f^d-n + - ^ p c ' 11 m f "dri .A1.8

A.!

=

+ I t should be n o t e d t h a t seme of t h e c o e f f i c i e n t s a r e d i f f e r e n t from t h o s e used i n r e f e r e n c e 2,

-42-B.

pVcf

^>y.

0

pV

2p

= ) \

o

ff

0

2 f dn

f2a^ + - - A

pv^-e^

. A l . 9 .A1.10

pfo'

J1 = ^

0^

fF dri +

₂

P° U

( i - i ) m fP dTi A1.11 fF d-n , A l , 12 K. 202

pvt

= A,

fP d-n . A l . 1 3 A.

pfc'

= n

n"i

fF dn + L/O

pc

( i - i ) m fP d-n A l . 14 U o B.

n^

A

2 2 c ^ ^

u

fP dri . A l . 1 5 fN^ 202

9n

- M, •0 fF dri .A1.16 «3 =

11.

01

P^c

4 = ^'ê

u

2 1 F'^dn + - ~ p c ' ^ ( i - 2 i i +;{^)m F^dri , . A 1 . 1 7 u' o

ril

pV^c

- n

F^d-n .A1.18 K,

A^

^3 = _pV^^c^ = ^if F^d-n + m, pV^^ 2P^2 •Al.19

I n the above formulae f\ •!, e t c . r e p r e s e n t t h e o v e r a l l aerodynamic f l e x u r a l d e r i v a t i v e s and jiv e t c , r e p r e s e n t t h e o v e r a l l aerodynamic t o r s i o n a l d e r i v a t i v e s . The r e f e r e n c e a x i s i s t a k e n a s the f l e x u r a l a x i s of t h e r e f e r e n c e section"*". The / e f f e c t . . . + The aerodynamic derivatives are functions of the plan fom, aerofoil section and the frequency parameter o). For preliminary calculations of the flutter speed and frequency sufficient accuracy is obtained if two-dimensional derivatives appropriate to the required frequency parameter are used.

4 3 -e f f -e c t of h y s t -e r -e s i s or s t r u c t u r a l damping h a s b -e -e n n -e g l -e c t -e d . I f the n o n d i m e n s i o n a l c o e f f i c i e n t s above a r e s u b s t i t -u t e d i n t o e q -u a t i o n A l , 5 and A1.6 and where ^1 ^2 ^3 ' ^o ^3 " ^? \ = ° / v 0)C " = — ' '^ % =

"h

=

^2 = % = ^ §1

" 3 ^ 3

^ ^1

^3 ^3

^ ^

a 3 k 3

b , k^

b^3

+ + + ^1 §1

h^3

\

J1

b ^3

^1 J'l

C3 J3

°1 s

°3^3

°1 ^1 « 3 8 3 i.1.20 .A1.21

The phase difference and the amplitude ratio betv/een the torsional and flexural motions can be obtained as follows. (See reference 2 ) .

If in the moment equations A1.1 v/e put 0 = 0 6 " ^ ' io)t

e

then it can be shown that the amplitude ratio is given by , Q ~ io)t

and 0 = 0 e

£-lf - ^°3 " Vl -^^^^3 ' Vl^

^ ^ 5^^^ - 3^^^ - ü)"(j^g3 - J3g^)

.A1.22

and the phase difference Y from

t a n Y 0) _{(133 j . , - b^ J3)} (k^ b3-k3b^) - 0)' (b3g^ -b^ g3) ,A1.23 Since

(11

=1 = ^

f dn +

[£

^3 = ^^ O o •^1 u o pV 2^3 2 F dri + m^

pV^f

20 2 /we can . . ,

-44-we c a n e l i m i n a t e V and d e r i v e t h a t where k3 = a + pc^ . A l . 24 a = \i,

f

Co 01 F^dTi - ^ \ f dT) 2 m^

P = - 2 F

°

^0

I f k , i s e l i m i n a t e d from e q u a t i o n s A l . 2 0 and A1,21 v d t h t h e a i d of e q u a t i o n Al, 24, we o b t a i n t h e f o l l o v d n g q u a d r a t i c e q u a t i o n i n terms of the unknovm c. , v i z ,

where and R^c. + R^c. + R, = 0 1 1 2 1 3 ^1 ~ ^1^2 ~ P ^ R2 = r ^ r 3 + r 2 r ^ - a, q R3 = r 3 r ^ + k^C3 '^ r^ = b^P + J3 . A l . 25 1-2 = (q.,g3 - qQJ3) + 3(q^a^ - q^b^) ^3 = q.,(b^J3 - b3J^ - a3k^ - C3g^) + %(t>^i^^ + C3J^) + a l

^4 = ^ ^ - "^3^ - °3Ji

I f t h e s o l u t i o n t o e q u a t i o n A l , 25 i s v / r i t t e n

^2 i / ^ ^ ~

^•^^-1c _2R.

2 ^-r)

11.26

then the corresponding value for the critical flutter speed V i s o

°ic - \ 1

f^dn

<2 i\1.27 / a n d , , , + c . and k , a r e unknown s i n c e V i s i n i t i a l l y unknov/n.