REPORT Nc. 19

August, 1948

THE COLLEGE OF AERONAUTIC S,

CRANFIELD

Flutter of Systems with Many Freedoms

-by-

W.J. Duncan, D. ac., F.R.S.,

Professor of Aerodynamics at the College of Aeronautics, Cranfield.

--o0o--

SUMMARY

Experience has shown that it is often necessary

to retain many degrees of freedom in order to calculate

critical flutter speeds reliably, but this entails much

labour. Part I discusses the choice of a minimum set of

freedoms and suggests that this should be based on the

equation of energy and the use of the Lagrangian dynamical

equation corresponding -to any proposed additional freedom.

The methods for conducting flutter calculations so as to

minimise labour are treated in Part II.

DMS

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11. The Method of R.A. Frazer ... IP • •

12. Methods Based on the Use of Matrices

-2-

CONTENTS

Page

1. Introduction e • • • • • • • • • • • ... 3

2. Some Exactly Soluble Problems of the

Flutter of Elastic Wings. m.0 066 4

Part I

Choice of the Degrees of Freedom

3. Types of Generalised Coordinates ... 5

4. The Equation of Energy 5

5. General Discussion of Couplings ... goo 8

6. Comparison of Inertial and Aerodynamic

Couplings. ... ••• ••• • 0 • ••• 11

7. Principles Governing the Choice of

Freedoms • • • ••• • • • ••• ••• 11

8. Illustrative Examples on the Choice of

Freedoms • • • ••• • • • • • • • • • 12

Part II

Methods for Predicting Flutter

Characteristics

9. Non-Dimensional Form of the Equations

of Motion. ••• OOO 4160

006 400 11+

10. The Orthodox Method of Solution. •• • 15

13. The Method of Duncan, Collar and Lyon. ... 16

14. Another Inverse Method. ••• ••• ••• 17

List of Symbols • •• ••• • • • *00 0011 18

List of References ... • •• SOO 111110 21

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_3-

1. Introduction

The aim of this paper is to discuss methods foe- calculating the critical flutter speeds and the nature of the motion at these speeds for systems with a large number of degrees of freedom. This problem is becoming increasingly important since it is now recognised that reliable estimates of critical speeds can, in many instances, only be made when many independent kinds of motion of the structure are admitted, However, the labour in the calculations increases exceedingly rapidly as the number of degrees of freedom is increased. Hence two principal problems arise:-

(a) The choice of a minimum set of dynamical coordinates or degrees of freedom which leads to calculated results of adequate accuracy.

(b) The choice of the method of conducting the calculations after the dynamical coordinates have been chosen.

These problems are separately considered in Parts I and II of the paper.

It is concluded that a particular freedom F must be retained when the balance of energy at a critical flutter speed is sensitive to its inclusion, unless it can be shown that the amplitude of F is very small. This amplitude will be very small when one or both of the following conditions is satisfied:-

(a) The coupling terms in the Lagrangian dynamical equation corresponding to F are all very small.

(b) The impedance for F at the critical flutter speed and for the flutter frequency is very large.

As regards the energy balance, it is shown that large skew-symmetric components in the aerodynamic stiffnesses are of particular importance. Inverse methods appear to be the most advantageous for the calculation of critical flutter speeds when there are many freedoms.

There are a few known special cases where /exact? calculations of critical flutter speeds can be made for elastic continuous systems having infinitely many degrees of freedom. Such systems throw much light on the general problem of the choice of freedoms and they are briefly considered in the following section,

/2. Some •••• auras •?/

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-c-

_4._ 2. Scme Exactly Soluble Problems of the Flutter of Elastic Wings

Exact solutions of two problems on the flutter of continuous elastic cantilever wings have already been published and the details are as follows:-

(a) Uniform rectangular wing supported by a pair of uniform parallel spars. The covering has no proper torsional stiffness and both the flexural and torsional stiffnesses are provided by the uniform spars. The inertial and aerodynamic coefficients are independent of spanwise position (See references 1, 3 and 4).

(b) Uniform rectangular wing as for case (a) but now the torsional stiffness is provided by the uniform covering or by a torque tube and the flexural stiffness is assumed to be negligible

m

(See the Appendix to reference 7).

Both wings have the following special dynamical characteristics. The modes of oscillation in vacuo or in an airstream occur in pairs which, in general, have both flexural and torsional components. The displacements in both flexure and torsion for the r.112 pair are proportional to the displacements in the rit mode of the purely flexural oscillation of a uniform cantilever beam for case (a) and to the displacements in the r±1.2. mode of the purely torsional oscillations of a uniform cantilever for case (b). When the wings are exposed to an air current the two modes of the rIL pair become coupled together aerodynamically, but remain uncoupled to all other modes. The flutters. which occur are therefore effectively binary and the earliest flutter occurs in the modal pair corresponding to the fundamental free mode.

These results are not, in fact, isolated and other similar cases will be discussed in a separate paper. At present we wish to emphasise the remarkable simplicity of the solutions, which were obtained in an orthodox way by solving the differential equations of the problems. An independent explanation of the results is given in §6 by means of an' analysis of the various couplings between the modes.

/ PART I

This is of course an artificial case but is not without interest since the flexural stiffness does not greatly influence the flutter.

XX ANth Or (r - overtone.

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-5-

PART I

CHOICE OF THE DEGREES OF FREEDOM

3. Types of Generalised Coordinates

The generalised coordinates should be chosen so as to secure some advantage of simplicity or convenience. There appear to be two particularly advantageous types:

(a) Those for which the specification of the uisplacements is specially simple.

Examples. The purely flexural and purely torsional coordinates of the classical theory of wing flutter.

(b) Those which secure the greatest simplification of the dynamical equations, as by the vanishing of certain coefficients.

Examples. Normal coordinates for which the inertial and elastic coupling coefficients vanish,

In the further discussion we shall adopt normal coordinates for the main structural deformations. These correspond, strictly, to the free modes of oscillation of the structure in vacuo, but differ little from the modes of oscillation in still air. These must be supplemented by certain general rigid body freedoms, for the resultant force and moment on the system are not, in general, zero when the oscillations occur in an airstream. Lastly, there will be one coordinate for the angular movement of each control surface which is concerned in the flutter. The main problem is to decide which of the normal modes and general freedoms must be retained in the calculations.

For a conventional aircraft, having a fore-and-aft plane of symmetry, and in straight symmetric flight, the small oscillations of symmetric and antisymmetric types will be independent. For each type only the corresponding normal modes, general freedoms and control movements must be retained. For example, in symmetric flutter a single symmetrically placed rudder takes no part and the body freedoms are pitching, normal, displacement and fore-and-aft di3plaeement; the latter will usually be negligible.

In order to determine what normal coordinates arc to be retained the influence of an added coordinate on the energy balance must be examined. We accordingly pass next to the equation of energye

4. The Equation of Energy

For the sake of conciseness and perspicuity we shall use the matrix notation. Thus the whole set of the eouations of free motion will be written

Aef B + Cq 0 (I)

/Where

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-6-

where A, B, C are the square matrices of the inertias, damping coefficients and stiffnesses, respectively, and q is the column matrix of the generalised coordinates. A, B and C are independent of the generalised coordinates and of time. The equation of energy is obtained by premultiplying the last by 4, , i.e., by the row matrix'''. whose elements are the generalised velocities; and it is accordingly

+ c1' B Cq = O.

Now A is always symmetric and the kinetic energy of the system is

=2 t A 4 i

(3)

a quadratic form in the generalised velocities. Therefore

d A,4 iv, A4 at

since A is independent of time. But

.41A4 ,VA'cr

by transposition, since A' = A, and we obtain

dt

For a system subject to aerodynamic actions neither B nor C is, general, symmetric. We may always put

B = B1 + B2

C = C1 + C2

where the suffixes 1, 2 are applied to symmetric and skew-symmetric matrices respectively. Then

B1 = 2 (B Bt)

B2 = (B - Bt)

and there are similar expressions for C. and C2, Let the dissipatio function be

?11334 = B1 4

since xtYx = 0 (3)

when Y is skew-symmetric and x is arbitrary, Also let the 'potential' be

V. 2q1Clq

so that

- q. (1C)

dt 1 /The ..-

An accent appended to the symbol for a matrix indicates that its transposed is to be taken.

( 2)

(7)

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-9-

(P.)

'9)

c)

-7-

The equation of energy (2) can now be written

dt + 2F+ 4IC2q = C.

The final term in the last equation is of great importance and must be discussed in detail.

The equation of energy is valid for any free motion of the system, but particularly illuminating deductions can be drawn when the motion consists of a single constituent, corresponding either to a single real root or to a conjugate complex pair of roots of the determinantal equation.

Case (a) Real exponential motion (Dilmrgerce or subsidence).

Here we have

q= q e>it

with 2\ real and go a column of real constants. Then

4 = Aq

and 4102q = XqIC2q = C (13)

by equation (8). Therefore the equation of energy (11) becomes

dt (T+v) + 2 f0. ---- (10

If F is negative, obviously the motion can grow. But suppose now that is -Positive during the motion'. Then we get

1 (ir+Y) < 0 d

at all instants. So lona 9.siVis essentially positive this implies decay of the motion. However, V is not wholly of elastic origin and at higher values of the relative wind speed it may cease to be essentially positive. ThenT may increase, i.e., the motion may grow, through dA, being nEptiv. At a critical speed for divergence ), dt V and ̀rare zero and the equation of energy becomes

Ar= constant, (15)

but the displacements need not be zero as they can be supposed to grow exceedingly slowly to finite values. This equation can be made the basis for calculating the critical divergence speeds, but we shall not pursue this question here.

/Case (b)

When the values of the aerodynamic damping coefficients for divergent motion as given by two-dimensional vortex sheet theory are used, the condition that the dissipation function shall be essentially positive is not satisfied for any value of the non-dlmensional divergence parameter Ac/V. The dissipation function will be essentially positive when there is enough structural damping.

(12)

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atTI Wox aeq.emuaud eoup2aaATp TuuoTouamTp-uou GT4 JO enTun. Sue aoj paTjoTq.ns q_ou sT GATTTsod SauTq_uessa act ITutis uoT;DunJ 110WedT0sTP GT14- 1-g- uoTP-TPu00 auk ePosn oxa E-101,4

aaus xaq_aoA TuuoTsuomTp-omq. Cq usATZ su uo-p_om Imp0aoATp JQJ WIOTOTJJ000 BuTdmup oTmuu oaar auk jo senTuA uaum

(q) asap/

•paoll uoTI.sanio 275(1. onsxnd q.ou IT-Ggs am q.nq 'speeds aouaZaaATp Tso-P,Tio aqq- 2uTq.u-Eno-po JOJ sTsuq

aqq. epum eq Tiro uoTq.unbe sTuI •sanTuA al:cuTJ off. SimoIs SI2uTpaaoxo moa2 0. pasoddns aq use SaliT on oaez aq Tou paeu uquemeonidsTp auq. q.nq

(c0 q_uuq_suoo =A.

samooaq SS.IGTI* o uoTvenbe am. pus oaez ere pug

;1) eNceouaaaATp JOJ peals Tromao u 6eATq_n eu 'fiTeci. p q2no1m. 4wa2 Kau uoTWILT euq. "D.T t°svoiouT

•0.A.Tq_TS0d STIVTI.U0SOD eq oq_ POU00 Z1.3M q_T paads puTm ;-,ATlniaa aw.i. Jo senTrA aoLOTti qu. pun uT2Tao oTv3u/a jo q-011 sT 4aaAamoH suoTwm atm, jo

laioDP s*TicimT 0AP:1:sod STreTP-upssa sTAsu ot IDS •squvq_suT Tre q.0 q-P

° > Cilt+10

q-02 am, um-,111 '3„110T1.0m aukauTarP aAT1Tsod sT q.811q. ALMA asoddno 'moa2 IMO Otlq. SIonoTAq2 eGAT;132'3u uT JI

(in)

(c0

samooaq (L4) Z2aaue Jo uoTqrnba aul aaojaaola *(R) uoTq_rnbe Aq

0 = -001ToN( = -001 Puu

b1 T.)

uma esq.urq.suoo Traa Jo umnioo n ab pur Traak6 TiorTm

o = T.)

any{ pa oaoH

e(eouopTsqns .to poupzimiATa) uoTq.om Tr-p_uouodxe reaH (r) asuj

•uoTlunbe ivq.unuTmaalap o•q. Jo swoa Jo aTnd xaTdmoo al.r8ncuoo a o. ao q_oo reaa

aTUT/Ts 13 01 ammo PuTpuodmaaano equenlm.suoo aTOuTs u jo sq.sTsuoo uoTlom auk umim umuap eq Tiro suoTq_onpop arTq.uuTmnTIT STasTuoprand ornq ‘Dreq_sSo all; Jo uoTwm eaaj ArC JOJ pTT-eA sT EVaaua Jo uoTvenba ata

'TTI3;01) uT PassnosTP aq q_snm pun souvq.lodmT q.uaa2 jo sT uoT;i3nba srT puor uT maeq. TruTj

•0 = golt +jz (A440

ueql_Tam aq AOU uno (z) SOasua Jo uoTq.unbe auk

-L-

-9—

where y, z are the rectangular cartesian coordinates of the element of mass 6' m; and the summation covers all the elements of mass for which the expression in the bracket does not vanish. The complete coefficient Acontains, in addition, the correspondingvirtual inertia or reversedrs acceleration derivative.

When qr, and q are normal coordinates the coefficient A .vanishes, The same is true when q, is a normal

. croordlnae and q is one of the general rigid body freedoms. This

follows from the fact that motion in any normal mode can persist independently of any other motion.

Pm- a cantilever wing with rigid support at the root we may usually neglect the chordwise and spanwise displacements in the flutter, Then the structural contribution to A is rs

z a z _ sAve' -

2 cis

Further; when the wing is straight and not swept back or forward and it is assumed that each fore-and-aft strip moves as a rigid body"K, this expression can be reduced to the form (see the list of symbols)

A s rs — TS F

where d (c

2

0-7,_(1) 6 , dq dq. c s

d d1P d 3 d c

do dq dq dq 0 ".2 S S r o

dni• d (c 4

j dqv dqs

(b) Cross or Compound Damping Coefficients

These are mainly or wholly of aerodynamic origin and the array of the coefficients is, in general; not symmetric, i.e.,

Brs ;4 Bsr, in general, We may, if we please, analyse these

coupling coefficients into symmetric and skew-symmetric parts (see equations 5A, 5B) and we may call these pure dampings and gyrostatic terms respectively, This analysis, however, does not appear to be particularly helpful.

For

- If we assume that the virtual inertias are the same for a finite air speed and for still air, then qw and q, should be normal a•oordinates for oscillations in still atr.

mm To avoid circumlocution we shall henceforward call this a type C 'Ting

(23)

3

(24.)

2uTK. 0 0014 Tivo panmaojoouall 'Palle an uoTV100171101T0 PT°" °I -x

'aye IT-m.s UT suoweiTToso aoj sonrnuTpaocc, Inmaou aq pTnogs 44b pule -lb UOUq. ~u ITTls aoj pun poodo aTy

aq.TuTj 10J GUMS am. axe onTwouT Int4aTA oqq. q.nuq. amnson oa x

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pun sBuTdmnp aand osota no Jaw ore. pun (Lc 417C sooTvariba 17F) sq.and oTaq.ommiCs-aaxs pun oTaq.allimWs oq.uT oquoToTjj000 2uTidhoo

, asom. am/CT-ewe gosnaTd am 3T '&771 GjA .inaouo2 uT as g sa

'ea.T loTa4ommSs treaouo2 uT csT ovioToTjjoco ot.p.„ jo Seaas ouq_ pun uTSTao oTumuSpoaan jo ST-toga JO kLUTOW axe osaqi

sq.uoToTjjao0 )1uTdmea punodmoo xo soap

.(00) qra, Abp

1/477 1.;:T (0o cbp sbp s

bp - EA)

c

of

sbPbP ()

gP

sa

3 aaaga

(sioquuce jo q.sTi oqq. 77) Two; oq_ peonpoa aq u.so uoTscoadxe sTql Spoq p-c2Ta n se sCIAOITI dTals q.j-s-pue-oaoj 7,1013O q--M4 you 11a" 57T IT 3E*.

pun paymaoj ao 3Ionq q.daAs pun q:(10T13.4s sT EuTa auk uaqa 4aatiorana

-

be la 9 .........—

ze sa sT off. uoTornqTawoo Teanlonav am. Tama 'aownTj

alit uT equomooyidsTp osTmunds pun asTapaoTlo 1,00iVou ST-Elation A'sm oa looa q.aoddns pT2Ta I.ia. rAt 2uTa aoLoiTq.uno n J.oa

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spa •SWap09JJ A:v)g pT2Ta reaaueS asp. jo ouo sT b pun oAruTpaco TYmaou sT -b upqa anal. oT owns otg,s 7w0G4STUUA—Ir = V ofuoToTjj000 aqq. soq.nuTpa000 Tywaou ,9113 b pun ab uotm

'boATq.eATaop uo-p.-exaTapon sapasxoAaa JO nT1 lauT 7.714-T TA SuTpuocTsoaaoo aqq. cuoT4.Tppy uT osurequoo yq.uaToTjj000 oq.aidmoo etTI

dlIsTunAyou swop lo)ronaq aqq. uT uoTssoadxo TioTqa aoj 20SM

JO sTuatuaIa 014 TTn saaAoo uowsularnv otiq. nu-s y ssau jo quomota 014 jo soveuTpa000 unTsag.ano anTnSunqoaa oqq. axe o 'S 4x axatia

-6-

o /LP S P P

SA V

0

-10-

For the type C wing the general formula for an aerodynamic damping coefficient is

B rs

e V c 3 o where

0 (T) d3 (c'\

rs V w dor dq t00 s

2 d S &If/. d cc)

c4. m dqr dqs w dq, dqr _

m R des '14' (0 3 dqr dqs c.0).

(26)

This expression is equivalent to that given i :M nAppendix 3 of R. and. 1904

6

( 0)

Cross or Compound Stiffness Coefficients

When normal coordinates are used these coefficients are entirely of aerodynamic origin and their array is, in general, not symmetric. For a type C wing

C rs

2 2 60 = Crs (P" DV co -7

(

6 where: in accordance with R. and M. 1904

dn..ft c ors dqr

s4 0

d-1,0" d'. 0

'11421. dqs (co)

2

. (28)

( 27)

/6. Comparison

o -61/E) -61-P

• 30) 9bp abp

(Le) )( 0 Ad

hex 0

4-Pk"t)

(0) s;u0ToTjjoo0 ssaujjT;s punodmo0 Jo ssoa0

(9;)

• ( 00) arop ,)e)

° bet> P 4.14P I. Ill

°O abp °bp ANn 2bp ab-p )6)+

is )

A a Si

uosTasdalo0 '9/

O0) 5bp by_ sa • -Jr

oUp £P

I 4, roD6 6w pus 0H u;Ta aouspa000.0 uT t.1917,X

9

2upa 0 adE; la .10,E •0Ta4GMM4e 4011 guaaua2 uT gsT 4N:ire aTati; pus uTBTao oTmsuApoas JO STa.rT;ua

ars s;usToTjjaoo asom pasn axe sa;suTpa000 Ismaou uaia

• 9to6 nw pine •g JO ( xTpuaddy uT uoATS ;-au; off.;uareATnba sT uoTssaa.42La sTia

(00) 21:3P '14 sa (kg')

° P SP

sT vaaToTjjaoo 2uTdmp oTmearcpoaGs us aoj sinmaoj Isaaua auk Sure 0 adS; aq; aoa

(R)

z

6. Comparison of Inertial and Aerodynamic Couplings

When normal coordinates are used the inertial couplings are zero and in certain instances a comparison of the expressions for the inertial and aerodynamic couplings shows that the latter also vanish. More generally: the comparison may enable us to show that certain aerodynamic couplings are small.

The special uniform wings mentioned in g2 provide examples of what has been said. Here c/co is unity, the various coefficients are constants, and

dqr r dqr

where k is constant for all r; this expresses the similarity of the normal and torsional displacements in every natural mode. Hence

A d.V (1 s rs / A. „ r- 4

( 4-t r

A. pk s r k 0 s d11 ....(30)

Brs k k +,• k k tde dq d .... (31 ) s

eVc034,1 w r s r w s -r

rs e d19-' d

2 01.,. kr .. d -( PV2c

o

(32 )

and all three coefficients vanish when

(1131-,1 dqr aq s

7. Principles Governing the Choice of Freedoms

Suppose that a critical flutter speed V has been calculated on the assumption that a particular set of freedcmg is effective and let it be enquired whether some other freedom, F say, should be included in the calculation. The complete answer to ths question is given by repeating the whole calculation with the new freedom F introduced and comparison of the new and original critical speeds and modes of displacement. This method is usually extremely laborious and it is most desirable to find an easier basis of judgment. It is here suggested that judgment may be based on a consideration of the possible influence of F upon the energy balance at the critical speed Vc. If it can be shown that the introduction of F can neither add nor subtract more than a relatively very small amount of energy then F may safely be neglected.

( 29)

(33)

/The

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q.uam2pnr q.um p0q.s022ns eaaq sT •q.uomiepnr Jo sTsyq aeTeue uy puTj 0/ GITGaTOCT ;oar sT q.T pre onoTaoquI 1Stamaal.x0 STTunsn sT pom0m sT11,1

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u0AT2 sT uoTorsenb cm. 01 aGMSUIS PlOIdU00 GUI •uoTqyInoreo pm. uT pepnTouT aq pinoqs 'Sus a Imopeaaj aom.o MOOS aGLT/GTAL paaTnbua 9ci q.T

lei puu aATI.oejje sT gmope0aJ Jo has auTnoTq.ayd uop.dmnsou am uo pweinorso ua0q sm.,' A pads Joq-InTJ reo-P-Tao e Veil; aooddnS

smopeaaa JO aoToqD am VuTuaanoo saTdTouTad •/.

P-61,P

S bP oPp e0

ua-qm qsTuyA sq.uaToTjjeoo 00.114 -nu puu

( C) ) gbp abp )01 / )(I) 1?7°?I"

L.t )

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;

; i'M s 16,

M

a ar

s X N sa

P -Z<L

(00.... 1,1) sTE !LE Q

/ 13 5 \ S a it C + ,.. }.1 + 31) d 4- 31 N m: = +7

a

-61dP -6n.P j / s.,TV c

aouail •opom Tuamyu SaGAG UT SlUGMGOITFISTp TuuoTsaol pr IUMIOU am. Jo SvcautTmTs 011/ SGSSGJdX0 STLI/ fa 'Ty ao j q.uul.suoo oT N saaqm.

(6z) .......... = abp a abp

-pp

puu goluel.suoo oay sluaTUTJJeoo snoTauA eqq. eA o Tun oT eaaH •pTus uoaq suq vow & Jo

saIdmyxa apTAoad Z uT peuo-p.uam auTm maoJTun moado gq1

nums 0au TI(Inoo oTmnuAposou ureqa0o van. molls Q. sn 0-Equua Sum uosTaudmoo am 4SITua0uo2 0aow

•11sTuuA osTy a0q.TuT aqq. veqq. smous auTidnoo op:mu/Spc.10-0 puu TuTq./ouT oU/ so; suoTssaadxa 014 jo uosTaudmoo u saouvq.suT uTyl.aao uT puy 0a0Z eau auTidnoo re.p.aauT am pasn eau soquuTpa000 Tumaou uaqg

sVuTIldnoo oTmuuApoaaN. puu TuTwoui Jo uosTaudmoo .9

—14—

-12-

The influence of F upon the energy balance will certainly be small when the amplitude of F is very small. We may estimate the amplitude of F by forming the Lagrangian equation of motion maammlala to F and then assume that the other generalised coordinates have the values already calculated at the critical speed; this approximation will be justified if the amplitude of F is in fact small. The amplitude of F will then be given by

(Sum of coupling terms reversed)

(Impedance in F at the flutter frequency)

Clearly the sum of the coupling terms will certainly be small when all the coupling coefficients are small and it may be possible to see that this is so without calculating these coefficients. The impedance will be large when the flutter frequency (in the absence of freedom F) differs widely from the natural frequency for F alone, calculated at tha wind speed Ve (since the direct stiffness coefficient for F may depend on V). We conclude that it will not usually be safe to neglect F when its proper frequency is near that of the flutter. On the other hand: when the proper frequency of F is much higher than that of the flutter the impedance will be very large and the amplitude of F will be very small; hence F can be neglected.

When the amplitude of F is neither large nor very small the decision to retain or reject should be based on a consideration of the sensitiveness of the energy balance to F. If some of the stiffness couplings between F and the other degrees of freedom show large skew-symmetric components (see go then it should be retained.

8. Illustrative Examoles on the Choice of Freedoms

A few simple examples are added here to illustrate the principles put forward in V.

In what circumstances can the aileron coordinate t be

neglected in wing flutter?

The stiffness coupling derivatives L and Hi (zero) are widely different2; likewise M and Ho are widely different2 Hence the equation of energy shows that movement of the aileron may feed in much energy to the system, as is otherwise obvious. COnsequently the aileron coordinate ,g' must be retained unless it can be shown that the amplitude in:-; is vary small. This will be true when either or both of the following conditions are satisfied:-

(i) The aileron impedance is very high. This can only be at,tained by providing the aileron with a very stiff elastic constraint (irreversible control).

(ii) All the coupling coefficients affecting the aileron hinge moment are extremely small. These are the two aileron products of inertia, and the aerodynamic derivatives H&, Ho. The latter will all be small for a certain location of the aileron hinge and the products of inertia can be made very small by suitable disposition of masses.

(a)

/(h) ..4V. (q)/

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ssaujjT;o jo moo ji -a O aouvivq naeua aril Jo osauoAmsuos a7; Jo uoT;vaapTouoo v uo possq oq pTno7o goopaa ao uTu;sa o; uoTsToop

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sgT TIGIIM 3 ;03VOU O. Sy00 aq STrenon lou TIP q.T uuz opslIouoo of (A uo puodop Wvm j aoj ;uoToTjjaoo ssouJJT;s ;oaaTp Ott; oouTs) A paads

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(Wouonbeaj ao;;nu aq; ;v a TIT aouvpoolthi)

• (posaoAaa omao; 2uudnoo Jo mns)

Sct uoATB oq UG11; IITM a jo

aP/14TIcluro 0111 Q-ovJ uT oT a Jo aPrig-TIdau 0111- JT PaTJTq-onP aq TTTm uoT;umTxoaddu sTr4q. !pads iloT4Tao aq; ;v paqvinoTvo Apvaaru sonTuA

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a7; a;vmprso Sum alit 'Treys SJOA ST a jo opn;TIdtm aq; uotim Timm oq, STuTv;aoo ITTm aouureq RBaouo 07; uodn g Jo aouonijuT o7I

(e)

-13-

(b) Must we bring in overtone modes in considering the flexural: torsional flutter of a cantilever wing not carrying large ooncentraied massess?

We have seen in §2 that for a uniform cantilever wing with a certain special but quite reasonable elastic specification the t7o gravest modes are strictly uncoupled to the higher modes. For other straight cantilever wings of normal proportions the couplings will be smalls although not zero. Also the impedances in the overtone modes at the flutter frequency will be large on account of the great disparity in frequency. We conclude that the amplitudes in the overtone modes will be very snail and that only minor errors in the critical flutter speed will result frau neglect of such modes.

(c) Flexural-torsional flutter of a cantilever wing carryinE large concentrated mass

If we take one of the two gravest modes the inertial coupling coefficient with any higher overtone mode is zero. In the exeression for this 'product of inertia' the displacements at the large mass are clearly heavily 'weighted', but these displacements are not heavily weighted in the expressions for the aerodynamic couplings. Thus we have evidently no right to assume that the aerodynamic coleplin s of the modes are small just because the inertial couplings are zero. Hence the higher overtone modes can only be neglected when their impedances at the flutter frequency are very large, i.e., when the disparity of frequency is large. Hence we must expect to find it neceesaly to retain some of the lower overtone modes beyond the first

/ PART II

In general both involving both flexure and torsion; the first mainly flexural and the second mainly torsional.

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v7; put; mus SaaA aq TITra sopom auo;aaAo aq; uT soprv-rTidwv 914 VULT; gpliTOU00 OM •Souonboaj uT S;TavdsTp ;vaa2 aq; Jo ;un000v uo oBavi aq ITTa Sottenbaaj aollmij au; ;v sapow aaolaaAo 014 UT

soouvpodur aq; ()sty •O.IG2 you uZnottliv 'ITvms rq ITTa sBuTIdnoo au; suoT;aodoad rumaou jo auTa ao4eiT4ruvo ;TIZTva;s aaqw aog

.sopom aotpT4 ou; o; paidnooun ST;oTal.s aav sopom ;saivea2 °IL; ou; uoT;voTjToods qT;svia oiqvuosuoa a;Trib luq TvToo& uTv;aao v 'ma 2uTa

ao4o 14u10 maojTun v aoj ;v111. zp, uT uoas axeq. 0J

osossvm poqa;uaouoo a7avi 'AuTAaavo lou uTa aan.aiT;uvo v Jo aa;;nu-tvuoTsao; --T ranf77-7572uTaapTouoo uT gap= OUO;a0A0 UT aliq aa ;srLDI (q)

-14-

PART II

METHODS FOR PREDICTING FLUTTER

CHARACTERISTICS

9. Non-Dimensional Form of the Equations of Motion

It is convenient to reduce the equations of motion to a standard non-dimensional form. When all the displacements are proportional to exp (2.,t) the complete set of dynamical equations in the selected degrees of freedom can be written in the matrix notation

2 +DV

2 1-E 1 q=0 (

where A is the inertia matrix BY " " damping matrix E " " matrix of the elastic stiffnesses

DV2 " " matrix of the aerodynamic stiffnesses.

We shall now suppose that the m dynamical coordinates q are all non-dimensional (e.g., angles). Then, if p and 0- are air and material densities and 41. is a typical linear dimension, we may define a set of non-dimensional matrices of coefficients by the equations

( 34-)

(242)

(43)

A A:.5 a

B e 4 b

D = 6> 3 d

E = Cr-t3U2e

OOOOO O••

OOOOOOOOOO O 000 n0Q01201.0 0000000 (38)

where U is one of the velocities of propagation of elastic waves it a typical material of construction. Next, introduce the auxiliary non-dimensional variables

The equation (34) can row be rewritten in the non-dimensional form

a.)2 4.111DX+ d e)q= O.

At a critical flutter speed and for the flutter component we can put

~'- ± iw

/aAd • • • • > • •

( ) • go = b (a !7' +p + git+ vu\k

maoJ TuuoTeuemTp-uou au; uT uo;;Tamea act moil uuo (+TO uoTvenbo aqa,

(a)

(5-)

(LC)

= a

= a C

• • • a 00000000000000000 • • - • • • a • • •

V

...... me,

C) =

*6 110.p.ON JO SUOTQ:ellba. 014 Jo mod TuuoTsuamTa-uoN

•111•.4•0 pura/

lnd uuo am ;ueuodmoo aa;;nTJ at{; JOJ pare peads ao;;IliJ Tuomao u

saiguTauA TuuoTsuom7p-uou A-mu:mu et{; aoripoaluT gq_xaN •uoTlona;suoo Jo TuTaa;um luoTdE;

LIT SOAWIL OT1S1q0 JO uoTquaudoad Jo saT;TooTOA au; Jo 01I0 ST n aaatia

suoTlynba au; Sq. sluaToTJJ000 Jo sooTa;vm TuuoTsuamTp-uou JO ;as u euTJap Attu ast, guoTsuamTp auouTT TuoTdS; u sT puu samsuop TuTaequm puu aTn aan puy SIT .(saTiluy 1.2•a) puoTsuamTp-uou fly

eau b sa;muTpa000 TuoTusuAp m au; ;um. esoddns mou Maus ag

•sacsouJJT;s oTutuXpoaat au; JO XT.WSM , zha

s‘'ssouJJ-P-s 0T;svio auT Jo xTag-rm R xTaVem OuTdwaT xTaq-vw IFP-TouT au; sT V alotta

b (R+ zA CI + z\e..V

uoTlu;ou xTa;um au; uT ual;Tam OCI MD M01100.1J JO saaa!dop poToeias at{ uT suoT;unba TnoTmnuSp JO ;es a;aidmoo am. (vt) &O O. TuuoT;aodoad

axe squamooutdoTp au; 'Ty ueu •turaoJ inuoTsuomTp-uou paypuns m O. UOT40111 JO suoTluribe 21.q aonpoa o; queTuanuoo sT ;1

§0IISIHRIOVEVHO

URIand omizoiaaaa075501M

-15-

and the equation to be satisfied is

at.02 d+g e) 0.

(WO

The problem consists in finding r and W so that the set of linear scalar equations represented by the foregoing matrix equation shall be compatible. Methods for doing this are described below.

10. The Orthodox Method of Solution2.

We regard and the matrices a, b, d and e as known constants. If frequency-dependent derivatives are used this implies that sane estimate of the flutter frequency parameter has already been made.

The condition for compatibility of the scalar equations represented by (42) is the 'determinantal equation'

6-4)

(4-5)

where the expression on the left is a polynomial in X, in general of degree 2m, whose coefficients are functions of t . At a critical

.

value ofS this equation is satisfied by the pair of equal and opposite roots iCAJ. Accordingly the parts of6 containing even and odd powers ofw vanish separately and we derive a pair of equations which we may write

(t,.o 0 2 ,..•;\

(46)

= Os (47)

The result of eliminating 4,0 from these is

Tn-1 ( ) = 0

(2+8)

where Tn-1 is the penultimate Routhian test function and n is the degree in ).< of the determinantal equation. The last is a polynomial equation fora and when its roots have been found the critical flutter speeds can be Obtained from (40). Finally, the flutter frequency and the complex modril ratios can be calculated.

It is worthy of remark that the test function always vanishes when the relative density parameter 11 is zero. Thus Tn..1 contains a power of "7? as a factor and when this is removed we are left with a polynomial in 1 whose 'constant' term is the asymptotic form of the test function for low relative air density. Since the critical values of :F are then independent of ri , equation (40) shows that the 'equivalent' or indicated critical flutter speeds are independent of the densities, It appears that modern heavily loaded aircraft are in or near this asymptotic condition.

The orthodox method is systematic and yields all the desired information. Its drawback is that it becomes unworkably laborious when the number of freedoms is greater than 3.

/11. The

s0 =

•C uuq; aaquaaS sT smopeaaj jo aaqpnu aqg uolia snoTaoqui STquvomun compoeq rt votn. sT 3iouqmsap sqj .uommaojuT peaTsap

G111- TTy sPT0T-g pub 0T4vmo4-s-go) sT PO4%am x0P0T1-10 atiJ

•uo-p.Tpuoo oT;ofiximEsu sT14 aueu JO uT aau qjsaoaTs paps'oT STTAuaq uaapom weqq. savaddu gI ..samsuap 0111

Jo ;uapuadspuT *.re spends aaql_nTjjEseTI.Tao peq.uoTpuT ao ouareATnbal au1 vetn mow (01) uoTTunbo a tt,jo wepuadapuT uaqq. eau & Jo

senTuA TuaTqTao aqg eouTs .4Tsuep ars aATIA3Taa moT .xoj uoTwunj 1.sa; eq Jo mood oTwq_dmSsu eqq. sT 0.1u4suool asoq* L, uT mmouEiod

B tip.Tm ;.jeT ars empeAomoa sT sTLI; UGIIM puu aowuj B su 1 jo aemod

surs4uoa y-u

sma •0.10Z ST ti ae;amsaud SlTsuop anTluTea ot{g ualIm setisTuuA sSumrs uo-p.ounj ;se;

•peluTnoTuo eq 1177,10 SOTWP1 Tupom xoTdmoa aqq. pus Souaribeaj ael.q.nTj aqq. ISTrauTg .(0-1) MOJJ reuTul.qp oq use spaeds

aol4n1J ITE*T1-T-10 0111- Pun0J upoct anvil clooa O;T UGTA pus Iaoj uoTqxnbo TuTmouSTod B ST_ 1.suT etti •uoTq.uriba Tu;uuuTmaa;ep Girt JO >( UT GOJSSp

Sifl St u puu uoTaunj gsag uuTmAloE avemT1Tnued alp. St aaatIm

1.11,3una Jo ST-Raom- sT 1I

(0)

sT asam. MOJJ oq 911Tq.VUTMTTO jo ;Trisaa ata

(9t)'

e;Tam Sum am 1.107411, suo-p_snba jo aTvd u anTaap am puu STaquaudes tIsTusA 07J0 saamod

ppo puu UOAS 2UTUTT4U00 VJO sq.aud etTor STSuTpaoopy .o7T sqooa ao,Tsoddo puu Turiba jo aTud agq. Sq peTjsTorus cT uoTlunbeottig S' JO OnTBA

-POTIrTa° ;V 5:JO suoTorounj SIU SWGTOTJJ000 22011a .me aq

aaa2ep jo Tuaeueli UT 4 A UT reTWOUSTOCI u sT q.joi aqq. uo uoTesaadxe q, aaaqm

0 = (5‘)-(7-)C-7

uoTvenbG raq.usuTmaaq.eps ern. sT (V?) peq.ussaadaa suoTTE.-nba autaas otn. jo Sq.TTTqTveclutoa xoJuotgTpuoO ata

.apum ueeq. Spseare usq aalamuand Souanbaaj aeq.qtru atilt Jo Oq:SUIT`4_,TO °MOO 1U 7q.

seTidmT sTtig peon eau saATI.uATaap luapuedep-Souoriboaj ji •sq.uuqsuoo umou31 su a pus p lq 8.0 seaTagArm aqg pyre Li pauRaa am

• uoTornios jo Poi ow xopoma0 otlY G06

nmoTaq peuaosep eau sTql. atTop JOJ spotwN Gem;ydmoo aq TTuus uoTvenba xTaq.um 'BuTaaaoj aul Sq pawasaadoasuoTI.unboxsTuos

auouTT Jo q.es atilt q..VUI. OS ns putt BuTpuTj uT so„sTsuoo maTqoad

0 = b (a 2+ p q ; 30-c -)

sT paTjewas aq ol uowsnbe om. pus,

-g-

(m)

-16--

11. The Method of R.A. Frazer8

Frazer writes the equations (46) and (4.7) as

f(x,y) = 0

(49)

g(x,y) = 0

(50)

where x e -W2

(51)

and Y = •

(52 )

Since x and y are both real they could be found from the real intersections of the curves (49) and (50). Frazer shows that the full expressions for the functions f and g can easily be calculated from expressions in partial fraction form containing the values of (M) corresponding to a special set of points (x, y). These are the inter-sections of a standard set of straight lines satisfying the conditions

(a) No two are parallel

(b) No three are concurrent.

Much labour can be saved by use of this method when the number of freedoms is large. It is to be regarded as a special development of the classical method, designed to minimise labour.

Frazer and Bratt have proposed mechanical and electrical methods for the description of the curves (49) and (50).

12. Methods Based on the Use of Matrices

The determination of the roots W for a given g can be made to depend on the problem of finding the latent roots of a certain matrix (Duncan and Collar9 ,10). An iterative method for finding the latent roots without expanding the characteristic determinant is given by Duncan and Collar and other treatments are described by Hermann and Dtirrli. It is not considered that these methods have yet been developed to the stage where they would be advantageous from the aspect of labour saving in flutter problems.

13. The Method of Durw,_.2.a u]aara 7

This method is based on the fact that-the mechanical impedance of the system is zero at the critical flutter speed and for simple harmonic applied forces having the frequency of the flutter. The method consists, in effect, in calculating the impedance for a number of assumed air speeds and frequencies and determination of the condition of zero impedance by interpolation. Since the impedance is found merely by solving a set of simultaneous linear equations with known coefficients, the calculations are relatively simple. The method has the further advantage that the modes of oscillation in the flutter are obtained incidentally from the calculations.

/It .

Frazer does not use the special reduction to the non-dimensional form adopted here, but his method is essentially as stated.

•pavels se STIvT;uosso sT paq;am sp.; ;nq gaaoq pa;dopu maoj TvuoTsuati4p-uou aq; to; uoT;onpoa TvToods eq; osn-;ou coop aezvaa

•suoT;vinoTvo moaj Sav;uopTouT pouTv;qo GJV J9q111IJ aq uT uo WeTiToso Jo sopoom *11q- o2V;uvApv agq;anj at{; cut{ poqlem a1W •oTdmTe SToAT;vTaa are suoweinoTvo Ggq.

swaToTjjam umoupT 11;T& suoT;unbo avauTT snoauuvrnmTs jo Q_GS V BUTAIOC £q STaaompunoj eT pouvpodmT aouTs •uoT;uTodao;uT Sq etouvpadm7 oaaz jo uoT;Tpuoo aq; Jo uoTlvuTmae;ap puv saTouenbaaj puu spoads aTv pamnssv jo

aoqunu v aoj oouvpadmT aq 2uT;vInoTvo uT ‘;oajja uT gs;sTsuoo poq;om aq1 eao;;Tu aq; jo Souonbaaj SuTAvq sooaoj poTTddu oTuoulavq oTdmTs xo; pin poods ao;lnij TuoT;Tao aq; ;v OIGZ ST 111010S0 Ott; JO

aouvpadmT TvoTuvqoam aq;-;vq; ;ouj 914 uo posvq sT poqqam sTqI

L uo Z puv xt=TTo3 usoung Jo par. •CI.

esmsTqoad aa;;nTj u-c BuTAus anoqvi jo ;oadsv eq; mox; cnoo2v;uvApv act pi-nom Soq; aaaqa aSuls aq; ol

padoToAop uooq ;ag spoq;am osaq; veq; poaopTsuoo you sT ;I 0t4aaRa pub uuumaoll Sq poqTaosap oav sluamqvaa; aoqlo puv auTToo put: uvouna Aq

uoAT2sT ;uuuTmao;ap oT1sTaaflovivqo aqf 2uTpuudxo lnoq;pA s;ooa ;uo;vT at{; 2uTpuTj aoj poqqam oAT;uao;T uy .(006-113TT00 puu uvouna) xTa;vm

uTs;..too v jo eqooa ;uoveT at{; 2uTpuTj jo moTqoad au; uo puodop ol opvm aq uvo u3AT2 v .IOJ t< S400.1 aq; Jo uoT;vuTmao;op aqa,

2GOTJ;VE JO GOfl gq; uo posvg opt:glow 031.

.(0c) puv (647) soAano jo uoT;dTaosop aq; JOJ eparilam TuoTaloaTe puv TuoTuvqoam posodoad oAvq lluag puu aozvaa

•anoqui osTmTuTm o; pau2Tcop 4porp.om TuoTssuTo aq; Jo ;uomdoTeAop TvToads u sv popav2aa aq a; oT ori 'o2avT s• smopaaaj Jo aeqpnu aq; uoqm poq;am sTq; Jo asn Sq pis oq uvo anoqvi ton♦

•;uoaanouoo aav aaaq; oN (q)

ToTTured GJV o OH (V)

suoT;Tpuoo aqq BuTSjsT;us souTT ;q2Tuals Jo ;os pavpuu;s v jo suoproas -./.9;uT aq; oav osoqI e(S swTod jo ;as TvToads v o; 2uTpuodsoaaoo (11).4) a jo sonTvA aq; 2uTuTv;uoo maoj uoTlovaj TuT;avd uT suoTesaadxa

MOJJ polvInoTuo aq Sueva use 2 puv j suoT;ounj oq; .xoj OUOTSSOldX9 Tut; lvq; smogs xaweig *(0c) pus (6i) soAano at{; Jo suoT;oasao;uT

aq; moaj punoj aq pTnoo Soli; Tvaa qwq axe pus x aouTs

(c)

= s pus

(

.. (oC)

(61)

1

0

=

=

x

(Z‘x)2

(ic'x)j

G.Tattivi

cv (Lir) pir (94) suoT;vnbo aq; solTam IGZVJJ

9aGreas: •IT•E Jo potiq.ow ata '14

-17-

It is ourxnient to make the calculations in the following manner. A-cbitily assign the amplitude of one dynamical coordinate (say, put q_ = 1) and. use the dynamical equations corresponding to

free motion in the other coordinates to calculate the amplitudes o2 these for the assumed air rir.)eed and frev.ency; this requires only the solution of a set of simuaaneous linear eouations whose coefficients are in general: cc alex on account of the presenoe of the damping terms. Tile c..-Tlitudos of all the dynamical coordinates are now kno• and when tAece are inseted in the equation for the coordinate qv the expression on the left hand side is c&.ival to the amplitude of the Laneralised for,'e Q_ which must be apnlied in order to maintain the assumed :notions This is the impedance, in general cempie since q, is unity-1 The vanishing ca7:lax impedance implies two condaiuns and it thus becol7;es possible to find both the critical flutter speed and frequency by interpolation when the impedance has boon criculated for a few pairs of values.

cuick success of this method evi&ntly depends on making a good prel:iminay gess or estimate of the critfr.col flutter speed and frequency

14, Another InvQrse Method PM, ••■ c.n.•■••"..,11.-■1• -■•• -

7;e may preoeced exactly as in the last method until we ;.•each the final (:',yaas-nical equation correETending to the assigned coordinate or., ilc-7..eve, the expression on the left hand side of the equati7m is equated to zero (free motion) while ! in this is regarded. -unkne7,n, We shall thus derive a value IF' for P in general complex; the cnnditioli to be ,,atisfied is

which is to a pair of real ovations. The critical values of E and a_e to be obtained by tzlal interpolation, as in the pt.viousj_y described method.

L

*pou;Gm p‘qTaGsGp LTsnoTLId ou; uT su cuoTTuTod2G;uT pT. , TuFz; Ec pGuTplqo oq m T:Tru jo cr,nreA

Ivomao asuo-I4p,tho Tara Jo ed t o; :7:GinNTnbo oT goTuAk

sT TJoTcysT;vc:, Gq o; uoT;Tpuo Gm :XOTC:MO TC1OLM3 UT 6 GnipA p GATJap Gg po,712.Ga sT sTu; uT GITITA (uoTqc,11 aG...cj) o.x z 0q. po;prabG vT wTqx-nhe au; Jo

GpTs puptl ;JGT Gm_ up uoTssaxlxo .;:a3.,1011 9 0 ,g_puTpal000 pGu2Tsep Gm. o; 2uTpuodJGasloo uoTvenT:* 1PoTtLc.v.4:;? TouTj au; Tove,7 as tAuu q.etT mil. TIT ea Stquaxa Doad X'au

a6i4;51.7

'SouGnbGaj pup peGds aG;;IITJ Two;Tao Jo GI:sTlea .20 ssG2 JCpu1izi:TG.2(1. poo0 p 1rpjem U0 spupdGp ST;u;,:pTAs romaul =BTU- jo scGoons T)TnT) Gux

•sGnIp.A. JO saTpd PIRJ u JOJ pG11Ino-po uGoq sP11 GouvpadulT Gu; uoTTuTodaG;uT SouGnbGJJ pup pools aG;;TITj TpoTqTao

du; moq puTj a; GIqTssod SO%20D9q. snug_ ;T pun 11o71Tpuoo soTidiuT Gour.podmT xoTJ::co ^7:1 ,;H SuTTJ,sTunA GuA 1Z1 Tun bE Y i C faidtpo

Iulou92 UT Gm sT sTuI -41.1oTTow. pG,Tinsep -(1; uTpluTuu o; aopao uT pGTIddp Gq ;snm To-pqm ") pGsT-E•2Guo(3 uq; Jo

i4pn;T-cdme ou; o; TprIT)o sT 3-Ts pup, ;JGT Gm Tito aoTs7JoacTxo au; Lb a;puTpa000 Gm aoj uoT;v150 uT pG;J:GsuT Gan GZPV4 7e2. pikp u!Aou3T

AOU TEOTM-OU4 ou; Tin Jo sopn;TTa.::: G71 4sm2G; )1I-Fclmup om. Jo GouGsGd. Gq; Jo wu000P uo IcGT.d.00 ITv2ou02 UT sa.Ip

eWucTDTLT.Tv00 ae0TA slAcqq-unLa -7130uTT au00=;:lintETe go las. J0 uoTluToe Gm JCIuo soaTilboa sTm !./.7ouT)0,-r.J pun pGG(12 pownssp Gq; .Tod GS0T/q.

ro sopn;TIdmu Gul G;vinoTso o; sG;PuTpa000 a.)1qq.0 ou; uT uoNFiva Gaaj o; BuTpuodsGlaoo suoT;pri.bn TvoTa3u4 pup(; a'f.) ;u171

G;puTpa000 TpoTopu4 Guo Jo orm;TIduld Gu; u?Tssu GaGuuum 2uTa0IT0J atr4 uT r3u0*V-VaTuora 0::vm 04 ;T-TK-2A71.°D sT

-18-

LIST OF PRINCIPAL SYMBOLS USED

The symbols are in alphabetical order and the Greek letters follow the Roman,

An accent indicates the transposed of a matrix.

A dot above a symbol indicates its time rate of change:.

The numbered references are to equations.

A

matrix of inertial coefficients

A rs general inertial coefficient (see (21) )

Clrs See (24).

non-dimensional form of A. See (35)

B

matrix of damping coefficients or the same divided by

V. See (34)

B1 symmetric part of B. See 5A

B2 skew-symmetric part of B. See 5B

rs general drmping coefficient. See (25)

63 rs Sec (26)

b

non-dimensional form of B. See (36)

ratriw of stiffness coefficients

symmetric part of C

2 skew-symmetric part of C

Crs general stiffness coefficient. See (27)

rs See (28).

chord of wing

co chord at wing root

D

matrix of aerodynamic stiffnesses, divided by V.

S (34).

non-dimensional form of D. (37).

matrix of elastic stiffnesses

e non-dimensional form of E. See (38)

F

dissipation function. See (7)

distance of local flexural centre aft of leading edge as fraction of local chord.

/j. • • ill

• • •c/

•paouo ivoot jo uoTTovaj sv aloe 2uTp2aT Jo Tjv oaTuoo TvanxoTJ Tvoot Jo eouvTsTp

(L) oos •uoTTounj uoTTvdTseTp

(gc) eos •g Jo ULIOJ TvuoTsuomTp-uou

sossaujjTTs oTTsvio Jo xTaTvm 5-

.(LC) oos. 'a JO WJOJ TvuoTsuomTp-uou

'(-1c) 00S

' A A 'FTTATp 4sassaujjTTs oTmvuApoaov jo xTaTvm a

Tooa BuTm Tv paouo o

BuTm Jo paouo

s

°(sz) ov>s a

(La) :),9S •TuoToTjjaoo ssoujj-ps Tvaauo2 sa 0

0 Jo Tavd oTaTommics-lamTs G0

Q JO Tavel oTaTommSs

sTuaToTjjaoo ssoujjTTs JO -.-i:TaTv,a

(90 Gos •q Jo maoj ivuoTsuomTp-uou

(9z) oas sa

(g3) oes •q.u@ToTjj0.00 SuTdmap IvaaunS sea

SS aes •i" Jo Tavd oTaTommgs-likav Zs

vC eau •g Jo la-wl oTaTommSs

(-W) *GS 'A Sct pep-ps.Tp GUMS auT JO OIAIOTOTJJ000 SuTdmvp Jo aqaTum

(CC) aos jo w.xoJ TvuoTsuamTp-uou

'Oa) 00S saltO

( (1z) aos) TuaToTjj000 TvTlaauT Ivaauo2 av

sTuoToTJj000 Tv-maauT jo xTaTvm

•suoTTuriba off. 0.1-0 seolloaajaa peaaqmnu auZ

•auvuo Jo eTva amTT sTT saTvoTpuT '0qm/co v eAogp Top v

•xTaTvm P JO posodsuvaT ouT soTvoTpuT Tu000v

fuvmoa ouT Aiouoj saoTTei )1asa-D au; puv aspao reoTToqvudiv uT axe sToqmEs au'

assn SUN= avaIDNI-da dO ISI7

-19-

kr

local value of moment of inertia about leading edge, per unit span, divided by crek

See (29)

distance from wing root to reference section or other typical linear dimension. See also 'n.

Local non-dimensional aerodynamic w 0( derivative coefficients referred to

mss leading edge.

m number of dynamical coordinates

m

local value of mass per unit span, divided by

local value of mass moment about leading edge, per unit span, divided Lyac5.

q column matrix of dynamical coordinates, which are taken to be non-dimensional.

go See (16)

cr r--

th dynamical coordinate or its amplitude

kinetic energy of system

T periodic time of flutter

Tn-1 penultimate test function in Routhis sequence

t

time

iT a velocity of propagation of elastic waves in a typical material of construction

'potential' of system. Coe (9)

V true air speed

lateral distance from plane of symmetry or wing root

z normal displacement

y/?.. in (23), (25) and (27)

local twisting or pitching displacement

downward normal displacement of leading edge, divided by co.

non-dimensional speed coefficient, See (40)

. . .

tu

1-5 iab

ogril-TIdEgoe 01.T ao oVeuTP-1000 TeoTmnuAID--a

(90 aas

'relloTsuamTp-uou aq oq ua3ra; a-133 uoTtlm 4sapeuTpa000 reoTumu4 Jo xTa;sm utunToo

•cos)fq papTATp fuvds ;Tun 'ad 'ape 2uTpsai ;noTe 1.110MDM SS•M JO anTeA [soot

• oD dg papTATp 4trads ;Tun aad ss-am Jo anTEA reoot

>4/

(m) 199g G;uaTaTjj000 paads reuoTsuaturm-uou

'o

-gq PoPTATP cap° attweaT Jo Tuatuaot3TdsTp remaou pannumop

quemaatirIdsTp EuTtla;Td ao 2uT4sTm4 IvooT

(i2) we (gz) '(c3) uT Ws

luameoBIdsTp immou

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at;

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-20-

a non-dimensional frequency parameter. See (44)

coefficient of t in exp (A,t)

/"- real part of >s.

P air density

density of typical material of wing

coefficient of imaginary unit in > or. See (16), (43);

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-21-

LIST OF REFERENCES

No. Author Title etc.

1

Gates, S.B. The Torsion-Flexure Oscillations of TWO Connected Beams. Phil.Mag., Jan. 1928.

2

Frazer, R.A. The Flutter of Aeroplane Wings. and (A.R.C. Monograph).

Duncan, W.J. R. & M. 1155, Aug. 1928.

3

Frazer, R.A. Conditions for the Prevention of and Flexural-Torsional Flutter of

Duncan, W.J. an Elastic Wing. R. & M. 1217, Dec. 1928,

4

5

Frazer, R.A. and

Duncan, W.J.

Duncan, W.J.

The Flutter of Monoplanes, Biplanes and Tail Units. (A.R.O. Monograph) R. & M. 1255, Jan. 1931.

Note on the Flutter of Complicated Systems. A.R.C.7630 (Unpublished) April, 1944.

6 Duncan, W.J. The Representation of Aircraft Wings, Tails and Fuselages by Semi-Rigid Structures in Dynamic and Static Problems. R. & M. 1904, Feb. 1943.

7

Duncan, W.J., Oscillations of Elastic Blades Collar, A.R., and Wings in an Airstream.

and - R. & M. 1716, Jan. 1936. Lyon, H.M.

Frazer, R.A. Bi-variate Partial Fractions and their Applications to Flutter and Stability Problems. Proc. Roy. Soc., A, Vol. 185, p. 465, 1946.

9

Duncan, T.J. Matrices Applied to the Motions and of Damped Systems.

Collar, A.R. Phil. Mag,, Ser. 7, vol. 19, p.197, Feb. 1935.

10 Frazer, R.A., Elementary Matrices. Duncan, W.J. Cambridge University Press, 1938.

and Collar, A.R.

11 Herrman, A. Forschungsbericht and No. 1769, Feb. 1943.

DUrr, J. Translation by Miss S.W. Skan 'Methods of Investigating Flutter Behaviour when there are a Number of Degrees of Freedom: A.R.C. 10,439, March, 1947.

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