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MINISTRY OF SUPPLY

AERONAUTICAL RESEARCH COUNCIL

REPORTS AND MEMORANDA

n. & Mo No.. 2904(13,567)

A.R.C. Technical Report

-.~

A ote on the Dynamic Stability ofAircraft at High-Subsonic Speedswhen Considering Unsteady Flow

By

w. J. G. PINSKER

Crown Copyright Reserved

LONDON: HER MAJESTY'S STATIONERY OFFICE

1954PRI CE 7s 6d NET

A Note on the Dynamic Stability of Aircraft at High­Subsonic Speeds when -Considering ~nsteady Flow

By

w. J. G. PINSKER

COMMUNICATED BY THE PRINCIPAL DIRECTOR OF SCIENTIFIC RESEARCH (AIR).MINISTRY OF SUPPLY

Reports and Memoranda No. 2904*

May> 1950

Summary.-The effect of an increase in speed relative to the speed of sound on the unsteady flow round a harmonicallyoscillating aerofoil, is to increase the lag of the aerodynamic forces and moments behind the deflection when the frequencyis small. It is shown theoretically that this will result in a serious deterioration of the damping of both the lateraloscillation and the high frequency longitudinal oscillation with high Mach numbers. Use is made of derivatives calculatedfor flutter purposes to estimate the unsteady derivatives at aircraft oscillation frequencies. Illustrative examples arepresented.

1. I ntroduction.-Calculations on the dynamic stability of aircraft are generally made withquasi-steady aerodynamic coefficients, i.e., the forces and moments acting on each aerofoilsurface at any instant of an oscillatory motion are calculated on the assumption that the aerofoilis in steady motion under the conditions pertaining at that instant.

In the past flight-test results have confirmed the quasi-steady theory within the accuracypossible in such tests. Recent flight tests with the 11Jeteor, however, showed a marked divergencebetween the measured damping in yaw with rudder fixed and theoretical estimations based onquasi-steady aerodynamic derivatives ; this divergence increased with Mach number. Thedifference between theory and experiment is shown in Fig. 1, where the logarithmic decrement <5

of the directional oscillation is plotted as obtained from flight tests; quasi-steady theory andunsteady theory. It is seen that the observed loss in damping with higher Mach numbers cansatisfactorily be explained by the effects of unsteady flow.

Recent flight tests on a tailless aircraft at high speeds have also tended to throw doubt on thequasi-steady theory and some experiments in the free-flight tunnel at Langley Field by W. E.Cotter" seem to confirm the results of theory considering the effect of unsteady flow on thedamping in pitch of tailless models even with incompressible flow.

The effect on the damping of aircraft of taking into account the unsteady flow conditions wasfirst considered by Glauert'. In common with more recent publicationst'" this treatment dealtonly with incompressible flow. Under these conditions the effect of allowing for unsteady flowdoes not appear to be large for aerofoils oscillating about a point forward of the quarter-chord,at any rate for the range of frequencies likely to occur in flight.

In this report an attempt has been made to take into account the unsteady conditions incompressible flow. The treatment is confined to conditions when no shock-waves are present,and it is assumed that compressibility effects on the derivatives obey the Glauert Law.

* R.A.E. Report Aero. 2378, received 7th December, 1950.

(61801) A

Owing to the inadequacy of the basic data needed for this work, the results presented in thisreport are far from being complete, and the conclusions are therefore somewhat tentative. Thereseems little doubt however that with increasing Mach number the reduction in damping due tothe unsteady flow conditions become very much more marked, and there are importantimplications of the theory on'the damping of tailless aircraft at high speed. The purpose of thisreport is mainly to draw attention to these extremely important trends, and to suggest the linesalong which future theoretical and experimental work should proceed in order to investigatethem further.

After the completion of the work presented in this report an American paper" on the instabilityof the oscillation of an aerofoil in incompressible flow has been received which arrives at con­clusions very similar to those represented by the stability boundary for Mach number = 0 inFigs. 9 and 10 of this report.

2. The Physical Basis of the Theory of Unsteady Flow.-For detailed information on thesubject reference should be made to the extensive special literature, in particular to two recentpublications by G. Temple" and A. 1. Necrasov', For readers who are not familiar with theaerodynamic theory of unsteady flow this section is intended to give a very brief descriptionof the main physical phenomena involved. Generally the flow pattern and thus the pressuredistribution round an aerofoil moving in a frictionless fluid is determined by the geometry ofthe aerofoil itself and by velocities induced by the free vortices, mainly those in the vortex wakebehind the trailing edge of the aero foil. In steady motion there is only one starting vortexwhich is assumed to be curled up at an infinite distance aft of the trailing edge. •

Vortex sheet and starting vortex aft of an aerofoil in steady motion after a sufficiently long time.

Since during an unsteady motion of an aero foil the circulation round the aerofoil changes withtime there is a continuous emission of new starting vortices into the wake, the distance of eachvortex from the aerofoil increasing continuously with time as the aerofoil moves ahead. Forexample during an angular oscillation of an aerofoil the distribution of the free vortices aft ofthe wing will appear as sketched below.

a =ClO'l iwt

A=TV

Free vortices in the wake of an aerofoil oscillating with the period T and the amplitude IXO'

Since the flow pattern round the aerofoil is affected by the contributions of the free vorticesaft of the wing it must be expected that:

(a) the pressure distribution round an aerofoil in unsteady motion differs from that obtainedwith quasi-steady flow

(b) the main parameter describing the aerodynamic situation round a harmonically oscillatingaerofoil will be the ratio between the chord c of the aerofoil and the wavelengthA = TV of the oscillation. This parameter is usually given as the reduced frequency

2ncIt = TV' (1)

Matters become more complicated if wings with finite span are considered smce then theinduced flows from the unsteady trailing vortices have to be considered as well.

2

"

IILI

Another contribution to the forces acting upon a wing in unsteady motion which is not takeninto account in the quasi-steady theory is caused by the reaction of the air being accelerated.This can be interpreted as an increase of the apparent mass and inertia of the aerofoil.

Finally the speed of sound enters into the calculations, since all disturbances and inductionscan only be propagated with the speed of sound. Thus the velocity of propagation upstreamrelative to the aerofoil decreases with increasing Mach number and this will result in a delay ofthe aerodynamic forces at the aero foil. The occurrence of shock-waves interferes with thissimple physical concept. The calculations referred to in the next paragraphs only hold forcompressible flow without shock-waves at the aero foil concerned.

3. Available Numerical Results.-Numerical solutions for unsteady aerodynamic coefficientshave been obtained by many authors. The most comprehensive results" available are fortwo-dimensional flow and they cover a wide range of speeds including incompressible flowconditions, the high-subsonic and the supersonic range of speeds. Solutions exist too for theunsteady coefficients due to flap deflections with various configurationsv", Some authors':" haveinvestigated wings with finite aspect ratio up to the extreme case of an ocsillating circular wing";but these solutions are for incompressible flow only.

As far as is known there are no reliable solutions for bodies of revolution representing fuselagesor nacelles.

Another important limitation is that the majority of the theoretical solutions apply to constant­amplitude oscillations and they are therefore applicable strictly only to the boundary conditionsof oscillations where the damping is exactly zero. Jones has shown how the unsteady forces varywith the damping of the supposed motion. In order, however, to simplify the approach in thispaper, it will be assumed that the more extensive data on the derivatives applying toconstant-amplitude oscillations, also apply to damped motions. It must be remembered,therefore, that the solutions given in this paper are strictly true only at the stability boundaries.

In papers dealing with the flutter problem, the aerodynamic coefficients are given in theform of complex numbers, e.g., .

~=~+~ ~where llX is the rate of change of lift with angle r/. during an oscillation. The real part, lrxR' is thecomponent in phase with the deflection considered, the imaginary part is the component in phasewith the corresponding rate of deflection, thus representing a damping or undamping effect.The apparent mass effect is sometimes given as a separate term which as a real number can beincluded in the real part of the coefficient, that is in phase with the deflection, though strictlyonly as long as the oscillation is of constant amplitude. This term is however small at the low

ci frequencies to be considered and the errorin including the acceleration term in thedeflection term is small for the general,damped motion to be considered; therelationship between the two terms is ofcourse dependent on frequency but sinceboth are already functions of frequencyno complication is added by puttingthem together. An example of thephysical meaning of a complex coefficientis demonstrated in the figure, inwhich llX defined according to equation (2)is plotted in the complex plane. Theangle between the lift vector llX and the

Physical interpretation of a complex lift coefficient in vector form. deflection referred to IX is the phase lag of

the lift due to unsteady flow81 = ost,

where 81 is the phase-angle lag,t, is the lag in time.

(3)

3(61801) A2

The modes of oscillation generally considered in the compilation of flutter coefficients are anangular oscillation IXoe i wt and a translational oscillation zoe iwt of a rigid aerofoil; several modes ofstructural deformation are also considered, but we will not concern ourselves with them here,though strictly they should be considered when the frequency of the aircraft oscillation isapproaching the frequencies of the wing modes, etc.

The axis of reference for the moments and for the centre of angular oscillation concerned, inthe various papers is by no means constant. The figures extracted and presented in this report,however, have been corrected to refer uniformly to the quarter-chord point of the aerofoilconsidered. Most of the data has been extracted from the summary paper, Ref. 5; in this paperthe coefficients have been presented in the form

t, = (lz + tu, + A2l;) ~ ' etc. (4)

where Z is the lift force on the aero foil

z is the oscillatory vertical deflection, etc.

They have all been converted to the form given above

i, = lZR + ilz i

with the acceleration term l; included in the lZR term as explained above.

To give a few typical examples in the vector form, results are presented in Figs. 2 to 6 ofcalculations of the aerodynamic forces and moments of an oscillating wing with infinite aspectratio in compressible subsonic flow extracted from Refs. 5 and 6. The coefficients are plottedas vector loci with Mach number and the reduced frequency A as parameters. Fig. 2 shows thelift coefficient l" corresponding to a translatory oscillation; Fig. 3 the lift coefficient lrx due to anangular oscillation about the quarter-chord axis. The corresponding moment derivatives m, andm; are given in Figs. 4 and 5. Finally the unsteady flap effectiveness lTl for the oscillation of aflap with 20 per cent chord about its hinge axis is plotted in Fig. 6.

The vector loci of the lift coefficients l ; and lTl start with frequency zero (A = 0) at a finite realvalue which corresponds to the steady lift slope a1 (or control effectiveness a2 respectively).With increasing frequency these lift forces are reduced in magnitude and delayed. At frequenciesfar beyond the range corresponding to the aircraft oscillations considered (A < "'-' 0 '1) a dampingeffect becomes more marked until at frequencies corresponding to A > "'-' 0·4 the lift vectorscross the real axis and advance against the corresponding deflections.

With a translatory oscillation of an aerofoil the aerodynamic incidence is actually producedin the velocity phase z of the motion. Thus, when discussing l, in Fig. 2, the correspondingquasi-steady term is a vector at the positive imaginary axis which increases in proportion to thefrequency parameter A. Compared with these quasi-steady values, the unsteady lift l, behavessimilarly to t: and i;

4. Equations of Motion.-The assumptions made in developing the equations of motion totake into account the unsteady flow conditions are :-

(a) The oscillation will be assumed to be in two degrees of freedom, i.e., the effect of variationin forward speed will be neglected in the longitudinal oscillation, and the rolling motionwill be neglected in the directional oscillation.

(b) As explained in section 3, the derivatives obtained from constant-amplitude calculationswill be assumed to be applicable to damped motions and the apparent mass effect ofthe air will be included in the derivatives in phase with deflection.

(c) The unsteady derivatives are all dependent in some way on the frequency parameter, A,but in general the equations of motion will be derived assuming constant derivatives.If necessary the solution can be improved by successive approximation using theobtained frequency for an improved estimation of the derivatives.

4

When steady-flow conditions are being assumed the wing incidence (X( = wjv) and the rate ofpitch q are convenient variables to define the aerodynamic condition of the aircraft at any instantof the longitudinal oscillation. Similarly (:J(= vjV) and rate of yaw r are convenient variablesto define the directional motion.

With unsteady flow, however, the aerodynamic load exerted upon an aircraft with a giveninstantaneous incidence (X differs according as to whether this incidence is produced by a changein attitude or by a translatory motion of the aircraft.

This means that a great number of terms has to be used for describing the aerodynamic loadson the aircraft when unsteady effects have to be taken into account. To avoid confusion withflutter theory we shall use the flutter parameters (X* and z and their derivatives. (X* is the angulardeflection of the aerofoil during an oscillation and is identical with the parameter () usual inaerodynamic theory for longitudinal motion, or ?jJ for directional motion. Similarly z defines thetranslatory motion so that for longitudinal motion

and for directional motionZ= -yV

Z=-Y=-xV.

(5)

(6)

4.1. Longitudinal Oscillation.-The two independent variables describing the longitudinaloscillation are () and y as shown below.

-z

mean flight path

The downwash delay term m;, will be assumed to maintain its quasi-steady definition since itis unlikely that unsteady values for the downwash 8 are available. In order to express therequired variable ci. in terms of the above variables, use is made of the relationship

(X=()-y. (7)

Now the differential equations of the pitching moments M and the lift forces L can be writtenas :-

aM aM aM i . aM i . aM l . .. ]ae () + ay y + a(()ljV) v () + a(yljV) v y + a(ci.ljV) V (e - y) = Be I.aL sr. »t. i , »t. lae e + a:yy + a((}ljV) V () + a(yljV) v y = - m; = mVy

5

(8)

Expressing the aerodynamic derivatives in the usual way (see list of symbols and section 5)and introducing, as usual

mttl = pSl '

this gives the determinant

me + ".: + m,;;) 2tll -.- A -;- -.- - At B t B t[l

and

(9)

The corresponding frequency equation is a cubic

,1.3 + A"A2 + B"A + C" = 0 , .with the coefficients:

A" == _ {~o + ~~ _ (1 + Zy)-l (Zy + ZIJ [":,y - ~~J)lt B t B ttl ttl t B t n f

(10)

Contrary to the corresponding solution of the orthodox treatment with quasi-steady coefficients(equation (16)) the solution of equation (10) contains an aperiodic mode of motion the dampingof which is described by the real root of approximately ,1.1 =:= C"/B,,, The existence of such anon-periodic motion contradicts the definitions of the derivatives used which have been derivedfrom an oscillatory type of motion. Thus the aperiodic solution ,1.1 can be expected to have nophysical reality. In fact the two terms in C" (equation (11)) when considering steady motionare identical with the quasi-steady coefficients,

(12)

Thus C" = 0 and consequently ,1.1 = O. The identity (12) holds with very good approximationfor unsteady flow as well, i.e.,

Thus generally C" =:= 0 and the cubic collapses into a quadratic

,1.2 + A"A + B" = 0 " (13)

with the coefficients A" and B" from equations (11). Damping time t; and period T', of the highfrequency longitudinal oscillation are then given by:

2lt: = A" (14)

mwhere l = pSV

2nlT" = V(+ B; - A,,2/4)

6

(15)

When comparing equations (11) with the orthodox solution with quasi-steady derivatives:

{ m; m; (1 Zq)}A 0:* = - --;- + Zw + -. + -'t B 'tB ttl

(16)

it is seen that apart from minor contributions the following coefficients are comparable withrespect to their effect on damping and frequency of the longitudinal oscillation

m; corresponds to m;

mqquasi-steady flow "

"

" unsteady flow.

[ m; " "m:w

Therefore it can be concluded that an approximated estimation of the effect of unsteady flow onthe longitudinal oscillation can be obtained by substituting the above unsteady terms for thecorresponding quasi-steady derivatives. This process may be advantageous when convertingexisting quasi-steady solutions of more complicated degree into unsteady solutions, in particularthe lateral stability case with freedom in roll, and also for treatments of control free stability.

4.2. Directional Oscillation.-The two parameters defining motion with two degrees of freedomare yawing '/jJ and the lateral displacement of the e.g. : y. As shown in equation (16) and thefigure below, y can be expressed by the azimuth angle at the flight path x-

flight path

-----,--flighl palh X

Sidewash can be neglected. Then the differential equations of the yawing moments N and theside forces Y for unsteady aerodynamic derivatives can be written:

oN oN oN b. oN bo'/jJ '/jJ + ox X + o(1pbj2V) 2V '1jJ + o(ibj2V) 2V i = C1jJ (17)

av oY oY. b oY ba;;; '/jJ + ax X + o(1pbj2V) '/jJ 2V + o(ibj2V) 2V i = my = mVi . (18)

Introducing as usual l, tt2 = 2mjoSb and ic = 4C jmb2 and performing the same operation as withthe longitudinal stability equations one obtains a cubic

A3 + A",A 2 + B",A + C", = 0

7

(19)

with the coefficients:

(20)

Again the identity

In.xy ,,, I == \n."'Yx I == n.V

y v .. (21)2c w=o 2c w~o 2c

makes C", = 0 and reduces the frequency equation to a quadratic with the coefficients A",and B",from equation (20). Damping time t,p and period T", of the directional oscillation are then

2ct; = A .. (22)

'"2nc

T", = V(+ B", _ A",2j4) . (23)

The coefficients of the corresponding quasi-steady frequency equation read

A/ = - {~: + Yv}

B",* = + {fl2 "!;!!(1-.2!) + ,,:r yv }2c fl2 2c

(24)

Comparison between equations (20) and (24) again indicates that there are main terms whichcan be substituted with an approximated treatment:

n; corresponds to - n;

quasi-steady flow n,

Yv"

"

"

"

n; unsteady flow.

Yx5. The Formation of the Aerodynamic Derivatives for the Complete Aircraft.-Having obtained

the solution of the equations for longitudinal and directional oscillations in section 4, all thatremains to do is to substitute the values of the unsteady aerodynamic derivatives into the ex­pressions for frequency and damping. A brief explanation is given in this section of the methodused to obtain the particular derivatives for an aircraft from the general flutter derivatives.

Now as explained in section 3, the flutter derivatives used in this report are given in the form

l" = l"R + il"i' etc.where

3CL R f( wC )l"R = 3()(* = A = V ' Mach number, A

t: = 33:L*i = f (y, Mach number, A )

and ()(* is the angular deflection of an aero foil during an angular oscillation.lift due to a translatory oscillation of an aerofoil is given by

3CL R 3CL;lZR = 3(zjc) and t; = 3(zjc) .

8

(25)

Correspondingly the

(26)

The moment coefficients CM about the axis of reference of an individual aerofoil (in this reportthe quarter-chord line is used throughout) are represented analogously by complex numbers:-

oCMR oCM im rzR = oe<* mrzi = 30:.* (27)

oCM R oCM im zR = o(z/c) mzi = o(z/c) . (28)

All these coefficients are functions of Il, Mach number and aspect ratio A, Numerical figuresfor the flutter coefficients in the form needed for the estimation of the stability derivatives aregiven in Tables 1 and 2. These values are computed from Ref. 5 and hold for two-dimensionalflow with a thin aerofoil. If a1 is the lift slope of the actual aerofoil with an aspect ratio A thenwe can obtain an approximated estimation of the three-dimensional value by multiplying thethin-aerofoil flutter derivative by a factor a1/2n. This value of a1 can either be taken from steadywind-tunnel tests or it is approximately given by the relation

Aa1 = A + 2 ao • (29)

5.1. Longitudinal Derivatives,-We have to evaluate the terms:

Zo, Z$, Zy, Zy

mo, m$, my, my, m;".Dealing with the 8 and eterms first and remembering that the flutter e<* is identical with the 8used in the stability equations, consider the general case of any aerofoil (wing or tailplane)hinged at a point lA forward of the quarter-chord point.

L

Then the total lift in terms of the derivatives given for the quarter-chord point is

L - 1 V 2S (l i 1 1A i 1A )- "2P ArzR8 + t rzi8 + zR8 + t zi 8 , , •CA CA

(30)

(33)

(32)

Now remembering that we are evaluating the derivatives for undamped oscillations only we have

e= iw8 (31)

and this can be used to transform equation (30) into

L 1 V2S (l 'l e lA i e lA)= 7)p A rzR8 + t rzi-, + l zR8 - + t zi -, - •- t.o» CA tw CA

This equation contains real terms only and can be split up into the partial derivatives

_ 1 oCL _ 1 SA { lA 1 }Zo - - "2 08 - - "2 S lrzR + C

AzR

9

and

_ 1 oCL __ .lSA{l.V l.lAV}Zo - - "2 O( Ol/V) - 2 S 0" lo: + ., CA wl •

Substituting A = WCA/V this gives

__ .1. SA {CAlai + lA lZi}Zo - 2 S l A C

AA .

(34)

(35)

(36)

Examination of the derivative data, however, shows that lai and lzi vary approximately linearlywith A over the range 0 < A <,..... O·08 which is roughly the range to be considered in dynamicstability problems. The equation (35) can therefore be rewritten in the form

Z _.- _ .1 SA ~A {ala; + lA Olzi }o - 2 S 1 OA CA OA .

The moment M about the e.g. of an aircraft as produced by an aerofoil oscillating at an arm lAfrom the e.g. is given by

(37)

Adding these two contributions the moment derivatives with respect to 0 and 0 are obtained:

_ ~ OCM = 1_ SA(CA)2{omai + (omZi _ alai) lA _ Olzi (~)2}m« - 2l a(Ol/V) 2 S lOA aA aA CA 3A CA'

(38)

(39)

For the determination of the stability derivatives with respect to y and y the kinematic relation­ship between zand y (equation (5)) must be remembered. Further corresponding to equation (31) :

ZZ=-;--'

~w '

yVZ=--·-.

~w(40)

The total lift L acting on an aerofoil A during translatory oscillation of the e.g. of the aircraftis given by:

L_ l V2S {1!' '1!'}- "2P A z R + Z zi .CA CA

(41)

(42)

Using equation (40) this can be transformed into an equation having real components only:

L = - i-PV2SA {-lZR~ Y2+ u; V .r }.CA W CA~w

This gives the partial derivatives required:

1 oCL _ 1 SA Olziz; = - "2 oy -"2 5 TI (43)

_ 1 oCL

Zy - - "23(y1/V) (44)

10

As seen from Table II l'R (and m.R as used later for my in equation (47)) is small and changesapproximately with A2- mainly because it represents as a major contribution the apparent massof the air. Thus it is felt that the use of a constant derivative for ol'R/oA2 and later om.R/oA2

does not involve any appreciable inaccuracies and the equation (44) can be rewritten:

(45)

(46)

(47)

For the determination of the corresponding moment derivatives equation (37) has to be appliedwhich gives finally:

C OCM _ 1 SA CA{u; t, om.; 1my = 21 oy - 25 T ar C

A- ----aT J

C OCM _ 1 SA (CA)2 { om.R Ol'R lA}my = 'Il o(pl/V) - 25 T 0,12 - 0,12 ~ .

If the equations (33 to 47) are applied for the evaluation of the contributions of wing., andtailplane- the geometric parameters SA, lA, CAin these expressions have to be substituted by thecorresponding values of the wing (5, lw, c) or tailplane (ST' l, cT) respectively.

In dealing with the effect of downwash on the tail we will as usual divide the total effect intotwo components:

(a) First, assuming that the magnitude of downwash at the tail corresponds to that whichwould be developed in a steady state with the angle of wing incidence at the samevalue as that at the instant considered.

(b) Then allowing for the lag in downwash produced(i) by the lagging of the wing lift behind the wing incidence due to unsteady flow

(ii) by the time delay in the flow reaching the tail from the wing.

This part constitutes the m; term.

The first effect of the downwash reducing the actual incidence of the tailplane can be providedfor by multiplying all terms with the tailplane derivatives which can be associated with asimultaneous change of incidence at the wing with a factor

(48)

These are the coefficients l(J. and m; in the derivatives with respect to () and (j and the wholeexpression for the tailplane contribution to the derivatives with respect to y and p.

This need only be demonstrated by two examples:

I I - 1. ST(CT)2( 1- dB) {om.R_ol. R J...}my T - 2 5 1 do: 0,12 °A2 CT .

(49)

(50)

From the discussion of equations (11) and (16) and the comparability of the momentcoefficients m.; === me, mq === m$ it can be concluded that in the modes of oscillation present duringthe longitudinal motion considered () and yare of quite different order and y obviously does notcontribute much to the pitching moments, i.e.,

() === r:x; y < () .11

Thus for the determination of the downwash delay, which is defined as a derivative with respectto Ii, only the lag of I; against iX* := () will be considered. The downwash, e , arrives at the tailplanewith a phase lag Ct against () which can be added to the above discussed two contributions

l Ilai re, = osi, = W V - l- .aR WING

This can be used for determining the derivative

-. __ ~ ~'- = 1. (l ) Sy dc { 1 _ ~ (ola/a).) }m; - 2lo(liljV) 2 «« r S do: l laR W'

(51)

(52)

5.2. Directional Derivatives.-The onlv aerofoil surface to be considered with lateral motionwith two degrees of freedom is the fin.~ Remembering equation (6) and the conclusion fromsection 4 the following identities can be used for the determination of the derivatives:

flutter terms

L:=Y

]J,![:= N

z ==-y

iX* == 'lfJ

stability terms. (53)

Sidewash can be neglected. Performing the same transformations as in section 5.1 the lateralstability derivatives are obtained.

_ 1 oCy _ 1_ SF { ':!: l }Y'P _.. "2 0'lfJ - 2 S laR + CF zR ..

1 oCy . 1 SF olz;Yx = "2 ax =- - -25 aI

_ 1 oCy _ SF 0:. OlzRY;< - 2" 2(Xbj2V) - S b 2).2

3CN _ SFCFtolz; IF omz; 1n -- --- ----x - ox S b 0). CF 0).

OCN _ SF 2CF { OmzR 0: _ OlZR If.}n;< = 0(xb/2V) - S b 0).2 b 0),2 b .

(54)

(55)

(56)

(57)

(58)

(59)

(60)

(61)

5.3. Fuselage and Nacelles.---If unsteady aerodynamic coefficients are known for bodies ofrevolution such as fuselages and nacelles, they can be expressed in the form of the abovedetermined derivatives. At present no such solutions exist. Thus one is confined to the use of

12

known quasi-steady values. These can best be expressed in terms of the above defined unsteadyderivatives by using the following identities which hold for infinitely slow motion (w~ 0).

(62)

YIP == - Yv

v, ==Yr

z; == - Zw my == -'- m; Yx == Yv

Zy = my = Yx = nx = 0

6. Numerical Examples.-The consequences of unsteady flow on the dynamic stability ofaircraft will best be illustrated by numerical examples. For these specific calculations twotypes of aircraft have been chosen which can be expected to respond differently to the particulareffects of unsteady flow.

Example A is an orthodox tailed aircraft with a relatively long tail arm, which thus possessesbasically sound damping both in pitch and in yaw.

Example B is the tailless version of A with a fin at a relatively short arm. When discussinglateral stability this example will therefore be referred to as short-tailed version as comparedwith the long-tailed version A. There are two versions of the tailless aircraft B considered withlongitudinal stability. They are distinguished by different sizes of the fuselage.

Since there are no unsteady aerodynamic coefficients available for sweptback wings the wingsin both cases are assumed to be identical both without s\veep. It must be noted that for a wingwith sweep the damping in pitch with unsteady flow is probably much higher than for a straightwing.

The geometrical data for both aircraft are given in Table 3. The unsteady aerodynamicderivatives have been obtained with the formulae derived in section 5. The two-dimensionallift slope, ao, has been assumed as 5·65 for all aerofoils concerned; the finite aspect ratio of eachsurface has been allowed for by using equation (29). For the tailed aircraft A, the downwashslope, deld«, has been assumed to be 0·3. The fuselage contributions are dealt with as indicatedin section 5.3. In all cases considered the calculations have been done for 4 different Machnumbers (0, 0,5, 0,7, 0,8) and the results are presented in Tables 4 to 6.

Table 4 refers to the longitudinal stability of the tailed aircraft, A. First the unsteadyaerodynamic derivatives are given as wing contributions (suffix w) and tailplane contributions(suffix T)' The complete derivatives including fuselage contributions as well are given as sums(:S ...). The coefficients of the frequency equation (equation (13)) are given in the dimensionlessform A" and B". The fll and !-l2 used correspond to ground level.

The change of dynamic stability with altitude has been checked in a'll cases considered.Apart from the generally known decrease in damping proportional to p a slight change of thedamping coefficient A" (and A" for directional stability) is to be expected since some of the termscontain III or fJ2 respectively. These terms, however, are so small that no appreciable change ofthe stability boundary has been observed.

For the determination of the reduced frequencies Aas the important parameters for the unsteadyderivatives the indicated wavelength of the longitudinal oscillation is computed from equation (15)which gives

7 J( PlWljS )A"i = T"T i = 22·7 + B" _ A"Zj4 .

The corresponding expression for the wavelength of the directional oscillation is

(63)

(64)

Since the reduced frequency is the ratio of the individual chord e to the wavelength .1

(65)

the formulae (63) or (64) respectively can be used to evaluate the reduced frequency by consideringthe relationship between the indicated wavelength Ai and the true wavelength A

cA = 2n I Vo . (66)

"z

For the determination of the real parts of the flutter coefficients (Table II) needed, the value ofA for the various aerofoil surfaces has been first estimated by a quasi-steady approximation forthe periods of the oscillations concerned. The values for the reduced frequencies Agiven inTables 4 to 7, however, have been obtained from the actual frequencies resulting from thecalculations with unsteady derivatives. '

The corresponding figures lor the longitudinal oscillation of the tailless'aircraft B are givenin Table 5 for two different c.g. positions, i.e., for the different values of the distance l:v of thequarter-chord of the wing from the c.g. of the aircraft. Table 6 contains the correspondingfigures for tailless version with a smaller fuselage.

The corresponding figures for the directional stability calculations for both the long-tailedversion (A) and the short-tailed version (B) are given in Table 7.

The most important results, the values of the damping roots A", and £1'1' for longitudinal anddirectional oscillations respectively, are plotted against Mach number in Figs. 7 and S. Theresults of the calculations with unsteady derivatives are represented by full lines. Correspondingcalculations have been done with quasi-steady aerodynamic derivatives which are assumed tochange with Mach number according to Glauert's rule. The results of these calculations arepresented for comparison and shown by dotted lines.

A8 Fig. 7 shows, the results with unsteady derivatives compare very favourably with quasi­steady results tor the damping of the longitudinal oscillations of the tailed aircraft. The closeagreement seen, however, is mere coincidence. It results from adding basically different effects,namely damping in pitch m, and downwash delay me; With appreciable downwash present,this tendency can generally be expected for conventionally tailed aircraft.

The damping of the longitudinal oscillation of the tailless aircraft (B) may, however, be radicallyaltered when considering the effects of unsteady flow at high subsonic speeds. Much depends,with the simple theory used, on the value of lwle assumed, i.e., the distance between the c.g.and the aerodynamic centre of the wing in terms of the mean chord. With well-forward c.gs.with a lil/e of between 0·1 and 0·2 (such values correspond roughly with those of the DH.lOS),the destabilisation of the dynamic stability is very marked at high Mach numbers, neutralstability being predicted at Mach numbers between 0·7 and 0·85. With a lwle of 0·05, howeverthe damping calculated at high Mach numbers using unsteady-flow theory is very similar to thatobtained using quasi-steady flow; this value of lwle could only be obtained on an aircraft with avery small or negligible fuselage, for the destabilising effect of the fuselage would otherwisecause an unacceptably low or a negative static margin. Thus this admittedly approximate theoryof unsteady flow is predicting in some practical condition of aircraft layout, a negative dampingof the longitudinal oscillation at Mach numbers where the theory does not necessarily breakdown due to the presence of shock-waves on the wing surface.

The results at the calculations of the damping of the directional oscillation of the two aircraftconsidered are shown in the diagram Fig. S. It is seen that calculations with quasi-steadyderivatives give far too optimistic results. In both examples considered a rapid deterioration ofdamping due to the effects of unsteady flow occurs at Mach numbers beyond O·6 and an undampedoscillation would appear likely with the short-tailed aircraft at a Mach number just above O·S.

14

The aircraft with the fin at the longer arm has the better basic damping, and the undampingeffects of unsteady flow appear less severe; actual instability would not therefore be predicteduntil the assumptions on which the aerodynamic derivatives used in this paper depend break'down due to the presence of shock-waves.

. The main effects of unsteady flow on the aircraft oscillations are mainly represented in thedamping derivatives me and n,p respectively. They are generally smaller as compared with thecorresponding quasi-steady values of mq and n, respectively, and they will become undampingwith higher Mach numbers in most cases. The other major damping terms affecting the aircraftoscillations considered, i.e., z; and Yx for longitudinal and directional motion respectively, arehardly altered when considering the effects of unsteady flow. Thus dynamic stability of theaircraft oscillations is maintained up to Mach numbers above those associated with the changein sign of the damping derivatives me or n,p respectively.

7. The Stability of an Oscillating Wind Vane.--Another more simple way of demonstrating theeffects of unsteady flow is by investigating the stability of the oscillation of a wind vane as itchanges with the length of the arm of suspension and Mach number. This may also be interpretedas the stability of the longitudinal oscillation or directional oscillation in one degree of freedom.In this case the wind vane can be identified with the fin or tailplane (or wing for the longitudinaloscillation of a tailless aircraft) of the aircraft considered.

Dynamic stability of such a system is present if the imaginary part of the resultant momentexerted upon the hinge during the oscillation is positive, i.e., there is a component of the momentopposing the motion of the vane. If m"i 0 is the imaginary part of the hinge moment then thiscondition reads

acM Om"iO=~ > O.

uiX(67)

Using the flutter derivatives as before and remembering that the translatory deflection of theguar~er-~hordof the aerofoil surface of the wind vane of an arm l from the hinge axis is z = iX*l,m"i 0 IS grven as

(68)

The flutter derivatives are functions of Mach number and A, thus the stability boundary willalso depend on Mach number, A and the relative arm, l]c, of the vane. The results of calculationsfor two-dimensional unsteady flowcorresponding to the figures given in Tables 1 and 2 are plottedas stability boundaries in the lie-A plane with Mach number as a parameter, in Fig. 9. It isseen that even with incompressible flow an unstable region exists for the range of hinge axesbetween the quarter-chord point and O·7c ahead of the quarter-chord with relatively small reducedfrequencies. When suspended aft of the quarter-chord the wind vane is damped, but of course-­without any additional spring in the system-statically unstable. With increasing Mach numberthe unstable region extends towards longer arms l and higher frequencies }.. A cross-plotting ofthe stability diagram against Mach number with the arm ljc as parameter is given in Fig. 10.

In these calculations no reference is made to the relationship between the length of the arm land the reduced frequency II. which is treated as an independent variable. It will of coursedepend upon the actual dimensions of the aerofoil, the inertia of the system and, if applied toaircraft, upon other parts of the airframe contributing mass and stiffness.

It must be noted that the results of this approach towards the dynamic stability with unsteadyflow is more correct than the calculations of the preceding sections, since no additional lineari­sations have had to be assumed. On the other hand it should be emphasized that the unsteadyaerodynamic coefficients used are still subject to experimental proof.

15

8. Conclusions.-(a) It has been shown that if account is taken of the unsteady flow conditionswhich exist during the oscillation of an aircraft, the damping may very well be less than wouldhave been expected on the basis of quasi-steady flow theory.

(b) If allowance is made for compressibility effects assuming no shock-waves are present, anydeterioration in damping associated with the unsteady flow conditions becomes very much worsewith increasing Mach numbers.

(c) The deterioration in damping is likely to be particularly severe in the case of tailless aircraftor aircraft with short tail arms. The dynamic longitudinal stability of tailless aircraft isparticularly affected with forward e.g. positions with respect to the neutral point of the wing.This will generally be the case if some fuselage is present.

(d) The frequencies of the aircraft oscillations are not appreciably affected by allowing forunsteady flow conditions.

9. Future TVork.--This report aims only to give an indication of the probable consequences ofthe effects of unsteady flow on aircraft stability. The importance of the foregoing conclusionsto practically all aircraft with high subsonic speed suggests more thorough investigation of allaspects of the problem. There are four lines along which further research should proceed:

(a) Flight research on the dynamic stability of high-speed aircraft.

(b) Wind-tunnel or flight measurements on oscillating aerofoils. A simple experimentwould be observation of self-excited oscillations of a freely suspended wind vane asinvestigated theoretically in section 7 of this paper at high speeds.

(c) Theoretical analysis on the unsteady compressible flow round wings with finite aspectratio and sweep. An approach towards a solution of the unsteady flow round bodiesof revolution seems also desirable.

(d) More comprehensive treatment of the dynamic stability of aircraft with more than twodegrees of freedom, and of the problem of auto-stabilisation with unsteady aerodynamiccontrol effectiveness.

If the conclusions of this paper are confirmed the provision of means for artificial dynamicstabilisation of aircraft may prove inevitable in many cases and the development of such meansshould be urged.

16

A

b

B

LIST OF SYMBOLS

b2jS, aspect ratio

Damping root

Two-dimensional lift slope

Three-dimensional lift slope

Wing span

Inertia in pitch

Chord of wing (tailplane, fin)

CL - Ljip V 2S, lift coefficient

Mjip V 2Sc, pitching-moment coefficient

Njip V 2Sb, yawing-moment coefficient

Yj}p V2S, side-force coefficient

L

t,

q

r

S(T,F)

TIX('P)

tlX(,P)

1 -

vV

W

w

(61801)

V-IBlml", inertia coefficient in pitch

Cjm(bj2)2, inertia coefficient in yaw

Lift

Distance between the quarter-chord of tailplane (wing, fin) and the e.g.

oCLjor.x.*, lift derivative with respect to r.x.*

oCLjo(zjc), lift derivative with respect to z

oCL!Orj, lift derivative with respect to 1} Flutter coefficients

2CMjor.x.*, moment derivative with respect to r.x.*

2CMjo(zjc), moment derivative with respect to z

Mass of the aircraft

Pitching moment

Pitching moment about quarter-chord of an individual aerofoil

Yawing moment

e, rate of pitch

1p, rate of yaw

Area of wing (tailplane, fin)

Period of the longitudinal (lateral) oscillation

Damping time of the longitudinal (lateral) oscillation

mjpSV, time unit

Lateral velocity component

Speed of flight

\Veight

Vertical velocity component

17B

LIST OF SYMBOLS-- continued

Y Side force

z Vertical displacement

a wjV, aerodynamic inclination

a* Flutter variable

f3 vjV, angle of sideslip

r Angle of climb

b Tjt, logarithmic decrement

E Downwash angle

E t Phase lag angle

Ii Control deflection

() Altitude in pitch

}. wcjV, reduced frequency

A TV, wavelength of an oscillation

III mj pSl, relative density (longitudinal)

Il2 2mjpSb, relative density (lateral)

p Air density

X Azimuth angle at the flight path

1p Attitude in yaw

()) '2nIT, circular frequency

aCm cm; aex 2l

aCM ca(<iljV) 2l

aCM ca(qljV) 'Il

aelI cm, a() 2l

aCM ca(OlIV) 'Il

aCM cay 2l

Quasi-steady pitching-moment derivatives

Unsteady pitching-moment derivatives

aCM c11t;. a(yljV) 'Il

aCN

af3

n r

aCN

a(rbI2V)

Quasi-steady yawing-moment derivatives

18

aCNn", - a'ljJ

aCN

nifJ - a(ip bj2V)

aCNnX - ax

aCN

ni - a(ib/2V)

1 acy

Yv - "2 ap

1 acy

y. "2 a(rb/2V)

LIST OF SYMBOLS-continued

Unsteady yawing-moment derivatives

Quasi-steady side-force derivatives

(61801)

y",

YifJ

Yx

Yi

1 Cy

"2 a'ljJ

1 Cy _

"2 a(ipbj2V)

1 Cy

"2 ax

1 Cy

"2 a(ibj2V)

1 aCL- "2 aiX

1 aCL

- "2 a(qljV)

1 aCL

-"2 ae

1 aCL

- "2 o(el/V)1 8CL

-"2 oy

1 oCL

- "28(yljV)

Unsteady side-force derivatives

Quasi-steady lift derivatives

Unsteady lift derivatives

19B*

No. Author

H. Glauert ..

2 W. E. Cotter

3 G. Temple ..

4 A. I. Nekrasov

5 Frazer, Schade and Diezc

6 Th. Schade

7 K. Krienes and Th. Schade

8 W. P. Jones

9 B. Smilg

10 H. Goland

REFERENCESTitle, etc.

The force and moment of an oscillating aerofoil. R. & M. 1242.March, 1929.

Summary and analysis of data on damping in yaw and pitch for anumber of airplane models. N.A.C.A. Tech. Note 1080. May,1946.

Modern developments in fluid dynamics. Vol. III, Chapter IX.Unsteady Motion. A.R.C. 11,745. May, 1949.

Wing theory for unsteady motion. Section 4, Chapter XI, Summaryof formulae and bibliography. (Academy of Sciences, U.S.S.R.)Translated by S. W. Skan. A.R.C. 11,795. September, 1948.

Tables and curves of two-dimensional subsonic aerodynamicderivatives. Unpublished data by Structures Dept., RoyalAircraft Establishment.

Numerische Losung der Possio'schen Integralgleichung derschwingenden Tragflache in ebener Unterschallstrowung. U. & M.3209. 1944.

Theorie der schwingenden Kreisflache auf potential-theoretischerGrundlage. Luftfahrtforschung, Vol. 19. No.8.

Aerodynamic forces on wings in non-uniform motion. R. & M. 2117.August, 1945.

The instability of pitching oscillations of an airfoil in subsonicincompressible potential flow. ]. Aero. Sci. November, 1949.

Stick-fixed short-period stability based on the Wagner air forces.Reissner Anniversary Volume. Contributions to applied mechanics.J. W. Edwards and Ann Arbor. Michigan, 1949.

TABLE 1

Derivatives of the Unsteady Aerodynamic Coefficients with respectto A, or ,12 respectively (Referred to quarter-chord)

Mach ola; s; oma; omz; olzR omzRnumber 0.1. 0.1. aT aT 0.1.2 0.1.2

0 - 7·12 6·06 -0,76 0 8 0·390·5 -12,87 6·90 -1,07 0 12 0·560·6 -16·85 7·38 -1·25 0 16 0·710·7 -23·86 8·14 -1·58 0·03 21 0·940·8 -41·00 10·00 -1,90 0·08 47 1·40

20

TABLE 2

Real Parts of the Unsteady Aerodynamic Coefficients withApparent Mass Effect Included

Mach I it i: l'R mrzR Z;Z,Rnumber

0 6·284 0 0 00 0·04 6·065 0·0163 0 0-0006

0·08 5·849 0·0483 0 0·0026

0 7·256 0 0 00·5 0·04 6·906 0·0261 0 0·0009

0·08 6·566 0·0763 -0,001 0-0037

0 8·798 0 0 00·7 0-04 8·148 0·0445 -0,003 0·0015

0·08 7·545 0·1261 -0,007 0-0061

0 10-440 0 -0,010 00·8 0·04 9·575 0-100 -0'040 0·0030

0·08 8-652 0-190 -0,080 0·0049

TABLE 3Dimensions and Data of the Two Aircraft

Considered in the Numerical Example

I Aircraft A I Aircraft B

Wing area S 320 ft2span b 40 ftwing load WjS 40lbjft2 35lbjft2

chord c 8 ftarm lw 0 2 ftaspect ratio A w 5

f!1 (ground level) 21·7 57·1f!2 (ground level) 26·1 22·85inertia in yaw i. 0-15 0·11inertia in pitch iB 0·12 0·50

Tail area Sp 48ft2 0chord Cp 4ft 0arm l 24 ft -aspect ratio A p 3 -

Fin area SF 24 ft2

chord CF 4 ftarm IF 24 ft 16 ftaspect ratio A F 1·5

{(mw)FU8 0·050 0,15*quasi-steady fuselage (nv)pU8 -0,070 -0,045

derivatives (Yvhu8 -0,20 -0,15

* 0·08 for the smaller fuselage.

21

TABLE 4

Unsteady Derivatives and Stability Coefficients of theLongitudinal Oscillation of the Tailed Aircraft A

---------,------,-------,-----

Machnumber

o 0·5 0·7 0·8

o)p -0,170 -0,192 -0,221 -0,254y)P 0·177 0·200 0·235 0·289

'ma -0,120 -0,142 -0,171 -0,204my 0·127 0·150 0·185 0·239

o)z. -0,216 -0,223 -0,222 -0,217o)w -0,027 -0,038 -0,057 -0,068

'mo -0,243 -0,261 -0,279 -0,285y)p -0,038 -0,058 -0,101 -0,229y)w 0·014 0·020 0·034 0·050

'my -0,024 -0-038 -0,067 -0,179o)p -0,178 -0,205 -0,246 -0-291o)w -1,88 -2,12 -2,43 -2,78zo -2,06 -2,32 -2,68 -3,07y)P 0·172 0·196 0·231 0·284

'.y)w 1·95 2·22 2·62 3·22,2·12 2·42 2·85 3·50zy

o)p -0·216 -0,223 -0,221 -0,205o)w 0·764 1·380 2·542 4·400zO 0·548 1·157 2·321 4·195

z;>)z. -0,039 -0,058 -0,101 -0,226y)w -0,860 -1-290 -2,260 -5,050Zy -0,899 -1,348 -2,361 -5,276IV -0'099 -0,132 -0,187 -0,272« 4·90 5·59 6·52 7·43

'" 29·5 37·4 49·7 80·0",ft 779 687 605 471w 0·064 0·073 0·083 0·106T 0·032 0·037 0·042 0·053

-mABAAA

(z(zE(z(zI­(z(zE((zE

(m(mI­E(m(mI­(m(mI-

22

TABLE 5

Unsteady Aerodynamic Derivatives and Stability Roots of the LongitudinalOscillation of the Tailless Aircraft B with Large Fuselage

Mach Inumber

o 0·5 0·7 0·8

o......o

II

oC'l

oII

<.>

::I

mo -0,189 -0,212 -0,245 -0,304my 0·195 0·222 0·252 0·296mo pus 0·150 0·173 0·210 0·2502:mo -0·039 -0,039 -0,035 -0,0542-my 0·045 0·049 0·042 0·046m$ 0·018 0·070 0·259 0·707my -0,130 -0,212 -0,373 -1,060Zo -1,885 -2,115 -2·434 -2,785Zy 1·950 2·220 2·620 3·220Z$ 2·48 4·36 7·96 13·50Zy -2,57 -3·86 -6,75 -15,13A 0'. 1·819 1·913 1·734 0·736BO'. 4·62 4·72 4·32 7·02.110'. 1467 1478 1520 1090Aw 0·034 0·034 0·033 0·046

mo -0·377 -0,424 -0,489 -0,585my 0·390 0·445 0·515 0·6192:mo -0,227 -0,251 -0,279 -0,3352:m y 0·240 0·272 0·305 0·369m$ 0·136 0·395 0·956 1·906my -0,390 -0,593 -1,850 -2,675Zo -1·888 -2,120 -2,440 -2,800Zy 1·950 2·220 2·620 3·220Z$ 1·904 3·660 7·15 12·57Zy -2,57 -3,86 -6,75 -15,13A", +1·568 +1·222 +0·200 -2,096E", 26·39 29·42 33·07 42·15A", ft ·567 535 501 447Aw 0·089 0·094 0·100 0·112

23

TABLE 6

Unsteady Aerodynamic Derivatives and Stability Roots of the LongitudinalOscillation of the Tailless Aircraft B with Small Fuselage

Mach Inumber

o 0·5 0·7 0·8

me -0,101I

-0,117 -0,142 -0,168my 0·098 0·111 0·131 0·135mOFUB 0·080 0·094 0·112 0·1302:mo -0,021 -0,023 -0,030 -0,0382:my 0·018 0·017 0·019 0·005me -0,132 -0,143 -0,129 +0·041my -0,003 -0,013 -0·035 -0,306Zo -1,885 -2,115 -2,434 -2·785Zy 1·950 2·220 2·620 3·220ZIl 2·20 4·04 7·55 13·04Zy -2,58 -3,87 -6,77 -15,15

A" 2·302 2·666 3·218 4·14B" 2·43 2·71 3·61 5·64A" 2720 2990 2840 2470l.w 0·0185 0·0168 0·0176 0·0203

twlc = 0·05 corresponding to e.g. at 20 per cent of the wing chord.

24

TABLE' 7

Unsteady Aerodynamic Derivatives and Stability Coefficients of the LateralOscillation of the Long Tail A and Short Tail Version B

Mach Inumber

o 0·5 0·7 0·8

A

B

(n"')F -0,106 -0,121 -0,142 -0,167(nX)F 0·105 0·120 0·142 0·174z«; -0,036 -0,051 -0,072 -0,094L:nx 0·035 0·050 0·072 0·104n.p -0,102 -0,100 -0,087 -0,067nx -0·028 -0,042 -0,073 -0,162(y",)F +0·089 +0·102 0·120 0·142(YX)F -0,088 -0,100 -0,118 -0,145L:y", 0·289 0·302 0·320 0·342L:yx -0,288 -0,300 -0,318 -0,345y.p 0·085 0·083 0·072 0·055Yx 0·023 0·035 0·061 0·136A", 0·974 0·977 0·892 0·816B", 6·45 8·96 12·53 16·60A", 1318 1114 936 811/IF 0·019 0·022 0·027 0·030

(n"')F -0,070 -0,079 -0,092 -0,108(nX)F 0·070 0·080 0·095 0·116L:n", -0,025 -0,034 -0,047 -0,063L:nx +0·025 0·035 0·050 0·071n.p -0,040 -0,035 -0,021 +0·001nx -0,018 -0,028 -0·049 -0,109(y",)F +0·088 0·099 0·117 0·136(YX)F -0,088 -0,100 -0,118 -0,145L:",y 0·238 0·249 0·267 0·286L:yx -0,238 -0,250 -0,268 -0,295y.p 0·050 0·043 0·025 -0,003Yx 0·023 0·035 0·061 0·136A", 0·604 0·564 0·457 0·285B", 5·26 7·15 9·94 13·40A", 1265 1080 912 785}'F 0·020 0·023 0·028 0·052

25

M~

FIG. 5. Unsteady moment co­efficient ma due to an angularoscillation of an aerofoilagainst frequency parameter

}. and Mach number.

'171.z i

FIG. 4. Unsteady moment co­efficient m, due to a translatoryoscillation of an aerofoil againstfrequency parameter ). and Mach

number.

1'0

directional oscillationComparison between

theory and unsteady

0·6MACH N9

o

B

l ~/ u'l-'o\Cl' 0

<>o~ 0

--0- -,,"-0 0..... 0

00 -,0'

~~'1:>

(

o ,:to\~

o ''0'~

\\\

0·4

\?>-=o-a/

o

FIG. 1. Logarithmic decrement (j of theof the Meteor against Mach number.flight-test results (0) and Quasi-steady

theory.

FIG. 2. Unsteady lift t,due to a translatory oscil­lation of an aerofoil withthe amplitude (zlc) againstfrequency A and Mach

number.

FIG. 3. Unsteady lift lrx due to an angular oscilla­tion of an aero foil about its quarter-chord line withthe amplitude a* against frequency A and Mach

number.

FIG. 6. Unsteady lift coefficient IT) due to a harmonic flap deflectionwith the amplitude II against frequency parameter }, and Mach number

(flap chord cT) 20 per cent of the wing chord c).

1'0

ol-----L----L--__L- ~----'

·~ ~---~---~-

c-e

\

-.~ 1-----1-----1-----1---

-'4 -----

0'4. 0'2. 0'6 \'0MACH N9

-2.'0~~~3Z~+__~t~~uJ"",--, O~05J I

'-QUASI-STEAO'l'

UNDAMPED

a-0 ~---~---~---~-----Jf-----.----,

-4'0

-6·0 I----l-----j_

-8,0 l------4-,....----+--~+_--~+_--__I

-Aet.

-12.'0 1-----1--

-14.0 l- l...- L-__........L.- L-__-"

FIG. 7. Damping root Act of the longitudinal oscillation of twoaircraft as affected by unsteady flow.

\\\. - - -------------_ ...

\\\\\\\

L- L- L -"__. "--__---'

FIG. 8. Damping root Alp of the directional oscillation oftwo aircraft as affected by unsteady flow.

O·4r----.,----........---...-----r---~

Mo~

o 1'0

v

,,0 3'0 (lie) 4·0

FIG. 9. Stability boundaries of the oscillation of a wind vane against Mach number.Arm of the aerofoill and reduced frequency A.

c,.) -------_J__------_J__

0·4r-------,----- I-I

1·00'80·4 0·6WCH NQ

o

(:Joe f------l-

, .<11 C" TV

FIG. 10. Stability boundaries of the oscillation of a wind vane (cross-plotting from Fig. 9).

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