Flutter and divergence of sweptback and sweptforward wings

TECHNISCHE HOGESCHOOL

VUEGTUIGBOUWKUNDE

REPORT No. 39

19 Sep. 1950

THE COLLEGE OF AERONAUTICS

CRANFIELD

JÊL Kluyverweg 1 - 2829 HS DELFT

FLUTTER AND DESIGN OF SWEPTBACK

AND SWEPTFORWARD WINGS

by

A. W. BABISTER, M.A., A.F.R.Ae.S.

of the Department of Aerodynamics

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

TECHNISCHE HOGESCHOOL VUEGTUiGBOUWKUNDE REPORT NO. 39 JUME. 1950.

T H E C O L L E G E O F A E R O I J x i U T I G S

G H A W F I E L D

F l u t t e r and Divergence of Svreptback and

Sv/eptforiTard Wings

-by-A.¥. E a b i s t e r , H,A., A.F.R.Ae.S.

S U M li A R Y

In t h i s note, the equations of the f l e x u r a l - t o r s i o n a l

f l u t t e r of a sv/ep+ v.dng are e s t a b l i s h e d , assuraing the wing t o

be semi-rigid and fixed a t the r o o t . The general effect of

sweepback, wing s t i f f n e s s and p o s i t i o n of the i n e r t i a axis are

determined. The c r i t i c a l speeds fccr f l u t t e r and for v/ing

div-ergence are determined (i) for incompressible flow ( i i ) for

compressible flov/, assuming a modified Glaucrt c o r r e c t i o n .

The c r i t i c a l f l u t t e r speed i s in general higher for

a sweptback wing having the same v/ing s t i f f n e s s as the uns'vvept

vfing; for a swept foi-ward wing, divergence \7ill occur before

f l u t t e r .

2

-W O T A T I O N

Dimensions and Displacements of 7ing (see Figure 1)

c = chord a t distance y from root chord ( p a r a l l e l to the

root chord)

c = root chord

o

c = mean chord

m

c, = t i p chord

d = 0.9 s

f and F define the f l e x u r a l and t o r s i o n a l modes of o s c i l l a t i o n

gc = chord-wise distance from leading edge t o i n e r t i a axis

he = chordwise distance from leading edge to f l e x u r a l a x i s

j c = chordwise distance from flex-jral a x i s to i n e r t i a a x i s

f^ = 0.7 z = perpendicular distance from wing root to f l e x

-u r a l centre of reference section

s = perpendicular distance from v/ing r o o t to t i p

s' = distance from wing r o o t t o t i p , measured along f l e x

-uro.l a x i s

y = perpendicular distance from wing root to a given

chord-v/isc clement

a = angle of incidence of wing

0 = nonupJ displacement of f l e x u r a l centre a t a given

chordv/ise element

Y^ = slope of f l e x u r a l a x i s a t a given chordv/ise clement

6 = angle of tv/ist of a given section perpendicular to the

flexuroJ. a x i s .

P = angle of sweepback of flexural a x i s .

Density

^ = air density/wing density = p/o"

p = a i r dcrisity i n slugs per cubic foot

cr = \7ing density = v/ing mass per u n i t area/mean chord, in

slugs per cubic foot.

S t i f f n e s s c o e f f i c i e n t s

^ , - clastic moment about perpendiciilar to flexviral axis _{for unit displacement}

0 at the reference section m^ = elastic moment about flexural axis for unit

displace-ment 6 at the reference section r

B

V J p

m

V = forward speed of aircraft V = critical flutter speed

-5-Introduction

In this note, the equations of the flexural-torsional flutter of a swept v/ing are established, assuming the v/ing to be semi-rigid and fixed at the root. The general effects of sweepback, vri.ng stiffness and position of the inertia axis are determined. The critical speeds for flutter and for v/ing divergence are determined (i) for incompressible flov/ (ii) for compressible flow, assuraing a modified Glaucrt correction. Data and Assumptions

General A straight tapered swept wing is considered (Figure 1). The flexijral and inertia axes are taken at given constant

percentage chord distances behind the leading edge. Principal Dimensions

s = span (root to tip), perpendicular to root chord d = perpendicular distance from root to 'equivalent

tip section' = 0.9s

JC = perpendicular distance from root to flexural centre of the 'reference section'

c 0

'^t

c m = = = = 0.7s root chord tip chord mean chord

he = distance of flexural axis aft of leading edge (measured parallel to root chord)

gc = distance of inertia axis aft of leading edge (mcasvired parallel to root chord)

1-T = taper ratio = c,/c . ^ t' o

p = angle of sv/eep back of flexural axis.

corresponding distances along the flexural axis arc indicated by dashes; thus

s' = span measured along the flexural axis.

Axes Ox, Oy are talcen ]parallel ajid perpendicular to the root chord through the point 0, v/here the flexural axis meets the root chord. Axes Ox', Oy' are taken perpendicular to and along the flexural ojcis.

Modes of motion and displacement coordinates

The v/ing is assuivicd to be semi-rigid, the modes of displacement in flexure and in torsion being talcon to be inde-pendent of the specdj all displacements of cither kind arc taken to be in phase v/ith one another. The modes of displace-ment arc taken to bo linear in torsion cmd parabolic in flexure; this apprpxiraatcs closely to the natural modes of the system.

-4-The displacement coordinates arc defined as

follov/si-The flexural coordinate 0 is the flexural displacement of the flexi;iral centre at s. given section divided by y' (positive for dov/nv/ard bending).

The torsional coordinate 0 is the angle of tv/ist of a given section perpendicular to the flexural axis m.ca.sujred relative to the corresponding root section Ox'^ (positive v/hen the trailing edge moves do-//n relative to the leading edge). Ö and 0

are the flexural and torsional coordinates of the reference section, (the section perpendicular to the flexural axis at 70 per cent of the span, measured along the flexural axis).

The xring is supposed "to be placed at a small angle of incidence in a uniform airstrcam of speed V (iviach number M ) and the wing root,is supposed to be rigidly fixed. •

The displacements Ö . 0 of any point arc related to the corresponding displacements at the reference section

Q , 0 by the equations

I'fM

P(TI)

r y' ' e, where 11 = y/t =

y'/•?'-The symbols used in the equation of motion conform with those in references 1 and 2.

Elastic stiffness coufficicnts

.The flexural and torsional coefficients arc denoted by -i.^ and m„ respectively. The non-dimensional fl\itter

speed coefficients o.re plotted against the modified stiffness ratio r defined by

^//'ing density

The v/ing density cr i s defined t o be the t o t a l '^ -^ CO

v/ing mass i n slugs divided by the product of the v/ing area i n square f e e t and the mean chord i n f e e t .

Also f = p/cr_^ I where p i s the a i r density i n slugs per cubic f o o t .

Let (T , o' be the v/ing d e n s i t i e s for a sv/ept and (3 'o

for an unsv/cpt wing of the sai'ne area a.nd mean chord. The sv/ept v/ing v/ill have a l a r g e r v/cight due t o i t s l a r g e r span, measured along the f l e x u r a l a x i s . I t can be shov/n on t h e o r e t i c a l grounds

-5-that the weight of a swept wing should vary approximately as sec (3.

cr = cr sec B

C 0 _ OJ

P o

and

6 =

_sec

'-^. é

The inertial coefficient:

To find the inertial coefficient, we replace the given wing by the wing ABB'A', considering the section Aii' to be rigidly fixed to the fuselage,

As in references'! and 2, we assume that the mass per _2

unit span (measured along the flexural axis) is m„c where c = local chord perpendicular to the flexural axis and m is constant for a given angle of sv/eepback.

We have approximately c = c cos |3

v/here c = local chord measured // to the line of flight. ^S' 2

Total wing mass = 2m^ \ c dy'

2m,, cos B c s

(3 "^ o 1 . .

-F o r t h e unswept v/i.ng, t o t a l v/ins mass = 2 m c' " ' ^ 0 0

i - T + 3 Assuming as above that the v/ing weight varies as sec (3,

m„ = m sec B .

P o '^

F o r b o t h sv/ept a.nd unsv/ept wings, t o t a l v/ing a r e a = 2s c and mean chord = c

Nov/ cr 1 -

c-a

wing mass CO cr cr 0),, Ü)

v/ing a r e a x mean chord 2

m

•p m

3

3-3T^+T

r

'To also assume (as in references 1 and 2) that the ro.dius of gyration kc of a chord vvrise section about a transverse axis tlirough the inertia ccnti-e of the section is a constant percentage of the chord. (k = 0.294)»

Lot 6m = vrt of wing element 5x' 6y' at point (x',y').

As in references 1 and 2, the inertia coefficients are given by the follov/ing

6 -A, =

Z-.-^(|J

o s ' m s e c p, G cos p -^' f dy' =

aWl

m c^ / ^ f^ sec^p dT) o •• p i c a . ^ o 1 where m / \2

^ J V

o Ü o ^ ' f^ sec^P dn / 2— ^ o " er and y ' - r\f\ o S ü i i l a r l y A^ = G^ = ^ 6 m x ' y ' / ^ l ] ( ^ ) n s ' U m 3 e c ^ p . c ^ c o s " p . ^ ' t T . j e cosp dy' A10/7 ü i?2 3 5 ';2 2 P ^ % °-3 m c^ i jfP s e c p dn = — ^ ^ o where ^3 = S-^ = O m co o U >.10/7 - 1 fP sec^p d-n "o

and the c e n t r e of i n e r t i a of any s e c t i o n i s d i s t a n c e j e b e h i n d t h e f l e x u r a l a x i s .

= S 6m

A l s o G, 3 ,..2 / 6 0 r\s" U 3„ 2 2^ ^2 \ 2 2 2„ , , m sec p, c cos p.F . A ^ cos p dy'

r^-10/7 Ü 4 -xS S>^'?- . P "^^o ^3 m ^ c ^ / \ I F dTi = TT «i m. ii o where g , - "^\' 3 c o U

^no/7

,4 p dn

X

2 = k 2 .2 / T h u s

Thus

a. varies as sec p , a, (= g.) as sec p and

4o „ ^„ „ ^ 2,

g is independent of p.

The aerodynamic coefficients

Yfe cxinsider the forces acting on a chordwise strip

of the v/ing (parallel to the line of flight). The geometrical

angle of incidence a and the downward displacement of the

leading edge of this chordwise strip are given by

a = 0 cos p + 'V' sin P

z = 0 'n ^' - 6 he cos p

v/herc

'~'y^

is the slope of the flexural axis at the section

considered, and any chordv/isc change of camber has been neglected.

0 = 0^ f/n

> = 0^ öf/ö-n = 0^ f •

e = O F r

f and P b e i n g f u n c t i o n s of "n = y/-^ .

Por the a e r o f o i l c l e m e n t , the l i f t and moment c o e f f -i c -i e n t s r e f e r r e d t o t h e l e a d -i n g edge a r e g -i v e n by 30^. ac^ ac^ ^ Jj , L , L C^ = a + a + z da Oct ÖZ ac ÖC ac ^ m . m , ii J = a + a ^- + z aa aó, a a

v/here a i s t h e g e o m e t r i c angle of i n c i d e n c e and z i s the dovmward v e l o c i t y of the l e a d i n g e d g e .

I n t h e s t a n d a r d n o t a t i o n , the downward nori'iial f o r c e i s g i v e n by

6Z = - -g- p V^c 6C -^d-n

- P V c -^ (a V -^ + s i*. + d c f . ) d'n

,'. substituting for a, d and z ,

- ——: = (e P cos P + 0 f'sin p) V £

„ ƒ? •, r ' ^ r ^' a

p V-t c dn

+ (^ -f'f - *e he P cos p) i .

_{r r ^ 7 .}

+ (e P cos p + ^ f'sin p) r.c

= e P cos p Vé^ + 0 f' sin p V -/

r a ^r c

+ 0 P cos p i-'. c - he F iP. cos p

r L "^ "a z K,

+ ^^[f'sin p-fl^c +^'f'4]

8 -S i r a i l a r l y i f 5M i s t h e p i t c h i n g moment on t h e s t r i p , about t h e f l e x u r a l c e n t r e , 6M = ^ D V^i*c^ (6C + h 60^) d:r\' '^ ' " in L' p V-fc (a Vm_ + z m!_ + d cm a a h a V P + h 2 i^, + h d c -t\) d r i ' . a

Substituting for a; d and z,

6M

^ = (e P cos p + 0 f'sin p) (Vm + h V ^ )

„ ü 2 , , r '^r '^' a ''a

p Vtfc dn'

•!• 0 .-F'f (m. + hJ?.) - Ó he P (m. + h-F.) cos p

' r z z^ r z z'

-i- (ö P cos P + ^ f'sin p) (m, c + he ^.)

Considering the work done in a given displacement,

Let 6 L = increment in the flexural moment

a

&M = increment in the torsional moment

a

Then 6L = i?f 6Z sec p -i- f' sin p &H'

6l\i = P olvl'cos p. P V-f c s e c p f " " ) 0 P cos p V i^ -;- jZl f' s i n p V -F a r a s i n p ' coo P f , o - he CCS p P ^ . + '0 f ' s i n p f . c + / ' f -f. I t dn'

, ^ L ^ d f

/™ j f' j 0 P COS p (Vm + hvl^ ) + 0 f ' s i n p(A^m + h v / ) + 0 P COS p (m. c -»• he ' t . ) - he r j '^ c<. a'' + ^^ p - ' s i n p (m^c + he f ^ ) + ^ ' f (m.^ + h,^^) M and V|c

mi

7y -- COS p P / — V J Ö P COS p (Vm + ml ) + 0 f' s i n p (Vm +mi)

2 c / '% r '^ a '"a'^ '^r *^ a a ' + 0^ P cos 3 (m. c + he -f.) - h e F cos p (m. + h -2.) \ " L a a z 2 J + 0 f ' s i n p (m.c -•- he -f,) + i ' f (m. + Yil)\ i dn' Now L = 6 L„ + 0 L^ f 0 L' + 2f Li/ a r ö ' r ^ r 0 • r 0 and M = 0 ]VU + 0 IvI^ •- 0 M- -i- '0 Mt,

-9-. *-9-. the non-dimencional aerodynamical coefficients are given by

nio/7

dri' ^ =

±

p V f c

o U f — ff. sec^p + f' t a n p 2' %" c '"z * * '-a .£ ^: \-10/7 U o

••(t)

f t a n p (m. + hi;'.) + § f ' s i n p( m. + hK _{0 '^^ a a} dtV 0 0

n^O/7

V2f3 - f

Ü 0 ff' — t a n P > dn' _{c ' ' a} 0 rNlO/7

-Cfj s.Aj^ r'(rjK*<^

dn' r\10/7 J^ = -i2 2 i^ o f a •v o /

(Pf. - hpl'O do'

•10/7 sin p cos p A10/7 Ü o c \ ^

(m. + hR)-h(m. +h'F.)

a a z z dn k , = -.

p Y^fo

o U A 1 0 / 7 -f' P f — dn' - - ^ s i n p a c .P ^ o U f ' ( ^ \ F cos p

(m + h P ) dn'

_{a a"^} M, b , = c , =

-i.

_{vi'2 2}

p n c

o a

•sWl M,

_±

Hk

f (m. + h ^ . ) + f ' s i n p cos p(m.+hK) § z z a a f j A 1 0 / 7 o

P v2£2

-•jr s i n p cos p 1^ n. / c _{— I f' (m + ht:') dn} a a U o H j Jx =

-pvfc5

2 (MO/7 3

1 ï^^^—V m. + h^ - h (m. + hi)

t c a a ^ " " '

Jo \ °/

^-z ^-z dn' M, k , = - _{Prt 2~ = " ° ° ^ P}

Ï Ï

Cj •N10/7 2 P Y M

• (m H- hl^

) d n ' . I t i s t o be n o t e d t h a t c, and c , a r e n o t i n g e n e r a l zero f o r a sv/ept bade v/ing.

For the asswied juodes of f l e x u r e and t o r s i o n , 9

-10-Derivation of critical flutter speed

As i n r e f e r e n c e s 1 and 2, the e q u a t i o n s of ir.otion a r e

kj + B J +0.0 + G , 8 * + J . ê + K , 0 , ^ = 0

1 r 1 r r r 1 r i r 1 r

A , ^ + B J + Cj + G,0 + J , 0 + K^0^ = 0 3'^r 3 r 3 r 3 Ï" 3 ^T 3 r

Let 0^= f e^^ , 0^=0e^^

S u b s t i t u t i n g and e l d x i i n a t i n g ( H ) , b we g e t

(^a^ x'2 + b^yr^ X + 2)j^g3>N'' + J3J^A' H- Y )

v:hcre >N' = / S C IYJ^ 0 / o Y = ^ — r = ^—^ + k , = Y' + k , ,,2j7 2 -.2_p 2 3 c J

p V f c p V

-tc q2 i . e . q ^ X ^ + q^X'^ + q^X' + q^X' + Ci^ = 0

v/hcre q^ = a^g^ - a^g^

q^ = ( a ^ j ^ - a , j ^ + t ^ g j - 'b3g>i)^^o j a ^ Y - a^k^ + (b^ J3 - b ^ j ^ ) C'^ + Xg, - c ^ g j j q^ = (b^ Y - b^k^ + X J^ - c ^ j ^ ) ^ ^ ^ q^ = XÏ - c^k^ 2 2 The t e s t f u n c t i o n i s T, = q. q^q., - 1 1-, - <!-, I, JJ 1 <i 3 0 3 1 4 T , = 0 a t the c r i t i c a l f l u t t e r speed V . 3 c q = 0 i s the c o n d i t i o n f o r vd.ng d i v e r g e n c e . 4 E s t i m a t i o n of the aerodynamic c o e f f i c i e n t s

Using r e f e r e n c e s ^ and 2, the aerodynamic c o e f f i c i e n t s f o r i n c o m p r e s s i b l e flov/ over an unsv/ept wing a r e g i v e n bjr

t^.^.5. 4 = !.«-, 4 = 1.6

~ ra. = 0 . 3 7 5 , - n . = 0 . 7 , - n = 0 . 4 _{z ' a a}

1 1

-Tho values of these coefficients have been derived from experiip.entally determined derivatives for a wing of finite span.

The aerodynaraic acceleration coefficients have been neglected in compp.rison v/ith the structural inertia coefficients.

Por the calculation of v/ing divergence speeds, quasi static values of the dcrivo.tivcs are used.

There is very little experimental data on the var-iation of the derivatives with sv/eepback and v/ith Mach number. For incoinpressiblc flov/ v/e assume that the coefficients vary

as cos pj for the swept v/ing, applying the Glaucrt correction as for the quasi static condition, the derivatives arc laulti-plied by the factor

1

Results

( I ^ l ^ ) ^ 0 - M 2 c o s 2 p ) ^

The c a l c u l a t i o n s v/crc pcrfon^ied for a. v/ing of aspect

r a t i o 5 G-nd taper r a t i o j . For the unsv/cpt v/ing, é vrns

talcen as 0.10, giving a v/ing density of 0.765 l b / f t . ' ' The

f l e x u r a l axis v/as taken a t 0.4 chord and the i n e r t i a a x i s a t

(i) 0,5 chord, ( i i ) 0.4 chord. The sweepback of the flexural

a x i s v/as varied froi:i + 60 to - 60 .

The non-di;,iensional c r i t i c a l speed coefficient

B

i s p l o t t e d for various angles of swccpba.ck and sv/ocpforv/ard,

showing the c r i t i c a l f l u t t e r speed and the c r i t i c a l speed for

v/ing divergence.

Curves are drav/n for tv/o values of the non-dii:iensioixil

s t i f f n e s s r a t i o ^

r = f e ,

Figures 2 and 3 "^-rc cirav/n for incompressible flowj

f i g u r e s 4 ^nd 5 for compressible flov/ (u = 0 . 8 ) .

Conclusions

C r i t i c a l f l u t t e r speed. Effect of sweepback o.nd sv/cepforward

Fror.i figi.u:^es 2 , 3 , 4 •'^-nd 5 v/c sec t h a t the miniiaum f l u t t e r

speed occ-urs for sv.^cpback angles of 5 to 20 . For highly

swept-back or sv/eptfonvard ",/ings the f l u t t e r speed i s doiiblc t h a t f o r

-12-unsv/ept v/ings vdth the same v/ing stiffness. (lIOTE; In these calculations v/c havo neglected the effect of any rigid body freedoms of the aircraft e.g. pitch and. vertical troaislo.tion. Recent theoretical and Gxperi!".ionta,l v/ork (reference 3) has

shov/n that v/hon these body freedoms arc neglected, the calcu-lated flutter speed is liable to be seriously overcstii-.ip.tcd. The calculations in this report can be applied to .an aircraft for which the fuselage is relatively heavy compared v/ith the v/ings. For such an aircraft, both the inertia effect of the fuselage and daiiaping duo to the tailplane tend to suppress the body freedoiiis in pitch and vertical translation).

Effect of change of flexural stiffness w and torsional stiffness m^

— — — — — Ö

The curves have been plotted against the non-dimen-sional parameter B for tv/o values of the non diiiicnnon-dimen-sional stiffness ratio r. Thus if the ratio of the stiffnesses is kept constant, the critical flutter speed is proportional to

y/u^ , and thus increases as the torsional stiffness increases. Over the range of stiffness ratios considered (r = 1 to 2) the critical flutter speed is increased slightly v/hen the flcxizral stiffness is decreasoö..

Effect of variation of the position of the inertia axis

The critical flutter speed increases rapidly as tine inertia a^cis is moved fonvard. The effect is less beneficial v/ith highly sweptback v/ings.

Effect of compressibility

In general, at a Hach number of 0.8, the critical flutter speed is some 15 p-r ce: ü lov/cr than in the incompress-ible case.

Win.g Divergence

Effect of sweepback and swecpfor\/ard

Wing divorgencc is not important for sweptback v/ings. The reverse is true for sv/ept forward v/ings, v/here for angl-s of sv/cop greater than 5 to 15 v/ing divergence v/ill occur at a lov/cr speed than the critical flutter speed.

Effect of change of flexural stiffness x\y and torsional stiff-ness n„

As in the case of flutter, if the ro.tio of the stiff-ness is kept constant, the divergence speed is proportional to jfrn^ , and thus increases as the stiffness increases. For highly

swept for\-/ard v.dngs, the v/ing divergence speed is almost inde-pendent of the torsional stiffness, while for unswcpt v/ings the

-13-divergence speed is independent of the flexural stiffness. Effect of variation of the position of the inertia _^axis

The wing divergence speed is unaffected by a change in the position of the inertia a:;cis, the flexural axis remain-ing fixed.

Effect of compressibility

At a Mach nujnbcr of 0.8, the critical speed for wing divergence is 15 to 20 per cent lov/er than in the incoj.ipressible case.

General conclusions

Prom the above results it is seen that the critical flutter speed is in general hig^hcr for a sv/ept back wingj for a swept forward "./ing, divergence v/ill occur before flutter.

R E F E R E N C E S 1. YJ.J. Duncan and

H. Lyon

2. '7, J. Duncan cj.id C.L.T. Griffith 3, E.G. Broadbent

Calculated flexural torsional flutter characteristics of soi'.ie typical canti-lever v/ings.

A.R.C. R. and H. No. 1782. (1937). The influence of v/ing taper on the flutter of cantilever v/ings.

A.R.C. R. and li. No. 1869. (l939). Sor.io considerations of the flutter problcr.is of high speed aircraft.

Proceedings of the Second International Aeronautical Conference, 1949,

pp. 55^ - 581.

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L O bl J

FIG 2

VARIATION O F C R I T I C A L S P E E D F O R F L U T T E R A N D WINCr O I V E R & E N & E F O R S W E P T B A C K A N D S W E P T F O R W A R D W I N & S ( ^ I N C O M P R E S S I B L E F L O w ) F L E X U R A L A>tlS /^T O-A- C H O P O — W \ N ' & D I V E R G E N C E

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C O L L . & & I : OF AERONAUTICS REPORT N o 3 9 .

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FIG 4

VARIATION OF CRITICAL SPEED FOR FLUTTER AND WtNG DIVERGENCE

r O R SWEPTBACK A N O SWEPTFORWARD W\NGS.

( C O M P R E S S I B L E

F L O W ; n ' Q » 8 )

r U E X U R A L . A X I S . AT 0 « 4 - C H O R D

m

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r

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c

g c = CHORDWISE POSITION

OF I N E R T I A AXIS

6 0 45 3 0

A N G U E O F S W E E P F O P W A R O ( O E O )

15 ? 0 4-5 6 0

ANOL.E O F SWEEPBACK ( o E G )