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REPORT NO.
SD-TDR-62-75 TDR-1 69(3230-11 )TN-2
-AC
91
Aerodynamic Influence Coefficients
from Piston Theory:
Analytical Development
and Computational Procedure
15 AUGUST 1962
Prepared by WILLIAM P RODDEN and EDITH F FARKAS
Aeromechanics Department
Aerodynamics and Propulsion Research Laboratory
and
cy.. , HEATHER A. MALCOM and ADAM M. KLISZEWSKI
C Computation and Data Processing Center
Laboratories Division
Prepared for COMMANDER SPACE SYSTEMS DIVISION
UNITED STATES AIR FORCE
Inglewood, California
LABORATORIES DIVISION * \ )>I'\(I ( ' )1I1 \ 1,,DDC CONTRACT NO. AF 04(695)-169
Report No.SSD-TDR-62-75 TDR- 169(3230- 11 )TN-Z
AERODYNAMIC INFLUENCE COEFFICIENTS FROM
PISTON THEORY: ANALYTICAL DEVELOPMENT
AND COMPUTATIONAL PROCEDURE
Prepared by
William P. Rodden and Edith F. FarkasAeromechanics Department
Aerodynamics and Propulsion Research Laboratory
and
Heather A. Malcom and Adam M. KliszewskiComputation and Data Processing Center
Laboratories Division
I
AEROSPACE CORPORATIONEl Segundo, California
Contract No. AF 04(695)-169
15 August 196Z
Prepared for
COMMANDER SPACE SYSTEMS DIVISIONUNITED STATES AIR FORCE
Inglewood, California
J
2 730
Report No.
SSD-TDR-6Z-75 TDR-169(3z30-11)TN-z
AERODYNAMIC INFLUENCE COEFFICIENTS FROM
PISTON THEORY: ANALYTICAL DEVELOPMENT
AND COMPUTATIONAL PROCEDURE
Approved by /
J. 9' Logan, Di'rect6rA rodynamics and PropulsionResearch Laboratory
B.A. TroeschDepartment Head,Computation and DataProcessing Center
AEROSPACE CORPORATIONEl Segundo, California
ABSTRACT
In this report we present method for calculating the aerodynamic
influence coefficients (AICs) based on third-order piston theory with an
optional correction to agree with Van Dyke's quasi-steady second-order
theory. The AICs are computed assuming the airfoil to have a rigid chord
with or without a (rigid chord) control surface. The influence coefficients
relate the surface deflections to the aerodynamic forces rough the following
definitions.'Inthe oscillatory case,
IF1 = w-ZbZs[C1f[ ] r r h i
and in the steady case,
1Ff (1/2)PV 2 (S/) [Chs] h/
The piston theory is limited to high Mach number (or high reduced frequency),
but Van Dyke's quasi-steady correction extends the validity to some lower
supersonic Mach number at low reduced frequency.K.
/The Acrospace IBM 7090 Computer Program umber HMll provides
the AICs from this theory in both a printed and an optional punched-card
output format. The program capacity is 25 surface strips, 15 Mach numbers,
and 20 reduced velocities for each Mach numberI -
iii
!CONTENTS
ABSTRA CT ....................................... 1ii
SYM BO LS . ... ... ... ... . ..... ... .. ... ... ... .. .... .
I. FORMULATION OF PROBLEM ..................... 1
A . Introduction ............................... 1B. Sign Convention ............................. 1C. Derivation of Equations ....... 1 . ............... 2D . References ... ...................... ...... 20
II. GENERAL DESCRIPTION OF INPUT ................... 21
A . U nits . . . . . . . . . . . . . . . . . . . . . . . . . ... . . . . . . . . 21
B. Classes of Numerical Data and Limitations ............ 21
III. DATA DECK SETUP ............................. 24
A. Loading Order ............................. 24B. Input Data Description ......................... 25C. Example Keypunch Forms ..................... 28
IV. PROGRAM OUTPUT . . ................................ 2
A. Printed Output ............................. 32B. Punched Output ............................ 47
V. PROCESSING INFORMATION ....................... 48
A . Operation ................ ............... 48B. Estimated Machine Time ........................ 48C. Machine Components Used ..................... 48
VI. PROGRAM NOTES . .............................. 49
A . Subroutines ............................... 49B. Generalized Tapes'........................... 49
VII. FLOW DIAGRAM ............................... 50
VIII. SYMBOLIC LISTING ............................. 51
iv
ISYMBOLS
a Ambient speed of sound0
b Local semichord
b Reference semichordr
Ch Element of oscillatory aerodynamic influence coefficientmatrix
Chs Element of steady aerodynamic influence coefficient matrix
C Coefficients in expressions for pressure coefficientn
C Pressure coefficientP
c Local chord
c Control surface chorda
c Mean aerodynamic chord
d Distance between forward and aft control points
F Cont::ol point force
g Airfc U1 semithickness
gx SloF, of airfoil, gx = dg/dx
h Vertical deflection
I ,i JThickness integralsn n C
K Coefficients in expressions for oscillatory aerodynamicn coefficients
k Local reduced frequency, k = wb/V
k Reference reduced frequencyr
L h , L a , LPo Oscillatory leading edge lift coefficients
hv
0 ~o
V9
IL Lift referred to leading edge motion
0
M Free stream Mach number
M h 0M ,MPo Oscillatory leading edge pitching moment coefficientso o 0
M Pitching moment about leading edge referred to leading edge. 0 motion
p -Surface pressure; p0 is ambient pressure
q Free stream dynamic pressure
rh, rt Ratios of hinge-line and trailing-edge thicknesses tomaximum thickness, respectively
S Wing area
s Wing semispan
Th ,T ,T Oscillatory leading edge hinge moment coefficientsO 0
To Hinge moment referred to leading edge motion
t Airfoil maximum thicknessmax
V Free stream velocity
V/b r w Reference reduced velocity, V/br w = 1/kr
v Unsteady component of downwash velocity
w Downwash velocity
x Chordwise coordinate; x 0 is coordinate of pitching axis; xmis coordinate of maximum thickness point; xh is coordinateof hinge line
a Angle of attack; a is initial angle of attack0
P Control surface incidence; also, p = (M 2 - 1)1/2
y Specific heat ratio of air, y = 1. 400
Ay Strip width
vi
!A Leading edge sweep angle
Dimensionless chordwise coordinate, = x/c
p Free stream density
T,Th T t Airfoil thickness ratios at point of maximum thickness,hinge line, and trailing edge, respectively
Circular frequency
(-) Bar denotes term depends on flow characteristics normal toleading edge
[ ] Square matrix
Column matrix
vii
ISECTION I
FORMULATION OF THE PROBLEM
A. Introduction
The pressure on a lifting surface is normally given by a surface
functional relationship. However, in the limit of a high Mach number (or
high reduced frequency), this relationship becomes a point function. As a
consequence of this limit, aerodynamic influence coefficients (AICs) may be
specilied exactly by a strip theory, and control surface and camber effects
may be determined in a straightforward manner.
The present formulation derives the AICs from third-order piston
theory for a lifting surface with control surface (both assumed rigid in the
chordwise direction; i. e. , no camber is presently considered). The deriva-
tion differs only slightly from that of Ashley and Zartarian in that in the
present case the third-order pressure coefficient is generalized to account
for sweep and steady angle of attack, and, following a suggestion of Morgan,2
Huckel, and Runyan, a correction (optional) is suggested to give agreement3.-
with the second-order quasi-steady supersonic theory of Van Dyke. This
quasi-steady correction should extend the validity of the piston theory to
lower supersonic Mach numbers at low reduced frequencies. The derivation
given here is taken from Ref. 4; further, the computational aspects of the
present report are an extension of the computing procedure of Ref. 4.
B. Sign Convention
The flutter sign convention is used in the oscillatory case: forces and
deflections are positive down; rotations are positive with leading edge up.
The aerodynamic sign convention is used in the steady case: forces and
deflections are positive up; rotations are positive with leading edge up.
C. Derivation of EquationsS 5
We quote here the development of Miles 5in obtaining the piston theory
pressure coefficient There are t _A=noLinterest. Th firs assumes
that the angle of attack is small enough that there are pressu perturbations
on the expansion side of the surface. The( second assumes that the angle of
attack is large, and that the expansion pressure approaches a vacuum and is
ineffective-in producing perturbations. Because of the difficulty in specifying
the transition from low to high angle of attack, we shall restrict the present
consideration to the first case, the low angle of attack.
'Hayes' hypersonic approximation states that any plane slab of fluid
initially perpendicular to the undisturbed flow may be assumed to remain so2
as it is swept downstream and to move in its own plane under the laws of
one-dimensional, unsteady motion. Thus, the problem of a wing haygn
arbires -otion ngrmal to its surface may be reduced to the
consideration of the ope,.direinlal motion of a piston into an otherwise
undisturbed flq~w. This problem is relatively simple if the disturbances
produced by the piston are treated as simple waves, for then the pressure
on the piston depends only on the instantaneous velocity there, w. and is
given by
P/P [ + (1/2) (y-l) (W/ao)] Z0/(N-i) (1)
where p0 and a are the values of pressure and sonic velocity in the
undisturbed flow.
"The result, Eq. (1), is exact for an expansion, but the presence of a
shock front(and consequent departure from isentropic flow) renders it only ap-
proximate for a compression. Lighthill has suggested a cubic approximation
Ito be adequate for practical application if lw/a 0 < 1. The series expansion
yields
P/po = 1+ -y(w/a 0 ) + (1/4) y(-y + 1) (w/a 0 )2
3+ (1112) -y(-y + 1)(w/a o) (2)
Lighthill has shown that this expression, Eq. (2), is within six percent of the
value given by either Eq. (1) or the exact solution with the shock at maximum
permissible strength. ,5
The pressure coefficient C = (p - Po)/q is found from Eq. (2) after2
noting that q = (-/Z) p 0 M
Cp = (2/M) [(w/a) + (1/4) (-y + 1) (w/a)2
+ (1)y( + 1) (w/ao) ] (3)
Following a suggestion of Morgan, Huckel, and Runyan, 2 we may generalize
this result, Eq. (3), by writing
Cp = (Z/M ) [ C1 (w/ao) + C 2 (w/ao)2 + C 3 (w/a ) 3 (4)
in which for piston theory
C 1 = 1, C 2 = (-Y + 1)/4, C 3 = (Y + 1)/1Z (5)
and for the quasi-steady theory of Van Dyke 3
C1 - Mfp, C2 [M 4 (Y.+ l)-4p ]/4P4 , C 3 = (+ 1)/1 (6)
3
IVan Dyke gives only the second-order solution so that the value of C 3 is
taken from the piston theory result. The use of the modified coefficients C
and C 2 could extend the lower Mach number limit of piston theory.
We may now calculate the lifting pressure coefficient from Eq. (4) and
the local piston velocity. The normal velocity (positive away from the
surface) on the upper and lower surfaces of a symmetrical thin airfoil having
thickness distribution Zg(x) and angle of attack a is given by
Wu = V (gx - a0 - v) , (7a)
w I = V (gx +a 0 +V) (7b)
where v is the unsteady component of the dimensionless downwash.
For the case of small angles of attack, the lifting pressure (positive
down) is
Cp = CPU - = -( 4/M) [(C1 + 2c2Mgx
+ 3C3M E ) (a° + v) + C3M (a° + v)3 (8)
If, consistent with the small perturbation assumptions of aeroelastic analysis,
only the terms linear in v are retained, Eq. (8) becomes
C =-(4v/M) [C + ZC Mg + 3c MZ(g2 + a (9)p 2Mg, 3 3
2 g x )]
Before discussing the swept wing transformation, it is appropriate to
review the limitations of Eq. (9). Ashley and Zartarian I have shown that the
piston theory is applicable if any of the conditions M > > 1, Mk >> 1, or
4
9k > > I is met. We see that for low reduced frequency the Mach number
necessarily must be high. However, if the reduced frequency is large the
Mach number is not necessarily large; in fact it could be transonic or even
subsonic. At this point it is apparent that any sweep correction introduced
to bring piston theory into line with linearized supersonic theory must be
considered as a low frequency approximation.
The result, Eq. (9), applies to the swept wing case if all quantities are
determined by the flow characteristics normal to the leading edge. The
expressions may be rewritten in the form
(4V/Vi) [" 1 + ?,r Ni- + 3M + vo)]Z (10)
The transformation from the normal values to the free stream values are the
following:
the Mach number
= M cos A (Ila)
the geometryx = x cos A (lib)
'G= b cos A (llc)
the angles of attack and slope
a = a /cos A (lId)
= p/cos A (Ile)
9- = gx/cosA (lf)
the dynamic pressure
- 2q = q cos A (llg)
5
!and the pressure coefficient
C C Cos A (llh)P P
We note that h and k are invariant. From the dimensionless downwash
v = (I/V) 1+ va + (x - Xo) a
+ [VP + (x - x h)PI 1(x - Xh) (12)
whic4 for harmonic motion becomes
v = ikh/b + [ 1 + i(k/b) (x - x ) a
+ 1 + i(k/b) (x - Xh)] (x -xh) (13)
we find the transformed value
v = ikh/S + [1 + i(k/'S) (x - ox .
+ [1 + i(k/) (x - Xh)]l (x - (14a)
= ikh/b cos A + [1 + i(k/b) (x - x ) c/cos A
+ [1 + i(k/b) (x - x h) (pIcos A) I(x -x h ) (14b)
= v/cos A (14c)
The transformed pressure coefficient becomes
C p p cos 2 A = [-4(v/cos A) cos 2 A /M cos A]p p
X T 1 + 2Zz(M cos A) (gx/cos A)
+ 3C 3 (M cosA) 2 (gz + a)/cos A] (15a)
-(4v/M)[[ +2 2Mg + 3 2CM (gx + a2 (15b)1 2 x 3 3 g+ 0 )
We note that the sweep effect shows up only in the coefficients C and _C 2
'for piston theory, there is no effect
S-C = 1, U z = c = (y + 1)/4 , (16)
and for the quasi-steady supersonic theory
I = M/(M - sec ZA)I/
2 = [M4 (- + l)-4 sec 2 A(M? - sec 2 A)]I[4(M - sec z A) z (17)
Equation (17) is seen to be the most general result. If sec A is taken
as zero then the piston theory results, Eqs. (16), are obtained; and if sec A
is taken as unity the sweep correction is not made in the quasi-steady super-
sonic result.
We next consider the integration of the pressure coefficients obtained
above. The oscillatory aerodynamic coefficients referred to the leading edge
are defined by the following equations,
7
9dL/dy = 4pw b 3Lh ho/b + Lao a + Lpo (18a)
dM/dy= 4pw b4(Mhoho/b +M ao a+ Mo (18b)
dT/dy= 4p2 b (Th o h 0 /b + T a * a+ T PO (18c)
The lift, moment, and hinge moment are found from the pressure coefficient
2bdL/dy = q f C dx (19a)
0 p
2bdM/dy= q f x C dx (19b)
0p
ZbdT/dy =q f (X - Xh) C dx (19c)o p
where the pressure coefficient is given by Eq. (15b).
C -- -(4/M) [ 1 + ZC 2 Mg + 3C 3 M (g2 + a2)
x likho/b + [l + ikx/b] a + [1 + ikx xh)/b P I(x xh)) (ZO)
and we have taken the pitch axis at the leading edge x ° = 0. We define the
following dimensionless thickness integrals
I1- (l/Zb) f gx dx (Zla)0
8
12 (1/4b 2) f b g dx (2ib)0
13 = (1/8b 3 ) Zbx gxdx (Z1c)
14 = (I/Zb) 2b dx (Zld)
2b 2 22e0
IS = (1/4b 2 ) f2 x g2xdx Ze
Zb= (1/8b 3 ) xg dx (22a)
2b(1/2b) J gx dx (22a)
xh33 (118/42) f Zb 2xd (22b)
J 3 = (1/8b 3) fbx' gx dx (22c)
xh
2b
34 = (1/ 2b) f gx dx (22d)xh
9 = 1/4b 2 ) 2b x g2dx (22e)xb h
xh
J= (1/8b 3 ) x b ~ gx . (2Zf)
,%
These thickness integrals are evaluated at the end of this section for a
typical airfoil. If we substitute Eq. (20) into Eqs. (19), make use of the
definitions Eqs. (21) and (22). of the thickness integrals, and identify the
resulting expressions with Eqs. (18), we obtain the oscillatory aerodynamic
coefficients
Lh = - iKI/k (23a)0
L a = - KI/k2 - iK2/k (23b)0
L = - K 4/k 2 - i(K5 - ZK 4 h)/k (23c)
0p4Y
Mh = - iK /k (23d)0
Ma = - Kz/k2 - iK 3 /k (23e)0
M~o = - K5/k 2 _ i(K 6 2K 5 h)/k (23f)0
T - i (K5 2K )/k (23g)
T1 = -(K 5 - ZK4 th)/k 2 - i(K6 - 2Ksh)/k (23h)
10
T o =-(K - 2K4 h)/kZ - i(K- 4 K5 h + 4K 4 Z)/k (3i)
where
gh = Xh/2b (24a)
K = (l/M) [-C + 2r MI I + 3C 3 M2 (1 4 + C) (24b)
* = (1/M) [C + 4- 2 MI2 +
3 C3 MZ( 2 15 + a 2 )] (24c)
* (4/3M) [ I + 6-CMI 3 + 3C3M2 (31 + a)2 (24d)
4 (/M) I?1( - h) + 2 2MJI+ 3c 3 M [J 4 + ao(l- h (24e)
K5 (I/M) l(1 - z ) + 4r 2MJ 2 + 3C3 M Z2J5 + a2(l - 9h) (24f)
5 1 h 2 3 5 0
K6 (4/3M) i(I - 93) + 6C 2 MJ 3 + 3C 3 M 2 [3J 6 + a2 _ -h) ] (2 4 g)
To conclude the derivation of the oscillatory aerodynamic coefficients,
we calculate the thickness integrals for the typical airfoil of Fig. 1. We
approximate the airfoil by two parabolas and a line. The equation of the
forward parabola that goes through the leading edge and is horizontal at
the point of the maximum thickness is
g 1 (X)/C = (T/2) (X/Xm) (2 - X/XM) (25)
The approximation by a sharp leading edge is consistent with the theoryhaving ruled out detached shock waves.
11
POINT OF MAXIMUM THICKNESS
HINGE LINE
g X
t/2 th /2
-X h Ch
C: 2b, C I
Fig. 1. Typical Airfoil Cross Section.
where T = t max/c. The second parabola, horizontal at the point of maximum
thickness and going through the hinge line, is
g 2 (x)/c = (T/Z) 1- 1 - rh) [(x - Xm)/(xh - xm)]21 (26)
where rh Th/T and Th = th/c. The line connecting the hinge line and blunt
trailing edge is given by
g 3 (x)/c = (ThlZ) [1 - (I - rt) (x - xh)/(c - Xh)] (Z7)
12
where rt = t/T h and T t = ti/c. By differentiating we find the desired slopes
g'(x)/c = (T/XM) (I - x/x m ) (28a)
gZ(x)/c = - T(1 - rh) (x - Xm)/(xh - XM) (28b)
g (x)Ic - (Th/2) (1 - rt)/(c - xh) (28c)
From the slopes, the thickness integrals follow immediately. Computing the
control surface integrals first yields
Id = -gd (liz) (-rh - Tt) (29a)
J2 = f t = -(1/4) (h - rt) (1 + th) (29b)h
3 = 2 =gd -(1/6) ( - (I + h + t (Z9c)3= f (h t) ~h hth
3 4 f I g d. = (1/4)(-h -Tt) ( - th ) (Z9d)
35 = f gd = (1/8) ('h - wt ) Z (I + h/(l - th ) (?9e)
J, =fIg( Tt) (
=~ + th~d + (112 t h) (29f)
13
The complete airfoil. integrals become
hf gtdc + J1 = Th/Z + Jl (30a)0
ahI f 2g d + J2 ('T/3 % + (T h/ 6 ) (zah + am) + (3 Ob)
0
14 = ah 2 + =[ + (1/3) (T -h)/(2 - m + J4(30d)t5fh .2
1 f g d + 3T h0
= T/I2 + ( J/) ( - Th) (3 ah + m)/ (h - am) + (30e)
ah16 t tgd + J"5
1+ 112(T - -( h+ )% -t)+J 3e
(T /30)Cm + (1/30) (T - Th) z (6 ' + 3 ahar n + am)/(ah am) + f 6 (30f)
where a = x/c, aM = xm/c, and ah = xh/c'
14
Having obtained the oscillatory aerodynamic coefficients, we are
now in a position to derive the AICs. We consider the given and equivalent
force systems in Fig. 2. The equivalent forces are arbitrarily placed at
the quarter-chord, the control surface hinge line, and the trailing edge.
The derivation must relate the forces F I , F 2, F 3 to the deflections h,
h 2 , h 3 through the given leading edge aerodynamic coefficients and deflec-
tions h, a, . We begin with the force equivalence.
0, L
b/2 (b/2 +-d) 2b F 2 M o 0 1)
0 '0 ca T
The loads and deflections are related through the definitions of the oscillatory
coefficients.
L 1 0 0 L h L La 0 h "
M 0 4p2b2y 0 b 0 Mh Mao M PO ba . (32)
T 0 0 b Th 0o T P b
The equivalence in the deflections is given by
h (1 + b/Zd) -b/Zd h 1
ba = -b/d bid 0 h2 (33)
b6 b/d -(b/ri + b/c b0c h 3a
15
1Lo T
(b) h
FD h3
F2 I F3
- 2b -
Fig. 2. Original (a) and Equivalent (b) Force
Systems and Geometry for Oscillatory Case.
16
SSubstituting Eq. (33) into (32), Eq. (32) into (31), and solving for the forces yields
F (1 + b/Zd) -b/d (b/Ca) (3b/Zd - 1)
F2 = 4pwzb 2 ~y 1 -b/Zd b/d -(b/ca) (3b/ Zd)
F3L o o b/ca
Lho Lao LPO (1 + b/2d) -b/ad 0
X Mh M MPO -b/d b/d 0 (34)o 0
Th Ta0 T b/d -(b/d + b/ca) b/Caj
From the definition of the AIC matrix
IFI pw-b 2 s [C hijh (35)
and by identity with Eq. (34), we find the AICs for a single strip.
(1 + b/Zd) -b/d (b/ca) (3bl2d - 1)
[Ch] 4(b/br) 2 (Ay/s) -b/d b/d -(b/ca) (3b/Zd)
0 0 b/c a
Lh L LL0 ° (1 + b/Zd) -b/2d 0
X Mh Ma MPO -b/d b/d 0 (36)o 0
Th T TPO b/d -(b/d + b/ca) b/ca a
17
IIn the absence of a control surface Eq.(36) reduces to
(I + b/2d) -b/d][C h I 4(b/brd2 (A Y/S)[ -b/ 2d b/dI
Lh L] ( /d -b/ 2d
j) (37)Mho Ma -b/d b/d
The complete AIC matrix for a surface of N strips appears in the partitioned
form
I_ _ I o I ..0 0 00
o C hl 0 ICrh' 0 l C tT (38)
--- 0
.1 I. .1I. I.I I III I I.
_ I I I. h
L IhN]
in which the first null partition is reserved for control points at which the
aerodynamic forces are negligible (e. g. , external stores) and in which the
remaining partitions are of the size 3 x 3 or 2 x 2 according to whether or
not the strip has a control surface.
18
The steady AIC matrix follows from the oscillatory solution as a
limiting case. If we compare the definition of the steady matrix
(1/Z)p V 2 (S/c [Chi thI (39)
with the oscillatory definition Eq. (35), we observe
[Ch] Z(sc/S) lim k 2[C h (40)Sk -0r h
r
From the previous section we find the limiting values of the oscillatory
coefficients to be
lim kk2Lh k 2M k h) 0 (41)k -0 o r ho rh0r
lim kL =- K(b /b)Z (4Za)r a 1 r
a00Zrr
limkr = - (K 5 - ZK 4 h) (brb) 2 (4 2c)
rk -0 ro
r
limkL = - K 4 (b/b) 2 (4 3a)k-0r 4 rk -0 0
r
19
Ilir k2 M K5(br/b) 2 (43b)
k -0 r Por5r
lim krT - (K - ZK 4 h) (br/b)2 (43c)k -0 r 03
r
D. References
1. H. Ashley and G. Zartarian. "Piston Theory--A New Aero-dynamic Tool for the Aeroelastician. " Journal of the Aeronautical Sciences,23 (1956), 1109.
2. H. G. Morgan, V. Huckel, and H. L. Runyan. "Procedure forCalculating Flutter at High Supersonic Speed Including Camber Deflections,and Comparison with Experimental Results. " NACA TN 4335, September1958.
3. M. D. Van Dyke. "A Study of Second-Order Supersonic FlowTheory." NACA Report 1081, 1952.
4. W. P. Rodden, E. F. Farlas, P. E. Williams, and F. C. Slack."Aerodynan Inuence Coefficients /)y Piston Theory: Analytical Develop-
ment afid'i 5r edure for- the-1-M- 7-090-Gem p+Aer. " Northrop CorporationReport NOR-61-57, 14 April 1961.
5. J. W. Miles. The Potential Theory of Unsteady Supersonic Flow.London: Cambridge University Press, 1959, pp. 184-185.
20
SECTION II
GENERAL DESCRIPTION OF INPUT
A. Units
Since all dimensional input is geometrical and the aerodynamic matrix
is dimensionless, only a consistent set of length units is necessary--inches or
feet.
B. Classes of Numerical Data and Limitations
The data required by the program are control and option indicators,
geometry, Mach numbers, and a set of reduced velocities for each Mach
number. The example problem illustrates their use.
I. Example Problem
We consider the four-strip wing shown in Fig. 3 at Mach numbers
1. 8 and 2. 5. We use reduced velocities of 4. 0 and 8. 0 for both Mach num-
bers, and compute the steady case for Mach 2. 5. The aerodynamic matrices
will be computed by piston theory and by Van Dyke's quasi-steady variation.
Strips 2 and 3 are considered to have control surfaces. The thickness
integrals will be computed for an assumed airfoil (constant across the span)
having 10 percent thickness, maximum thickness at 40 percent chord, and a
blunt trailing edge having 1. 5 percent thickness.
2. Program Restrictions and Options
a. The number of strips into which a wing may be subdivided
must be < Z5.
b. The number of Mach numbers must be < 15.
c. The number of reduced velocities used with any one Mach
number must be < 20.
21
IA
2b
low S
Fig. 3. Example of Four-Strip Wing.
Strip No. Ay(ft) b(ft) Ca(ft) d(ft)
1 4.7 12. 28120 0 11.9
2 4. 2 9. 50000 5. 25000 9. 0
3 3. 6 7. 06250 3. 99375 6. 6
4 3. 1 4. 96875 0 4. 5
Strip No. am ah T Th Tt
1 0. 4 (not used) 0. 1 0. 015 (not used)
2 0. 4 0. 72368421 0. 1 0. 050 0. 015
3 0.4 0. 71725664 0. 1 0. 050 0. 015
4 0.4 (not used) 0. 1 0. 015 (not used)
sec A = 1. 25 S = 554.0 sq ft
b = 6.5 ft c = 21.0 ftr
s = 15. 6 ft a 's (constant) = 5. 00
*N. B. The trailing edge thickness is listed as the hinge line thickness
in the case of no control surface.
22
d. If it is desired to compute the steady matrix [ Chs I , a zero
or negative value of V/br w must be supplied to the program. (S and c must
also be provided. )
e. Thickness integrals may be given or computed. If given
.they may be given only once with each deck and are considered constant with
strips, a 's may be constant or vary with strips (for each Mach number).0
(T, Tht Tt)'s may be constant or vary with strips. m and h may be
constant or vary with strips.
f. The control surface strips must be a continuation of the
main surface strips; e. g. , in the case of a partial span control surface the
inboard and outboard span statiois should be used as boundaries of the main
surface strips.
g. As many complete sets (decks) of input data may be
supplied as desired (one following the other).
23
SECTION III
DATA DECK SETUP
A. Loading Order
Input decks punched from keypunch forms are loaded behind column
binary deck HMll. The data for each deck should be in the following order:
(1) Heading Card 1
(2) Heading Card Z
(3) NTHRY, NTHICK, NALPHA, NTAUS, NZETAS
(4) ISZ, MSZ, NO PUNJ, JSZ 1 , JSZ;), .. J~zMS
(5) sec A, b, s, 5, c
(6) Ayl, Ay, 'S
(7) bit b 2 , . . . ,bis z
(8) c al' ,ca 2 ' . . ISZ
(9) dip d2 , . . . dr.z
(10) Mach1 , Mah. . . . Mach MSZ
(Ila) If thickness integrals are given:
(a) When all c ai= 0 tabulate only IV~ 12t t v s 1 6.
(b) Any c ai f 0 then include J' ,2 f - v , J 6 t and
t fl I thZ' * *. - hISZ (if NZETAS I only t hl is needed).
(11b) If thickness integrals are computed:
[ if NTAUS = Ilonly T I, T hit and T tl are needed; if c .i=0(.e
no control surface), the trailing edge thickness (T ti) is listed as
Thand the locationfor Tt may be left blank for these strips].
*Please, no remarks about our Greek!
24
(b) "m 1,h I ; m 2 h; . ; mISZ, hISZ [if NZETA = 1
only ml and ahl are needed; if Cai = 0, the program
uses h = 1. 0 (t for trailing edge), and the location
for ahi may be left blank for these strips].
(IZa) If alphas do not vary with strips:
1 a2' ' CLMSZ
(12b) If alphas vary with strips:
(a) al, a2 ' . . aIS Z for first Mach number
(b) al, az, , a S Z for second Mach number
(c) al, a2, a iS z for MSZ Mach number
(13) V/b w series
(a) (V/b r)l, (V/b r0)z, , (V/bro)JSZ for first Mach
numbe r
(b) (V/b r)l, (V/b W) 2 , (V/b )S for second Mach
number
(c) (V/b r )l, (V/br w) 2 , (V/b r) jSZ for MSZ Mach
number
B. Input Data Description
(1), (2) Heading Card I and Heading Card 2 may contain any characters
desired in Columns 2 through 72. These cards are convenient for
identifying the vehicie, surface, date, engineer, etc. Both cards
may be blank but must be included in the data deck.
(3) Control card: FORMAT (1814)
(a) NTHRY = 0, piston theory is used to compute CI and. 'C
25
INTHRY € 0, Van Dyke's theory is used to compute 1 and
C2 (If sec A = 0, then with either theory 1 and
-C are the same)
(b) NTHICK = 0, when thickness integrals are computed
NTHICK # 0, when thickness integrals are given (in this
case they are constant for the surface)
(c) NALPHA = 1, the alphas are constant (do not vary with
each strip)
NALPHA = ISZ, the alphas vary with each strip
(d) NTAUS = 1, the T, T h , and Tt are constant for all strips
NTAUS = ISZ, the T, T ho and T t vary with each strip
(e) NZETAS = 1, tm and th are constant for all strips
NZETAS = ISZ. m and th vary with each strip
(4) Control card: FORMAT (1814)
(a) ISZ = number of strips, < 25
(b) MSZ = number of Mach numbers, < 15
(c) NO PUNJ = 0, or blank, when punched card output is
desired
NO PUNJ 0, no punched output is desired
(d) JSZ 1 number of (V/br w)'s for first Mach number, < 20
JSZz - number of (V/br w)'s for second Mach number,
< Z0
JSZMsz = number of (V/b r)'s for last Mach number, < 20
(5) Single parameters: FORMAT (6EIZ. 8)
(a) sec A, secant of leading edge sweep angle
(b) b, reference semichord
26
B(c) s, wing semispan
(d) S, wing area.
(e) , mean aerodynamic chord
(6) Ay, series: FORMAT (6E12.8)
AYl " . AyIs z , strip widths
(7) b. series: FORMAT (6E12.8)1
b . b is z , local semichords
(8) c . series: FORMAT (6EI2.8)
cal c . caISZ' control surface chords; in the absence of a
control surface, c ai may be zero or blank, but a sufficient
number of cards must be included
(9) d. series: FORMAT (6El2.8)
d1 . . disZ, distance between forward and aft control points
(10) Mach number series: FORMAT (6E12.8)
Mach 1 MachMSZ , in any order desired, but the number
listed must agree with MSZ
(Ila) Thickness integrals given: FORMAT (6E12.8)
(a) . 1 , I , ..... 16 the complete airfoil thickness
integrals
(b) Jl' J 2. I6 the control surface thickness
integrals, use only when c ai 1 0
ghl' h ' . ' hISZ' dimensionless chordwise
coordinate (xh/.c) for the control surface hinge line
(lIb) Thickness integrals are computed: FORMAT (6E12.8)
(a) Tit Thit and Tti' airfbil thickness ratios (t/c) at point of max-
imum thickness, hinge line, and trailing edge, respectively
27
(b) mi and ahi' dimensionless chordwise coordinates for
point of maximum thickness and hinge line
(Ia) Alphas do not vary with strips (alpha is a , the initial angle of
attack). FORMAT (6E12. 8)
al1' CL 2 " MSZ (degrees) are tabulated in order for each Mach
number
(1Zb) Alphas vary with strips: FORMAT (6E1Z. 8)
a 1 , a, 2 .P lSZ (degrees) are tabulated for each Mach
number. The series for each Mach number, starts on a new line
(card).
(13) V/b w series, reference reduced velocity: FORMAT (6EI2.8)
There is a reduced velocity series for each Mach number;
each series starts on a new line (card), and the number
of V/b w's must agree with the JSZ for the respectiver
Mach number.
C. Example Keypunch Forms
Example keypunch forms are given on the following pages. Columns
73 through 80 are reserved for data deck identification. This space rilay be
used in any fashion; however, it is suggested that the last three columns be
used for sequencing. Only the cards with sequencing in Columns 73 through
80 are to be used in the sample data deck; the lines (cards) with Columns
73 through 80 blank are for clarification of input.
28
1-
0~It
29
fl _OD00 N W() a
00 0 00 __ ----
0 0 0 0 0 0 0 0 0 it00 0 0_ 0 0 0 _ 0 0 cz
x x - - -
2 2 2 ___ 2 2lit
z __ _ __ ___ II I _ to
0D r-n 4&-
0 - -
r-
A- ca * - __
40 3n
o 00 0 0 0 ~ 0- _
I1 1
-Aft0 f
131
SECTION IV
PROGRAM OUTPUT
A. Printed Output
1. All input data
2. Thickness integrals (Is and J's)
3. Each group of aerodynamics influence coefficients (comprising
a complete aerodynamic matrix), associated Mach number,
and V/br w
4. Sequencing numbers (Columns 73 through 80) of the first and
last punched cards (output) for each group (one V/br W) of
influence coefficients
5. Example problem printed output is shown on the following pages
32
0
0% a4010% 0n
11-4OD %
0 0% ON%*
0000
ii 0 0CO
wuJ 1D00 000 0000%00 0
- ~ ~ ~ L N NI00- 400 * 000 000zV 000
00U. m 0000
I -J 4z0 0 00 0 0cz 0 U-0ixcc ccczol00 L
L) LU 0 r -4 00 00
U. 00 0%0
0tf0
~~00
LU~~ QU00U. 0 0 0
0000
0 LU L LU30
# Ity NN
N LM
N
'00
00
N
0 00
VNN~00
-4
w. u
f 0-
- - y
at.
* *34
10
N
0 L0x 00p DO
Go NO NO W% 0,0 y -.4 4
m. P.-401
oj 1." C4z ULJu CI da U
(" 00 10 0I
P.- r O
U.1
44
0j I4 0
4 00
~~0.cc NyW-4. f
f4.Y
03
LU
0 -*
-t g-z 10
-- 0000
0 w~ N f
UNF-?U. co.4..4fn4.
-Lf -I UL 4"a.c .co f4N0
z c
z c 0m
0O I I
08 0
0- N
tnp
N .4
36
000
00
0 00
00
LL w0-. 0lu
0U 0
- 0
-4~ 000 u 0
0-
U, 000
LU-4c
LzU.
37
0 1
I.,
Ce00 0D w. I
0r-4 000
2 0
00 If 0
U.1 0 0
W W%0 f ~ N
0. 0 0 N0
0 N0 0
0U I' I - X-.-
W0 Q
U.)U
LL '0 U L-4p-
9 NN I
0
4 0LU'U
0% -4
II,
00 0 0
.. ....
38
0
u~j w
Zq 0
- 00 - 0 to CDO
00f I N
-j t-
-4 040 00 V
LL. WIz e
U)N'
2 00
001 00uj0U
uj ui000-0 00~00 000-0 00
000
0 000ll
.4 0O cy 0
0 0%0%'- 0' 0'
aLI- 0 0I
ON 4 44 0
0 i
U. -9 ui U0- 0% 0%
>.0P 0 0 0 6 00
-~0 000 mCh 0
.4 z 0 0 0% 00 w 0 V ( M 1
D cp% ON
LU O uj i 00 -
I- u o 00 0 c40
Go
0
'CY
000
000
o
N
uJ J 0N Ir
'441
00 0CI0 0
UUJI
m 00-4 IUN u0 It
0- 00 0
of -900C U. Lw Lu NvLU LU w4 0rCDP
N-0 I .00 li 4.
z .J .1 0 .0 0
!0 m0N0IN
L -. 00 II ;c0 00
CD 0
1:1 0 ou14CY
V)0 0 00 0 c0 0 1LU z 0UJ u L LU c
aD 0 t In1111
0 10 LA U0 f" 1
0 0Wf
0- N 00 0
LU 0
z u CKw C7 4 - -4 00 x -
Z 39 Q .0.j in u w0LLL. U LAJUJI t MN
ry 00 uP% .4
z 0 0 el 142
wwLf 0 OD
0
0
00 N
0 U. W I
- Do N
0-I-- 00 0 c00
0W 0of O . yN
- US 0 0 j.Lr0z x 0 a
It 00 0' 3
0 If Ln I'U.- o00 00 01.0 0iu 0 0 oL
oo
LU UY o
UUN LULA 0 t-' U.1L
-000
4> N 00r
00 0000
in- N ry N4 00 00 0
LW LUW
40m? .
4.L -P
NN3
00 000 0
mw w
000
00
4*
U 00 0c- - -
0 (n(In 00 0P-
w ww 0 4It.M 0 -
- Itu 1010 .0
c-o-
uN N q- 0 N0 0 0 0 00 0 c
0 N 00 M 0 INr N0 0 '0- -1
03 -- 11-f00L
LUpqP-N D N .0
0~~~ IA-oM ' 4P01 1 M 0 0 0* NA -4
Mi -M t
LLJ~~ ~ ~ a 0x-d0 0 0x -
z 3 P-4 0 0ON-4L
Z a - -4.0 f
.4> 10 c oz tn
01-
w-. -- 04 00 0
N.NIN
'CM N
IQ 4
00 00
-44
00 000 0
4 U.%
co 0
00 0;
00
00 0000 ; I w -
410 LM
1200 00 0a
-4n
U. . .0 c
- l- -44 0
0 00 g A
0il 110C
0: W I 4 -N -40 0oN0 0 00 0
Q0 0 L LL .1
U.41 !-440
00 00 0oA W,
ODt -t
00
Ug
0I- 0
0-.1
44
I ' C
0000 0
= 0 -4
00 0 0T
A Z 0 00 N N 0
a~~0 W * 4 4
0 ,,' 00 0
0 Q
0 N.O N NAD UI )-
z. C- -
U. tLJ U0I 0 L
U. ' f I -4 m 0 4
-j 0 -9 -J WiQo -4 X4
LA. W 0
0 .ry Z
C;.
0-
LU 0
44
I
B. Punched Output
1. A deck of punched cards (output) from this program is suitable
as an input deck to other programs requiring the use of AICs.
2. All punched output is sequenced in order on Columns 73 through
80 starting with HMI 10000. The data is punched in the following
order:
a. Card I contains (V/b rW) I and M FORMAT (6E12. 8)
b. Card 2 contains the size (number of control points) of the
AIC matrix and the number of strips:, FORMAT (1814)
c. The AIC matrix punched in column binary form and its TRA
card make up the remainder of the punched output for
(V/b r W) 1
3. The order of Statement 2 above is repeated for all reduced
velocities and associated Mach numbers per input deck.
4. Each AIC matrix is punched by columns. Column 1 starts in
Origin 1 and Column 2 in Location (1 + matrix size).
5. The oscillatory AIC matrix is punched in the order -- Column 1
(real), Column 1 (imaginary), Column 2 (real), Column 2
(imaginary), . , , Column N (real), Column N (imaginary).
In the steady case all columns are real and are punched in order.
47
ISECTION V
PROCESSING INFORMATION
A. 2e ration
STANDARD FORTRAN MONITOR system
B. Estimated Machine Time
T = time in minutes
ISZ = number of strips
JSZM = total number of reduced velocities
MSZ = number of Mach numbers
n = number of sets (decks) of input data
T = 1. 0 + .02 [(ISZ MSZ • JSZM)1 + (ISZ • MSZ JSZM) 2
+ + (ISZ • MSZ JSZM)n ]
C. Machine Components Used
Core storage, about 5300
Standard FORTRAN input tape (NTAPE Z)
Standard FORTRAN output print tape (NTAPE 3)
Standard FORTRAN output punch tape (NTAPE 7)
48
ISECTION VI
PROGRAM NOTES
A. Subroutines Used
RDLN, reads and prints title cards
AEROP4, punch AIC matrix
BINPU, column binary punch
All other subroutines are on library tapes
B. Generalized Tapes ,
Input, print,and punch tapes in this coding are defined as Units 2, 3,
and 12, respectively; however, these may be altered by placing the desired
units on symbolic cards HMll0060, HM1l0061, and HM110062.
49
tB A
w 50
SECTION VIII
SYMBOLIC LISTING
Some of the symbols used in the program are defined as follows:
FORTRAN Symbols Definition
NTHRY Option--theory used for CV C 2
NTHICK Option--thickness integrals given orcomputed
NALPHA Option--a's constant or vary
NTAUS Option--T's constant or vary
NZETAS Option--t's constant or vary
NO PUNJ Option- -punching or no punching
ISZ Number of strips
MSZ Number of Mach numbers
J SIZE (M) Number of reduced velocities forMach number
JSZ Number of reduced velocities for aMach number
SEC LAM sec A
BR br
S s
CAP S S
C BAR
*Please, no remarks about our Greek!
51
ISYMBOLIC LISTING (continued)
FORTRAN Symbols De finition
C BAR 1
C BAR 2 2
RAD DEG w/180. 0 (program constant)
DELTA Y(I) Ay for strip i
B (I) b for strip i
CA (1) ca for strip i
D (I) d for strip i
EMACH (M) m'th Mach number
EKR(J, M) 1/kr =(V/br w) for reduced velocity J,
for m'th Mach number
El (N) I series (thickness integrals)
EJ (N) J series (thickness integrals)
Al (I, N) I series for strip i
AJ (I, N) J series for strip i
ZETA H (I). th for strip i
ZETA M (1) m for strip i
TAU (1) T for strip i
TAU H (I) Th for strip i
TAUT (1) Tt for strip i
ALPHA (I, M) n for strip i, for m'th Mach number
EK (I, N) K series for strip i
52
SYMBOLIC LISTING (continued)
FORTRAN Symbols Definition
CONST (I) 4(b/br)2 Ay/s for strip i
A (I, N, K) Premultiplying matrix in oscillatory
coefficients matrix equation
G (N, K) Real, oscillatory leading edge
coefficient matrix
GI (N, K) Imaginary matrix
H (N, K) Postmultiplying matrix in oscillatory
coefficients matrix equation
Q (N, K) Working array
QI (N, K) Working array
P (N, K) AIG matrix, complex
The symbolic listing of the program is shown on the following pages.
53
0E l~ l,0oal 4 - 4l l 4INN NN m ne00 0 0 !CC0 1ISZ000Z 0 0 0 0-N 0N
0 01 3 000 1C1000 01l!HQ0 011 000 0 00 00
Z -M o
D0% 0
W.1-4 W N
I.- LL--U- QI -4 It Iz wU wi LA'
- LU U -9-4 --) a -. 1
N -. z t It
-i U. 04
:1!LU m - xNN C)- 0 101--n M Lu
u 4 z - -
I0- C;. L) r- Lu Z-. LU
HZC X- LA1
-4j -4 z
V)A It x .D CC - W
.- ~ ca * f/0-I xU x 10I
N z -. ) -4 -0o '0
10 NY Z LU N
U0~ cc - 4
LI, -I 0 x xO 0-
x 4 z fl IX44 x
M0 .NM D tIM(0 NI.
ft 1.U x( -0 -
Ln fn x 54
00 i1 10' . O ,I
oog 4: 00 000 00.00 00 000 000 000
1-4.4 -r4 4 r 1 11 4
-4-' a. " ~ HZH ZH 2:nx 3E
-~ cer Xj
4 0 -n -
X P4 CID
Z-0. --- I r 4A N
C.D P2 -
N 4
-0 II W
zU JJ 4Z - '
'2~~ t. S4uIt> _j- 4Ul
O t. x CC NN'a I .1O0~ V? I owU
Q'%. ' - -
0. - 4 .4* -. CDN Ny
£00. w t - -- riI m O C0
S) V)& 1-40 * - -4 NNN
cLO - '0 r4-N -. L aa0 CL
NIN .- X- WW144
Lx- enf a, 2.1 ZU .
if)) In CL' 'Ww
S- Nof l 0fI f w1)O C L m 0 0 L. C L
-~1 V) -4 -4 I - 0
-o X : : C 0 0 43 a-I0
ot a 4 -a 4 Q
00 000I .1 .1 . 0.
0 U. -- U. - owCCccc o -T~-
U.U U. U-
-4 .4~ I 4 NNNIj
55
P. 00 (V Cca, 0 n 00 000 t c00 0-(D00 000oc m 0 0.
0 000 00 0 00 0 00 Z000 000
x xz 3E3Exz z i c !1z
4-AUJ -
U.1,
- - 40 Luz
I CA11U.
CA
4. -dU.
- I
U-. U,:
44 U. We
O W 1- 0.4 rp
-, < , II LU (AA
tA1- CL 1 L -
LU L
00 0 N4 0N N N qC 1 y
LL'I -9000'-A .u
0 .40-LU0 LU -iL 2 IL N
56
M :2
00 00 0C)00C 0 00 0 00 000o 0001o1
Ni at ifiit
-4 n
-4
-c x
LUN- -I
-9- LU L
LU a LWCL N m 49i9IL d-
I4 Z
4 LCL I.- CLC
f9 4 : .1.- 4
ry 0 L z )0Q Ve
toI- 00 aJ 0-I
I.-0 0-" 0 49 -9-
0 90 0Uwc
v ON fm
NN N NY
57
co l o - 0 o 0-N 10r c N("s c0kA kn% 01 -P oc o c o0
aK ZZZ .4 . 4 4- 4 4Z * 4.00 00 00 C) 00 X Z Cc0 0 0
z
0 wU
z 010
-a I.
ON
z~ dc
I .- 4
II PQ 9 -9U.
z +
z 14
44W
z z zCL 0Cc .&- - .00
ui -zZm - - CYw If
02. Q0.0.U U. N w
Z II b.- (a- U
N-- 't I~ vs -eg cy cm r
- 44 WW WN W 58
N - 10 NO t c0 4u inN 10 C00 N N N N NN NJ (I iN NOo0 0 0 0 0 0 000!
0- I)
4) -4 Ijx
.0 nI WUJU
X4 V4
We+
- WW U. U.4U liWU M 0.0.L N I *W 1 IP!*~~I I. 4-IC7 ~ HFI -
-Nj I 4 1Z 1U' 1i1 ,J
4-~~ -<4 -j N---Ij I' .(AN% N L I U. U.1- xI~ . 040I1-T-I~ 4% C 0 1 .. > 4) X.) - Li X
o il, 11. 1- .- II(
Izwu 111. 1 W W:)D 1 4 )XiL ON 3z z~ %AI.;1 L
1- - I.-I 1 .111 V c
59
00 'tU1 Oo I 0 4N 'o - D M - c -0 .uo
NN N NN N NN N NN N NN N N N (i N04N0 00 00000 000 000 000
,4 - -4 -.. .4 .4- .4.44 . 4 -4 .
4-
P-4 CL
-u-
+0'0
ccW W 4
c. -01
4% U +
ZE - -c.
* * 0-
-NN
-1CL0 g
W 0- 10
NN NNt yre yN N f N ,IN N Nf (00 00 000 000 000 !1000 .0001-4 -4 -4 -d~ -4W-4 .44- W .44 w444 1--
.4 # *- d - -- "4 ..d .d .- 444.i-.-
Xmz xr1 Ix ZZZ'xx xx
-4 3d -
.4 .9 4
I.- k" 0-wj pq P4 WN
Ld **4/ 0 0 * -
*~i -- .
N 'N -
Ir.'j f"L -0 . JN "I *w I*
N~ ~ -4 - - : a -
39 u fn -N
t*I zN l N 4-44fE 6m N #- C
N N N
61
00 1 -4- .- 4. NN N
4.-4 - 4 q -4 4-4
IX six sr si 3rs S 1111 i3c zx X
If I
z-4- -4
--
U.1 N M
So- uJ 0. ff-~C aI .
z N 4L CL N z0 Z 0. Uu a .
0 00
I 62
9 0 00
00 ie a 0
.4
Li
m
cz
0~0
az'oil
06
I.-0 n 2Y%
,A 1) IN4A6w1
.OO 9- N- cc G %
C7 0 ^2 W 0 )w00 000Ln CL9
4~ ~ z ~ ~ iL
0 0(
2: 0 -. 1 I -1:N Coe0I I K1. 0 0m 0 FP.
0LC:N- y t 4 0 1 N2:P
4- 10 %- r It III tI
4A4 IOI 0 N tvNN :
LU 000 0 00 cmLU. IIIo I" JE 0 L-Cc
0 4i 09 1:1 1 cc~E -f
IA A t 0 NEI c cc
LU I p0 0 w
9-A ~4u 9Kn IAI-
0 CL WU0. '0 %- 0 tNi
49 1 %U.1 n 41
4o 11 02tI 4
o 00
IA LU j YeIA 0. ca
0e X'-.- 4A N- 0.' 14 .- a
4U 0J 000 0 2 00
0LL
10 P- 0 91
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