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REPORT NO.
SSD-TDR-63-14 TOR-169(3230-I1)TN.I
"0 403 03510
j Quasi-Static Aero-Thermo-ElasticCC) Analysis: Analytical Development
and Computational Procedure
1 MARCH 1963
Prepared by WILLIAM P. RODDEN and EDITH F. FARKAS
Aeromechanics Department
Aerodynamics and Propulsion Research Laboratory
Laboratories Divisionand
HEATHER A. MALCOM
Computation and Data Processing Center
Prepared for COMMANDER SPACE SYSTEMS DIVISION
UNITED STATES AIR FORCEInglewood, California
LABORATORIES DIVISIONS AE I ROSPACUL ( CORPORATIONCONTRACT NO. AF 04(695)-169
' € Report No.
SSD-TDR-63-14 TDR-169(3Z30-11 )TN-8
QUASI- STATIC AERO- THERMO- ELASTIC
ANALYSIS: ANALYTICAL DEVELOPMENT
AND COMPUTATIONAL PROCEDURE
Prepared by
William P. Rodden and Edith F. FarkasAeromechanic s Department
Aerodynamics and Propulsion Research LaboratoryLaboratories Division
and
Heather A. MalcomComputation and Data Processing Center
AEROSPACE CORPORATIONEl Segundo, California
Contract No. AF 04(695)-169
1 March 1963
Prepared for
COMMANDER SPACE SYSTEMS DIVISION( UNITED STATES AIR FORCE
Inglewood, California
ABSTRACT
A collocation formulation is used as the basis for a unified approach to
the various quasi-static aero-thermo-elastic problems. These problems
include rigid and flexible load distributions, divergence, estimation of rigid
and flexible static and dynamic stability derivatives, and the correction of
wind tunnel data measured on flexible models. The formulation utilizes
structural, thermal, and aerodynamic influence coefficients.
The Aerospace IBM 7090 Computer Program No. LD003A provides the
solution to the above problems. The program capacity is fifty collocation
control points and ten values of dynamic pressure.
-ii -
(
CONTENTS
ABSTRA CT ... .... .... .. . ... .... .. .. .. ... ... .... ...
SYMBOLS .......................................... iv
I. FORMULATION OF PROBLEM ....................... 1
A . Introduction .. ....... .... ................ ... 1B. Sign Convention ............................. 1C. Derivation of Equations ........................ 2D . References ...... ... .................... ... 15
II. GENERAL DESCRIPTION OF INPUT ................... 17
A . U nits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17B. Example Problem ............................ 17C. Program Restriction and Options .................. 20
III. DATA DECK SETUP .............................. 22
A. Loading Order .............................. 22B. Input Data Description ......................... 23C. Table of Option Requirements .................... 33D. Example Keypunch Forms ...................... 34
IV. PROGRAM OUTPUT .............................. 43
A. Output for Each Option ........................ 43B. Example Printout ............................ 44
V. PROCESSING INFORMATION ........................ 66
A . Ope ration ... ........... .. .............. ... 66B. Estimated Machine Time ....................... 66C. Machine Components Used ...................... 66
VI. PROGRAM NOTES ............................... 67
A. Subroutines Used .. .......................... 67B. Generalized Tapes ........................... 68
VII. FLOW DIAGRAMS ............................... 69
VIII. SYMBOLIC LISTING .............................. 75
-114
(-
SYMBOLS
a Element of flexibility matrix
aT Element of thermal influence coefficient matrix
b Reference semichordr
Ch Element of oscillatory aerodynamic influence coefficientmatrix
Chs Element of steady aerodynamic influence coefficient matrix
C Z Aerodynamic rolling moment coefficient
Cm Aerodynamic pitching moment coefficient
CZ Aerodynamic normal force coefficient
C Element of aerodynamic normal force coefficient distributionz matrix
C (e) Element of experimentally determined aerodynamic normalZ force coefficient distribution matrix
c Local chord
c Mean aerodynamic chord
c" Average chord
c• Local spanwise lift coefficient
F Element of force matrix
H Generalized reference deflection
H Reference deflection for initial deflection mode0
h Element of deflection matrix
I Element of unit matrix
K Flexibility matrix normalizing constant
k Refe'e.ace reduced frequencyr(
- iv-
Rolling moment
M Element of mass matrix; bending moment
Pitching moment
N Reference load factor
n Element of load factor matrix
q Dynamic pressure
q div Divergence dynamic pressure
S Surface reference area
s Surface span measured from root to tip
T Reference temperature; torque
t Element of temperature distribution matrix
V Shear
W Element of aerodynamic weighting matrix
x, y Cartesian coordinates
5z, YCoordinates of aerodynamic center
x° 0Coordinate of pitching moment reference axis
Z Normal force
A( ) Denotes incremental value
e Angle of pitch
X Eigenvalue
Control point location as fraction of local chord
p Atmospheric density
W Frequency
-V-
Subsc ripts
a Aerodynamic
f Flexible
H, T, N As subscript to CZ, Cm, and Cý, denotes partial differ-entiation with respect to H, T, or N, respectively, e.g.CZH = BCz/8H; as subscript to Cz, merely denotesreference to H, T, or N
Inertial; also used as dummy index, e. g., Ayi denotes
width of ith strip
m Model; moment
0 Initial value
r Rigid
Matrices
[ ] Square
l I Column
L I Row[ F- Inverse
r ]Diagonal
-Vi-
SECTION I
FORMULATION OF PROBLEM
A. Introduction
The most general numerical formulation that can be made of the various
aeroelastic problems is a collocation formulation expressed in terms of
matrices of structural, inertial, and aerodynamic influence coefficients.
Such a formulation of the flutter problem has been discussed in Ref. 1, and
of the static aeroelastic problems in Ref. Z. The present report reconsiders
the static aeroelastic problems by reviewing the derivations and extending
the computational procedures of Ref. 2 in treating the problems of rigid and
flexible load distributions, divergence, estimation of rigid and flexible static
and dynamic stability derivatives, and correction of wind tunnel data meas-
ured on flexible models.
The basis for the present generalized aero-thermo-elastic analysis is
found in recent advances in both aerodynamic and structural theories. The
aerodynamic developments have proffered aerodynamic influence coefficients
based on a number of theories, covering the entire Mach number range of
current interest. The structural developments have provided structural
and thermal influence coefficients for external and temperature loadings,
respectively, for surfaces of arbitrary aspect ratio. The current state of
the art of aerodynamic and structural influence coefficient analyses is sur-
veyed in Refs. 3 and 4, respectively.
B. Sign Convention
We choose the NASA body stability axis system as our coordinate
system: x forward, y starboard, and z down. Positive loads and deflections
are in the same directions, and positive moments are given by applying the
right-hand rule to the coordinate directions.
-1-
C. Derivation of Equations
The force distribution IF} acting on a flexible lifting surface or body
arises from two sources: the aerodynamic forces resulting from the attitude
(as specified by the deflection mode 1h.), and the inertial forces resulting
from a distribution of load factor InI. The aerodynamic forces are found
from the weighted (the weighting matrix must be derived from experimental
data on the force changes resulting from changes in attitude; see Refs. 5 and 6)
steady aerodynamic influence coefficients (AICs), and/or a set of experi-
mentally determined control point force coefficients
IFal = (qS/ )[W1[ChsI IhI + qS IC, r} (1)
The use of the experimental force coefficients is generally necessary to
account for the aeroelastic behavior arising from camber and/or wing-body
interference. The inertial forces are found from the mass matrix
JFiI = [M] Inj (2)
The total force distribution therefore becomnes
{FJ = IFJ + JFi] (3a)
: (qS/X)[Wl[Chs lh} + qSIC~zr + [M]1n} (3b)
-2-
The deflection mode IhI is composed of the initial deflection mode of the
rigid surface JhrI* the thermal distortion mode (which may be considered
to be initially built in) which is found from the incremental temperature dis-
tribution and the matrix of thermal influence coefficients (TICs), and the
deformation mode of the flexible surface {hf I
{hI = JhrI +[arTItI + {hfI (4)
The deformation mode of the flexible surface is found from the total force
distribution and the matrix of structural influence coefficients (SICs)
jhfI = Kaa] fF} (5)
where K is a normalizing factor to the SICs (introduced for convenience in
studying variations in stiffness levels). Combining Eqs. (3b), (4), and (5)
permits the solution for the deformation mode
IhfI = K[A][a] I Fr (6)
where IFrj is the force distribution on the rigid system (again considering
the thermal distortion to be initially built in).
{Fri = (qS/U)[WI[Chs](Ihr} + [aT] t 1) + qS{Cz)} + [Min} (7)
A distinction is made later in the derivation [see discussion precedingEq. (31)] between two sources of the initial deflection mode: that arisingfrom any built-in twist or camber and that arising from the attitude (some-times called the additional deflection mode). The program computes theinitial deflection mode as the sum of these, IhrI = Holhr/Ho} + HNh/Hi, and
both sets of data are required. The distinction is not made at this point
because it is not essential to the derivation.
-3-
whe re
[Al (fl - (KqS/T)[a][W][Chs])- 1 (8)
The total force distribution is found by returning to Eq. (3b) with Eqs. (4)
and (6)
IFI= [B]IFri (9)
where
[B] = ['j + (KqS/E)[W][Chs][A][a] (10)
For the purpose of structural analysis, it is often convenient to convert the
control point forces into the structural loads of shear, moment and torque at
particular stations. A set of transformation matrices may be defined to
compute
tY-e shear
lVi = [V/FlIFI (11)
the moment
IMI [M/F]{I•t (12)
and the torque
IT1 = [T/FI IFI (13)
-4-
Each element in the load coefficient matrices is the load at a particular
station due to a unit control point force.
In certain longitudinal problems, it is desirable to adjust the attitude
so that the flexible system sustains a specified total aerodynamic force. If
we denote the specified total by Z, then
1IJ(tFt - [MIlnI)-- Z (14)
and if we extend the definition of {hrl in Eq. (4) to include the change in pitch
attitude, we have
{hr = Ihrol - A61X1 (15)
The substitution of Eqs. (7), (9), and (15) into Eq. (14) leads to the solution
for the incremental pitch angle
Ae = (I/D)(C - Z) (16)
where
C = I]([BJLFro} - [M]{n}) (17)
D (qS/•-)1IJ [B] [W] [Chs]I x } (18)
and
lFroi (qS/•-)[W][Ch] (Ihrol + [aTltI) + qstc )t + [MZtnl (19)
-5-
The final force distribution in this case follows from Eqs. (7), (9), (15),
and (19)
IFI= [BI Frol- A9(qS/-)[Bl[Wl[ChslJjx} (20)
The final deflection mode follows from Eqs. (4), (5), (15), and (20)
h1= Jhro - AeIx} + K[aIIF} + [aT]1tl (21)
A variation of the foregoing is found in maintaining a constant total force
equal to that given by the rigid condition (without the thermal distortion). In
this case
Z = qS[Ij((1/'-)[W][ChsflhroI + Ic (e)}) (22)
and Eqs. (16), (17), (18), (20), and (l) provide the solution for the final force
distribution and deflection mode.
The possibility of divergence is seen in Eq. (6) as the inverse of the
matrix [A] approaches singularity. The critical values of (Kqs/-) that yield
a singularity are related o the eigenvalues of the matrix, and the divergence
condition is specified by the eigenvalues of the equation
%jhf = [a1w] [WChs lhf} , (23)
where
X F/KSqdi , (24)
from which the divergence dynamic pressure is found
q /div = X/KS (25)
-6-
Any positive value of qdiv shows a divergence possibility although
usually only the first or second values have any practical significance. The
corresponding eigenvectors of Eq. (23) are the divergence modes.
The foregoing development of the force distribution on the flexible
system provides the basis for estimating aerodynamic stability derivatives.
The aerodynamic coefficients are defined by the follnwing:
CZ = Z/qS (26a)
= (I/qS)LI(IF} - [M]In}) (26b)
Cm = f/qSZ (27a)
- -(l /qsE)[x - xoj(IFI - [M] ni) (27b)
C? = X/qSs (28a)
= (1/qSs)LyJ(IF} - [M]ln.) (?8b)
The center of pressure is found from
( X - = (29a)
= -C m/C z (29b)
y/s = /Z (30a)
= C, /Cz (30b)
-7-
The force distribution for use in the above equations is found from Eqs. (7)
and (9). To derive the stability derivatives, it is convenient to define a
reference generalized deflection H, a reference temperature change T, and
a reference load factor N. It is further convenient to consider an initial
deflection mode jhr- = Holhr/Ho} where Ho is a reference initial deflection as
distinct from that caused by the attitude change (additional deflection) specified
by H. * These modifications permit Eq. (7) to be rewritten in a more general
form
IFrI: qSC(zcer)r + (I/0)[W] [Chsi (Holhr/SHo
+ Hjh/HI + T[aT1It/TI))+ N[M] n/N1 (31)
The initial coefficients and the stability derivatives are defined by
CZ = CZ + C H+C + C ZNN (32)
C = C + C H + C T + C N (33)m m° mH mT mN
C? = C + CZ H + C• T + C| N (34)o H T N
where CZH, CZT, and CZN are examples of aerodynamic, thermal and
inertial derivatives, respectively. The substitutions of Eqs. (9) and (31)
into Eqs. (26b), (27b), and (28b) and a comparison with the coefficients of
H, T, and N in Eqs. (32) through (34) yields the following. The initial
SSee footnote, p. 3.
-8-
()
coefficients are
Cz L I][B] fCzr (35)0
Cm = -(l/Z)Lx - xoJ[BI GCz r (36)
C,= (l/s)LYJ[B I Czr (37)
where
ICZr I = I + (Ho/) WI[Ch]1h/H° (38)
rl Zr | F[l[hsihr(8
The aerodynamic stability derivatives are
Czs = L II [B] CzHr} (39)
CH -(/)L x - XJ[B] CzHr (40)
m H
CH= ( /s)LyJL[B]CzHrI (41)
where
jCzHr = (1/-)[W][Chs jh/H 1 (42)
The thermal derivatives are
SCZT =IJ[B]ICzTI (43)
(T
)
Cm -(l/-c-) x - XoJ[B]ICzTI (44)
CT =(1/s)L yI [B]I CZTI (45)
where
ICZTI = (1/U)[W] [Ch.] [aTI It/TI (46)
Finally, the inertial derivatives are
CZ L I] I CzNI (47)
zN
C m -(1/ / -7 " x 0ICzNI (48)
C1N = (1/s)L y I ICzNI (49)
where
ICZNI = (K/Z[W] [Chs [Al [a] [M] In/NI (50)
The center of pressure for each of these loadings is found from Eqs. (29b)
and (30b) using the appropriate coefficients or derivatives.
The rigid aerodynamic stability derivatives may be estimated from
Eqs. (39) through (42) by taking [B] = [ Ij (i. e. , q = 0). Equations (39)
through (42) may also be used to estimate dynamic stability derivatives by
using the (complex) oscillatory AICs which are defined by
IFI = pw b2bS[WI[Ch]IhI (51)
-10-
IBy comparing chis detinition to the steady definition of Eq. (1), we see it is
only necessary to replace [Ch.] by 2k(r (s/S)[Ch] and to permit {h/H} to
become complex in order to make all of the preceding development applicable
to the oscillatory case. Since this is a significant feature of the present
analysis, we digress to illustrate the calculation in some detail. We choose
as an example the estimation of the symmetrical longitudinal stability deriv-
atives. We begin with the general expressions for the lift and moment coef-
ficients for transient longitudinal motion.
C z C z a + CD (&z/2V) + Cz (6z-/2V)a ZDa q
+ CZD(ala-I4V ) +CZ (Dql /4V) +. (52)
C =C a + C (a+/2V) + C ( +9(/ZV)mm m ma mDa q
SC +4V)+C (-k C14V 2) (53)DD a mDq
If we truncate the series at the points shown in Eqs. (5Z-) and (53), and con-
sider the case of harmonic pitching (8 = a) with amplitude ao, then
Cm = aoCma + ik(CmDa +mq) -k(mDZ -a) (5
S~-11-
where the reduced frequency k = uZ/MV. In the case of harmonic plunging
(0 = 0, a = h/V) with amplitude ho, then
Cz =ho0(2 /)[likCZ - kCZDa ik3 CZDa (56)- ~- ik56)z
C = hol/r) ikCma kZ C D ik 3 C mDZa (57)
The lift and moment coefficients can also be found from Eqs. (39), (40), and
(42) by making the substitutions noted above (i. e. , B] [ IJ, and replacing
[Chs] by 2k(ZrS/S)[ChI).
Cz = HIIfJlCzHrI (58)
Cm - H(l/-c)Lx - x J I'CzHrI (59)
where
1CZHrl = -kr(s/S)[WI[Ch]lh/Hj (60)
To appraise the various longitudinal derivatives it is necessary to evaluate
Eqs. (58) through (60) for both motions, pitching and plunging. For pitching
about station x1 , the reference deflection is taken as H = a0 , and the addi-
tional deflection mode is Ih/H}= IxI - xj; for plunging, the reference deflec-
tion is H = h0, and the additional deflection mode is [h/Hl= III. The
computational sequence is now evident. The evaluation of Eqs. (58) and (59)
for the plunging motion yields two complex numbers which, when identifie4
with the real and imaginary parts of Eqs. (56) and (57), lead to the four
derivatives CZD), CmDa, CZDZ , and CmDZa (assuming, of course, that
the static stability derivatives CZ and Cma ha,.z hdready been determined).
i -it-
A similar evaluation of Eqs. (58) and (59) for the pitching motion yields
another set of two complex numbers which, when identified with the real and
imaginary parts of Eqs. (54) and (55), and with appropriate utilization of
the angle of attack derivatives determfned previously, lead to the remaining
derivatives CZq, Cnmq, CZDq, and CmDq. In a similar manner, many of
the lateral-directional dynamic stability derivatives can be found from
oscillatory AICs.
Our treatment of the stability derivatives provides a means of estimating
a dimensionless spanwise load distribution and the spanwise variation of the
chordwise center of pressure on a lifting surface. This calculation requires
the data on which the AICs are based, i. e., the location and number of th&
chordwise control points, and the widths of the strips into which the surface
has been divided. The load distribution coeffl ients are needed to make these
calculations. The flexible load distribution coefficients are given by
ICZf =[B]IlC , (61)
where JCz} denotes any of the various distributions for the different loads
except for the inertial loads; i.e. , .CzJ can be taken to represent any of
JCzr1 _ICzHr1' or ICzTj; in the inertial case, ICZN1 is used directly, i. e.,
IeCf1= JCzNI. The elements of ICzf, are listed in order for each surface
strip as derived in the AICs and can be used to find the local dimensionless
loading and center of pressure. Let us denote the sum of the load coefficients
on the ith strip by
AC Z cf (6Z)
1 strip i
and the moment about the leading edge by
ACm E .Czf (63)i stripi
-13-
where t is the dimensionless chordwise location of each control point aft of
the leading edge. Then the spanwise loading on the ith strip is given by
(cIcl/)i = -AC z.(sl/Ay) (64)1
where Ayi is the load strip width. The local center of pressure is given by
Ai= Cm./ACz. (65)
1 1
The problem of reducingwind tunnel data is our last consideration. The
purpose of a wind tunnel test is the measurement of the so-called rigid loads
on the configuration (unless the model is specifically designed for aeroelastic
measurements). But since no model is completely rigid, some aeroelastic
correction can be made to any wind tunnel measurement. We now consider
this correction. The situation is described by Eq. (61), which may be
written as
C~zf BmI Czr(66)
where {C~e)l is the set of force coefficients measured on the flexible model,
[Bm] is the aeroelastic amplification matrix based on the model properties
from Eq. (10), and IC () is the set of desired force coefficients on the rigid
configuration. Obviously the rigid force coefficients follow from the inverse
of Eq. (66)
czr = [Bm]I - Z1)f (67)
The rigid force coefficients so determined then provide the starting point for
the aeroelastic analysis of the prototype.
-14-
I
S(
D. References
I. W. P. Rodden. "A Matrix Approach to Flutter Analysis." Institute
of the Aeronautical Sciences Fairchild Fund Paper No. FF-23, May 1958.
2. W. P. Rodden, E. F. Farkas, and F. C. Slack. "Quasi-Steady
Aero-Thermo-Elastic Analysis: Analytical Development and Procedure for
the IBM 7090 Computer. " Norair Division, Northrop Corporation, Report
NOR-61-58, 14 April 1961.
3. W. P. Rodden and J. D. Revell. "The Status of Unsteady
Aerodynamic Influence Coefficients. " Institute of the Aerospace Sciences
Fairchild Fund Paper No. FF-33, January 1962.
4. R. H. Gallagher, I. Rattinger, and J. S. Archer. "A Correlation
Study of Methods of Matrix Structural Analysis. " Structures and Materials
Panel, Advisory Group for Aeronautical Research and Development, NATO,
Report to the 14th Meeting, 6 July 1962.
5. J. D. Revell and W. P. Rodden. "A Rational Method for Utilizing
Experimental Aerodynamic Data in Flutter Analyses Through the Use of
Aerodynamic Influence Coefficient Matrices. " North American Aviation, Inc. ,
Report NA-59-867, 30 January 1959.
6. W. P. Rodden. "An Empirical Weighting Matrix for Use with
Aerodynamic Influence Coefficients in Aeroelastic Analyses. " Norair
Division, Northrop Corporation, Report NOR-59-320, 1 April 1959.
7. R. L. Bisplinghdff, H. Ashley, and R. L. Halfman. Aeroelasticity.
Reading: Addison-Wesley Publishing Company, Inc., 1955.
8. W. P. Rodden, C. H. Hodson, and J. D. Revell. "Subsonic and
Sonic Aerodynamic Influence Coefficients from Unsteady Lifting Surface
Theory." North American Aviation, Inc., Report NA-57-1104, 25 October 1957.
-15-
X
• - - I-c- X' I•! - Mc . •oPITCH AXIS- - -1 - 2 35%YELASTnI Ais - -I - Ye10
ROLL AXIS
Fig. 1. Jet transport wing geometry.
S~-16-
75M em
SECTION II
GENERAL DESCRIPTION OF INPUT
A. Units
All units are taken in the pound-inch system with two exceptions: the
surface area is in square feet, and the dynamic pressure is in pounds per
square foot.
B. Example Problem
We consider the static aeroelastic analysis of the five strip wing whose
geometry is shown in Fig. 1. This is the jet transport wing analyzed by
Bisplinghoff, Ashley, and Halfman throughout Ref. 7. We will use the follow-
ing program options: trimmed loads with the structural loads suboption, di-
vergence, aerodynamic stability derivatives (due to angle of attack), and
inertial derivatives (due to symmetrical load factor). We proceed to assemble
the data required as input to the program when it is to perform these options.
The flexibility, mass, and aerodynamic influence coefficient matrices (all size
10 X 10) are printed in the program output example (pp. 45 through 48) and
need not be shown here. The flexibility and mass matrices are taken from
Ref. 1; the aerodynamic influence coefficients are developed from the subsonic
lifting surface method of Ref. 8 at a Mach number of zero. We shall list only
the additional data needed for the representative options. A basic list of con-
stants can be used for all of the options and is given below:
c = 225 inches
K = 1.0X 10-7
S = 564. 236 square feet
Z = -41,919 pounds (this is the lift on one wing required in level flightneglecting any tail load or fuselage lift, and taking one-half thefuselage weight as 17, 400 pounds)
H =1.0
-17-
T =0
N =1.0
s = 500 inches
x ;00
H 00
For the loads option we make the following assumptions: the aircraft is
to be trimmed at three values of dynamic pressure: q = 400, 800, and 1200
psf; the aerodynamic matrix needs no experimental correction (the weighting
matrix is taken as unity); no lift has been derived experimentally (lCze"il=10;,
size 10 1); there is no initial deflection mode (Ihr/Hol = 101, size 10 X
the additional deflection mode corresponds to zero angle of attack (lh/H} = 101,
size 10 X 1); and the symmetrical load factor distribution matrix is a unit
column (0n/Nj = III, size 10 X 1). The control point chordwise coordinate
matrix for use in the trim analysis is
20. 25'-81.00
17. 85-71.40
15.80IxI -63. ZO (inches)
13. 30-53. ZO
11.05-44.20
measured from the (unswept) 35 percent chord line as shown in Fig. 1. The
final data necessary to compute the structural loads are the shear, moment,
and torque coefficient matrices. Considering only the loads at the side of the
fuselage, we have
[V/F] = jI], size 1 X 10,
[M/F] = [45 45 141 141 223 223 323 323 413 413J,
[T/F] = [-20. 25 +81.00 -17.85 +71.40 -15.80 +63.20 -13. 30 +53.20 -11.05
+44. Z0].
-18-
I
The divergence option may be run simultaneously with the loads option
since it requires only a pc~rtion of the above information. We request two
divergence modes and dynamic pressures to be found.
The aerodynamic derive*•ves options will be carried out for the rigid
case (q = 0) in addition to the previous list of dynamic pressures. The addi-
tional deflection mode ih/Hj used in the aerodynamic stability derivatives will
be based on a one-radian angle of attack mode and is found from
-0. 353431.41371
-0. 311541.24616
Sh/HI = -{x/57. 2961 = -0. 27576 per degrec)1. 10304
-0. 232130.92851
-0.192860. 77143,
The calculation of the load distribution and the rolling moment coefficientrequires the following additional data shown in Fig. 1: the spanwise coordinate
matrix
Lyl = 190 90 186 186 268 268 368 368 458 458] (inches),
and the width of each strip and the dimensionless chcrdwise location of each
control point on the strip.
Strip No. AY gf a1 93 0.25 0.752 89 0.25 0.753 91 0.25 0.754 95 0.25 0.755 87 0.25 0.,5
-19-
It is possible to perform all or any selected number of program options
with one input deck if all data are real, although, in the example problem
taking jh/HI = 101 in the loads option does not permit calculation of the aero-
dynamic stability derivatives and taking q = 0 in the stability derivative op-
tion (to find derivatives for a rigid system) is not compatible with the trimmed
loads option which requires q > 0. To avoid computing irrelevant results, we
use two data decks, reproducing the cards (data) in the first deck that can be
used as a part of the second deck.
C. Program Restrictions and Options
1. Maximum matrix size is 50 X 50.
2. Maximum number of dynamic pressures per deck is 10.
3. Provision has been made to reserve a partition in the upper left-
hand corner of the total AIC matrix for control points (and the flexibility and
mass matrices must be compatible with this order of control points) whose )aerodynamic forces may be neglected or found from an alternate theory to
that used for the primary control points. This partition is termed the "external
store" region since external stores are an example of a source of additional
control points requiring such special consideration. The maximum number of
control points that can be reserved for external stores is 49.
4. Maximuni number of modes to be evaluated for the divergence op-
tion is 25.
5. For real and complex derivative calculations, the maximum number
of strips is 20; the maximum number of control points per strip is 10.
6. Multiple options may be performed with one deck except for the
following restrictions:
a. The structural loads option must be used as a suboption to the
other loads options.
-20-
b. In any one input deck, all data must be for steady case (real)
options or all data must be for oscillatory case (complex) options.
c. The divergence option cannot be used in the complex case.
7. Only one matrix of associated aerodynamic influence coefficients
may be included in any one deck. When external stores are present, the
matrix includes the AICs for the stores; however, this area may be set to
zero.
8. In the oscillatory case, the following data must be input in complex
form: AIC matrix, temperature .mode, experimental forces, initial deflection
mode, additional deflection modes, and load factor mode.
9. Program options and significant equation numbers as assigned in
Part C, Section I are listed below.
a. Loads for constant root angle of attack; Eqs. (31) and (9).
b. Loads for trimmed condition; Eqs. (31) and (20).
c. Loads for constant lift coefficient; Eqs. (22), (31), and (20).
d. Divergence, Eq. (23).
e. Initial coefficients; Eqs. (35) through (37).
f. Aerodynamic stability derivatives; Eqs. (39) through (41).
g. Thermal derivatives; Eqs. (43) through (45).
h. Inertial derivatives; Eqs. (47) through (49).
i. Experimental data reduction; Eq. (67).
j. Structural loads; Eqs. (11) through (13).
-21-
SECTION III
DATA DECK SETUP
A. Loading Order
Data decks are loaded behind column binary deck LD003A. There are
nineteen items that can be used in the data deck. The deck is set up with the
items entered in the following order:
1. Heading card
2. Data deck control card
3. Constants cards
4. Dynamic pressure card(s)
5. Flexibility matrix
6. Aerodynamic matrix /
7. Weighting matrix
8. Thermal matrix
9. Temperature mode
10. Experimental force column
11. Initial deflection mode
12. Additional deflection mode
13. Mass matrix
14. Load factor column
15. Chordwise coordinates
16. Spanwise coordinates
17. Strip widths and control point locations (percent chord)
-22-
18. Load coefficient matrices (shear, moment, torque)
19. Second and successive "additional deflection mode" columns
Items 1, 2, 3, and 6 must be present in all data decks; the presence (or
absence) of the other items is determined by the option and data indicators
used in the data deck control card (Item 2).
B. Input Data Description
1. The heading card may contain any alphanumeric characters desired
in Columns 2 through 80. It is normally used for job title, engineer's name,
and data.
2. Data deck control card (FORMAT 1814): This card specifies which
options are to be performed and also indicates the presence of certain data.
The eighteen fields of this card are used as follows:
Field 1. Column 4 contains a 1 if load option I (LOADS FOR
CONSTANT ROOT ANGLE OF ATTACK) is to be
performed; it contains a zero or blank if not to be
performed.
Field 2. Column 8 contains a 1 if load option 2 (LOADS FOR
TRIMMED CONDITION) is to be performed, a zero
or blank if not.
Field 3. Column 12 contains a I if load option 3 (LOADS FOR
CONSTANT LIFT COEFFICIENT) is to be performed,
a zero or blank if not.
Field 4. Column 16 contains the number of modes (25
maximum) of the DIVERGENCE option to be
performed, a zero or blank if not.
Field 5. Column 20 contains a I if the INITIAL COEFFICIENTS
option is to be performed, a zero or blank if not.
-23-
Field 6. Columns 23 and 24 contain the number of solutions
for AERODYNAMIC STABILITY DERIVATIVES
desired. This option permits varying the "additional
deflection mode" jh/Hi when the other data are
unchanged. The first mode is included as Data Item
12, and the second and successive modes are
included as Data Item(s) 19.
Field 7. Column 28 contains a 1 if the problem is complex, a
zero or blank if all data are real.
Field 8. Column 32 contains a I if the THERMAL STABILITY
DERIVATIVES option is to be performed, a zero or
blank if not.
Field 9. Column 36 contains a 1 if the INERTIAL DERIVA-
TIVES option is to be performed, a zero or blank
if not.
Field 10. Column 40 contains a 1 if a weighting matrix is
included in the data deck, a zero or blank if excluded.
Field 11. Column 44 contains a 1 if a thermal matrix and
associated temperature mode are included in the
data deck, a zero or blank if excluded.
Field 12. Column 48 contains a 1 if an experimental force
column matrix is included in the data deck, a zero or
blank if exluded.
Field 13. Columns 51 and 52 contain the number of elements
(control points) reserved for external stores, a zero
or blank if none is reserved.
Field 14. Columns 55 and 56 contain the number of dynamic
pressures (10 maximum) for which cycling options
(repeated for each "q") are to be performed.
-24-
Field 15. Column 60 contains a 1 if the EXPERIMENTAL
DATA REDUCTION option is to be performed, azero or blank if not.
Field 16. Column 64 contains a 1 if the STRUCTURAL LOADS
option is to be performed, a zero or blank if not.
Field 17. Columns 67 and 68 indicate the order of the system
contained in the data deck (50 maximum).
Field 18. Column 7Z contains a 1 if the flexibility matrix is
present in the data deck, a zero or blank if absent.
In the subsequent description of input data we find it convenient to
refer to above fields 1, Z, 3-, 4, 5, 6, 8, 9, 15, and 16 as Options identified
by the respective field numbers. The other fields contain control numbers
and will be referred to as Item Z, Field (respective number).
3. Constants cards (FORMAT 6E12.8): The two constants cards
contain the items listed in the following order:
a. CBAR: mean aerodynamic chord
b. FLEXK: flexibility matrix normalizing constant (or scaling
factor); when the matrix has not been normalized FLEXK = 1.0
c. CAPS: surface reference area
d. CAPZ: vertical force
e. CAPH: generalized reference deflection
f. CAPT: reference temperature change
g. CAPN: reference load factor
h. SMALS: surface span measured from root to tip
i. CAPXO: coordinate pitching moment reference axis
Sj. CAPHO: reference deflection for initial deflection mode
--25-
4. Dynamic pressures (FORMAT 6EI2. 8): If six or fewer dynamic
pressures are liste', one card is used; seven through 10 pressures require
two cards. The q card(s) are not required when performing Option 4(divergence) above, but unless Item 2, Field 14 is zero or blank, the q's
must be input and must agree with the number in this field.
5. The flexibility matrix [a] is input if its presence has been indicated
in the data deck control card (Item 2, Field 18). If not included as input, a
rigid case is assumed ([a] = [0]). The following card order and formats
must be used when the matrix is input.
a. Control card (FORMAT 1814)
Field 1. Columns 3 and 4 contain m, the number of rows in
the matrix (< 50).
Field 2. This field is not used; it may be left blank.
Field 3. Column 12 contains IFORM; if IFORM = 1, thematrix elements are to be input using FORTRAN
FORMAT 6EI2. 8; if IFORM = 0, the elements will
be input using column binary format.
Field 4. Column 16 contains IROW; if IROW = I, the matrixelements are input by row; if IROW = 0, the elements
are input by column.
b. Matrix elements (use format specified above)
If IFORM = I, and IROW = 1: use FORMAT 6EI2. 8 and input
the matrix element by row; each new row starts on a new card (line).
I If IFORM = 1, and IROW = 0: use FORMAT 6EI2. 8 and inputthe elements by column with each new column beginning on a new card.
If IFORM = 0, IROW must = 0: the. matrix elements are input
using column binary format. This format should be used only if the data are
available as punched-card output from appropriate computer programs. Theelements must be punched by columns; Column 1 starts in Origin I and
Column 2 starts in Location (1 + m). A TRA card must end the deck. (This
transfer card has a 7 and a 9 punch in Column 1, Columns 2 through 72 areblank, and Columns 73 through 80 will contain the characters used foridentifying and sequencing the deck.)
-26.
6. The aerodynamic matrix [Ch] is always present in the data deck.
It consists of two parts if external stores have been reserved in Item 2,
Field 13, and it is real (steady case) unless indicated as complex (oscillatory
case) in Item 2, Field 7. The [Ch] matrix must be entered as follows with
this exception: Items a and b (below) will be omitted when no external stores
are reserved:
a. Control card for store[ Ch] (FORMAT 1814)
Field 1. Columns 3 and 4 contain m, number of control
points reserved for external stores (S.49); m = 0
will direct the program to set the store [Ch] to
zero.
Field 2. This field is not used, it may be left blank.
Field 3. Column 12 contain IFORM as defined in Item 5a.
Field 4. Column 16 contains IROW as defined in Item 5a.
b. Store matrix elements (format specified in Fields 3 and 4,
above control card): Omit this input if mn = 0 is used in the control card.
When the external stores [ Ch] matrix is input, use the format in the manner
described in Item 5b. If the matrix elements are complex numbers
(oscillatory case) consider the matrix size as m X 2m (imaginary parts of
the elements form the even-numbered columns) and input as though all
elements were real numbers.
c. 1/kr card (FORMAT 6E12. 8): The reference reduced velocity
card is always present. For the steady case, it may be blank or contain any
number, but for the oscillatory case it must be the 1/kr associated with the
aerodynamic matrix.
d. Control card for surface [Ch] matrix (FORMAT 1814)
Field 1. Columns 3 and 4 contain m, the number of controlpoints on the surface. ( m = size of system minus
number of store control point.)
-27-
Field 2. Columns 7 and 8 contain L, the number of strips
(partitions) in the surface. L = 1 for a full
(unpartitioned) matrix.
Field 3. Column 12 contains IFORM as previously defined.
Field 4. Column 16 contains IROW as previously defined.
e. The input listed below must be repeated for each partition
(i = 1, L).
(1) Partition size (FORMAT 1814)
Field 1. Columns 3 and 4 contain n, the number of control
points on strip i (size of partition i). If L = 1,
n = total number of control points on the surface.
(2) Matrix elements in partition i (format specified in
control card, Item 6d).
When the matrix is for the steady case the elements in each partition are
input in the same manner as the elements in the [a] matrix (Item 5b). In
the oscillatory case, the elements are entered in the same manner as
described for the stores oscillatory [Chi matrix (Item 6b).
Note: Punched cards output from IBM programs based on the sources listed
below may be used (with no alterations) for Items 6c, 6d, and 6e when the
theory is appropriate to the problem. Punched cards output from the pro-
gram based on slender-body theory may be used for Items 6a and 6b if the
first card (1/k r) is removed from the deck. The list shown is complete only
at this writing; additional compatible sources are currently under development.
W. P. Rodden, E. F. Farkas, H. Malcom, and A. M. Kliszewski.
"Aerodynamic Influence Coefficients from Incompressible Strip Theory:
Analytical Development and Computational Procedure." Aerospace Corp.
Report No. TDR-169(3230-11)TN-5, 3 September 1962.
-28-
S(
W. P. Rodden, E. F. Farkas, H. Malcom, and A. M. Kliszewski."Aerodynamic Influence Coefficients from Supersonic Strip Theory:
Analytical Development and Computational Procedure." Aerospace Corp.
Report No. TDR-169(3230-11)TN-I, 1 August 1962.
W. P. Rodden, E. F. Farkas, H. Malcom, and A. M. Kliszewski.
"Aerodynamic Influence Coefficients from Piston Theory: Analytical
Development and Computational Procedure." Aerospace Corp. Report
No. TDR-169(3230-l1)TN-2, 15 August 1962.
W. P. Rodden, E. F. Farkas, and G. Y. Takata. "Aerodynamic
Influence Coefficients from Slender-Body Theory: Analytical Development
and Computational Procedure." Aerospace Corp. Report No.
TDR-169(3230-11)TN-6, 31 October 1962.
7. Weighting matrix: The weighting matrix [W] is input if its
presence has been indicated in Item 2, Field 10. The matrix is entered
with formats identical to those given for the [Ch] matrix (Item 6a, b, c, d, e)
with three exceptions: (1) the I/kr card (Item 6c) is omitted; (Z) in
repeating Item 6a if m = 0 the program will use a unit matrix for the store
[W]; and (3) the matrix is always real. It is not required that [ChI and [W]
be input with the same format in the same data deck; i.e., one may be in
column binary while the other uses FORTRAN format, and one may be
partitioned while the other is a full matrix.
8. Thermal influence coefficients matrix: The thermal matrix [aTI
is input when Item 2, Field 11 notes its presence. The matrix is required
input for Option 8. The formats for entering [aTI are the same as those
used for the flexibility matrix: Items 5a and 5b.
9. Temperature distribution column (FORMAT 6E12. 8): The temper-
ature mode It/Tj always accompanies the above thermal matrix. For
steady cases (real AICs), the It/TI matrix is entered as one column, six
elements per card. For oscillatory cases (complex AICs), the mode is
complex with the imaginary part considered as a second column; tabulate by
( column with the second column starting a new card.
-29-
)
10. Experimental force column (FORMAT 6E12. 8): The experimental
force column {Czr)r is input only if its presence has been indicated by Item 2,
Field 12. The column (real or complex) is entered in the same manner as
the column in Item 9.
11. Initial deflection mode (FORMAT 6E12. 8): The initial deflection
mode 1hr/Hol, must be present to perform Option(s) 1, 2, 3, and/or 5.
Again refer to Item 9 for entering the column (real or complex). If all
elements in the column are zero, a sufficient number of blank cards may
be used.
12. Additional deflection mode (FORMAT 6E12. 8): The additional
deflection jh/HI must be included for Option(s) 1, 2, 3, and/or 6. See
Items 9 and 11 for inputing this column (real or complex).
13. Mass matrix: The mass matrix [M] must be in the data deck
when performing Option(s) 1, 2, 3, and/or 9. The matrix is input as follows:
a. Control card 1 (FORMAT 1814): Field 1 must contain NRC,
order of the matrix (same as Item 2, Field 17).
b. Control card(s) 2 through 7 (FORMAT 1814): Field I and
consecutive fields in control card Z and successive control cards contain in
order LL1, LH1, LL, LH, ... ,LL NRC, LH NRC. LL. is the row
number where the first nonzero element appears in column i; LHi is the
row number where the last nonzero element appears in column i. If only
one nonzero element is in column i, its row number must be used for both
LL. and LH..1 1
c. Elements in [M] matrix (FORMAT 6EI2.8): The elements are
entered by column as shown below. Each column starts on a new line (card).
Column 1: elements LL1 through LH1
Column 2: elements LL2 through LH2
Column NRC: elements LLNRC through LHNRC
-30-
Any zero elements between LLi and LHi must be entered or their respective
fields left blank. If all elements ina column are zero, then at least one
element must be entered as zero (a blank card may be used); LLi and LH.
for this zero element must appear in the above control cards.
14. Load factor column (FORMAT 6E12.8): The load factor column
In/NJ is included if [M] is used (Option(s) 1, 2, 3, and/or 9). If no
correction is desired, input a unit column. The column is real for steady
cases and complex for oscillatory cases. (See Item 9 for column input.)
15. Chordwise coordinates (FORMAT 6E12.8): The control point
coordinate matrix fxJ or {xJ is input for Option(s) 2, 3, 5, 6, 8, 9, and/or
15. The matrix is entered only once (six elements per card) for any combi-
nation of the above options.
16. Spanwise coordinates (FORMAT 6E12. 8): A spanwise control
point coordinate matrix tyJ must be used with Option(s) 5, 6, 8, 9, and/or
15. The elements in LyJ are the control point distances from the roll axis.
17. Strip widths and control point locations (percent chord): The strip
widths Ayi and the control point locations gi, j (given as a fraction of the
local chords) are required for Option(s) 5, 6, 8, 9, and/or 15. These data
have the following input format.
a. Control card(s) (FORMAT 1814)
Field 1. This field must contain NSTRP = number of surface
strips (maximum of 20 for above options). Do not
include external stores as strips.
Field 2. Field 2 and successive fields contain NCPT. = numberI
of control points (L< 10) on strip i. (i = 1, NSTRP)
b. Ayi and 9i,j (j = 1, NCPTi) (FORMAT 6E12.8): These data
are entered by strips; each strip requires one line (card) when NCPT < 5 and
two cards when NCPT > 5. Start with the first card for strip 1 and input the
data as follows:
Field 1. Ayl: width of strip 1.
-31-
Field 2. 1f: location of first control point for strip 1.
Field 3. 2: location of second control point for strip 1.
Field (NCPT1 + 1). 91NCPTI: location of last control pointfor strip 1. (If NCPT > 5, then •16 would be in
Field 1, second card).
Repeat the above for each strip
18. Transformation matrices for structural loads: The shear, moment,
and torque matrices, [V/Fl, [M/F], and [T/F], are input only if Option 16
is selected in conjunction with Option(s) 1, 2, or 3. The elements in any
specific row of each matrix are the loads at a particular station due to a
unit control point force.
The matrices are input in the order in which they are listed (above
pp. ) with each using the same format (similar to the mass matrix but input
by rows instead of columns). The format for [V/F] is given below; the same
format is used for [M/F] dnd [T/F].
a. Control card 1 (FORMAT 1814): Field I contains NVMT, the
number of rows in [V/F]. Note the matrix size is (NVMT X NRC), where
NRC agrees with Item 2, Field 17 and NVMT < 20.
b. Control card(s) 2 through 7 (FORM.AT 1814): Field 1 and
consecutive fields of these cards contain LOW1 , LHIGH 1 , LOW2 , LHIGH.,
... I LOWNVMT, LHIGHNVMT. LOWi is the column number of the first
nonzero element in row i, and LHIGH. is the column number of the last
nonzero element in row i. (i = 1, NVMT)
c. Elements of [V/F] (FORMAT 6E12.8): The elements are
entered by row in the same manner as the columns in [M]. (Item 13c).
¶19. Second and successive "additional deflection mode" columns
(FOR MAT 6E12. 8): It is possible to vary the additional deflection modal column
Sh/HI to obtain two or more solutions for real or complex aerodynamic
stability derivatives. Use the input format given for Item 9 (or IZ) repeating
the format for each new column.
C. Table of Option Requirements
To aid the program user in data deck setup, the following table shows
which of the above described 19 data items are required to make up a data
deck for any option.
DATA ITEM
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19
1 P P P P P P OO O O P P P P A AA A
2 P p P P P P O OO O P P P P P AA -A
z3 P P P P P P OO O O P P P P P AA -A
S4 P P P A P P O A A A A A A A A A A A AS5 P P P P 0 P 0 A A 0 P A A A. P P P A A
06 "P P P P 0 P 0 A A A A P A A P P P A 0
8 P P PP O P O P P A A A A A P P P A A
9 P P P P O P 0 A A A A A P P P P P A A
15 P P P P P P 0 A A P A A A A P P P A A
16 - Set up for option (1, 2, or 3) being performed A P A
P = must be present; A = must be absent; 0 = optional inclusion;
-= present when Option 16 is used with Option 1, 2, or 3.
Note: The above table applies to options performed singly with the exception
of Option 16 which must be used concurrently with Option 1, Z, or 3. Where
multiple options are to be performed with one data deck and an item must be
present for one of the options and absent for another, that item must be in-
cluded in the data deck. Also any data item indicated as present in the con-
trol card (Item 2) must be included in the deck, even though it may not be used
in the particular option(s) chosen.
-33-
D. Example Keypunch Forms
Example keypunch forms are given on the following pages. Columns 73
through 80 are reserved for data deck identification and sequencing. Only
the cards with sequencing are used in the two data decks set up for the exam-
ple problem; the lines with Columns 73 through 80 left blank are for descrip-
tion of input.
-34-
on I I* in0 0 1 o 0 ey 2
1 0 *0 t:in it'
01 0al 2x I
00
81 1 1w Iuz
ccw
ab 0 iR
04 It'R S a
w
-5-
04fn -A-N ov 0 ai-D N- -In+ rl mnfn 0-- W-
cc1.- 14
0
0-- 0
InIn +a Fn 2cc 0G6
w
j
IL
in 0 0- o
X cy
0
......... . . .....
+
-35-
0 1
0 0
1 3
7j")
-36-L
Ix 9
a ft
I -0
ft-c 0 - - - -0 -y 0- -0 AyW 0 0 r
- - - - - -37-
I~~ ol I N ' - m
1 0
a 0 0- ei
Vr I
0 - ) N .
1W)F T W W
cm 0 0- 0 0 IVP 0 W 0)
0W W 0 0 W
O" to 0 _U N - 1W)
-38-
30 1
10 1a+
If
0 __
v aIr -18 i
S Nq0 1
+a
3-39
3I o
0 2_
1 N
2 01
3q 0 0 P
at fffmitN 0 0 I
A IR
-40-
ty in I 1 0 0 e4 in0 1 1 0- 0 1 1 0 40
1! 1 oil 1 0 v
T0
3121
0 0
040
w+00+
It
it
12In
j 40.9 N CY
0 0+ +
04
0
al o 0
6 z 0
0
a N CY N+ + It
0 0 0 0 0 0 0 aj 0 0 0 0 0 0 si
+ + + +
ccr-4 CY
co a In ALA 40 to IV P.
I o 0 0 0 0 0 Ao 0 0 0 0 0 a
ror C4
M-4 in4n
cyW 0 CY-j V CY N 44 N
+ +0 00 0
a. + +
W CYJIL - I" m2
0 cyx fn
in ty
iC Iaa AI
AH
- -U
U1
4 111A4
V4 0 .5 -
Va a -
a 0 1
2-42
SECTION IV
PROGRAM OUTPUT
A. Output for Each Option
In addition to all input data, the following is printed for each option.
1. Option 1 (for each dynamic pressure)
a. Deformation mode
b. Total deflection mode
c. Final aerodynamic force distribution
d. Total force distribution
2. Option 2 (for each dynamic pressure)
a. Incremental pitch angle
b. Deformation mode
c. Total deflection mode
d. Final aerodynamic force distribution
e. Total force distribution
3. Option 3 (for each dynamic pressure)
a. Aerodynamic lift
b. Incremental pitch angle
c. Deformation mode
d. Total deflection mode
e. Final aerodynamic force distribution
f. Total force distribution
-43-
4. Option 4 (divergence)
a. For each mode: mode number, eigenvalue, divergence
pressure, and number of iterations
b. Divergence modes (eigenvectors)
c. Check eigenvalues
d. Check eigenvectors
5. Options 5, 6, 7, 8, and 9 (for each dynamic pressure)
a. Distributed force coefficients
b. Aerodynamic normal force, pitching moment, and rolling
moment coefficients
c. Total chordwise center of pressure
d. Total spanwise center of pressure
e. For each strip: spanwise loading and local center of pressure.
6. Option 15 (for each dynamic pressure)
a. Corrected experimental force coefficients
b. Aerodynamic normal force, pitching moment, and
rolling moment coefficients
c. Total chordwise center of pressure
d. Total spanwise center of pressure
e. For each strip: spanwise loading and local center of pressure.
7. Option 16 (for each dynamic pressure)
a. All data listed under the associated option (1,2, or 3).
b. Shear, moment, and torque at load stations.
B. Example Printout
The printout for the example problem is shown on th. following pages.
Some of the results shown may be compared with similar calculations presented
in Ref. 7.
-44-
.00000000000
1L0~ ~~~ 0 V0N I LI0,N. aM0 00M&&&-.OQN4Q
* LU 0 *****0400 m
w V. *" %1 1( ,-
.Va. ~~~ 0 C 00f. 0 00 0N-t00
a- U. m000 w4 0.4 N N
000000.00000.
0-.~C 0 KONU4F&.P-0 O.E0 a
0o 0. 0 0P-N-a.0U 0 00 &4 N. 0 C00. 0 9 ,- 0. 0 0000000000
00% ItL 00N.m -N N -.1 4M 4 44g
-- 0 *0 O o o o o 0 F- ooe 0&Aooo
0 0 0 0
4000.00.0.0000 0.000.0.000.000
x ýr-L 0 - -N 0 XN400m4-d.O000
- j N P- P-.O-im0 m W -j 4 P- m WN N m mf
4. 0* ** * 0* **0* .Lu U K 0 0 000 0000000000;8 C 4 C00 a, 0
AI VIUN & 0
z 4u4n 49N io 0 ... MO0o0o00oo0 0.OOOOOOOOOOIA W - U'r 0 Ciez t -. 0 0 m w LULUwLUwLUUww wwwwwwwwww0f -l U 0 01 &~ 0 .00 .00.0.&0.0 00=000.0000
-0~* . '4A~ UO0OMQ0C0Cr00 000.000.0QOO1 0 0 w LU z UMOMOu'~l.0U zOO0QOo0%0ooO
. ON Ni z J 20 N m W% f- m P I0 211 0.0fMO MMLA 0000 de CL4 . = -4Um ~~VN 0N W m " Pf ~0 0Ar- 0 a, - f-.t.-4A-q 0 ~ ryN4N .JNONNNO.4O
LU." NZ.-:-0 m 'N~~ tu0 % c0uooo =ýrN4N.LI 0 - W fn p .0-'0 0 UNN0.0NNI14W14Wcy gO .90 C;OO C; * 00 0 *
1. P '- 2 0
l- 1- 00o44 1w LU. 0 NM0NMM er4MMM ~ 0I4444,
-a a, tO.0.INN NOOOOOOOOOO W000000000003 0 CPUULULWU WLUWLUUWL-6.
0 00 (P 4.0a- 0 0 cý0 0 000.ocp oo0. &00ý00000..00z I Kz CP w c 00.0.000..000 CIO00.000. CO 00
0.4-0I -N W N r- -r xO0.0.ON0.0N00. Z0.000000.0 00 ~ oz op 0. 40 1-0 Q - &N N 0.-m9-N Z.4-0 O.N 0"m -i oLAm ? N ýN 0u 0 0 N t- -1 4 M 0 N4u0 ~N'0 N f-4
o LU 040NN0MUNMOO Ot40U4 r 00WN 1N.4r-
O an 0
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SECTION V
PROCESSING INFORMATION
A. Operation
FORTRAN II MONITOR system
B. Estimated Machine Time
Because of the number of variables involved such as options, order of
system, number of dynamic pressures, etc. , it is not possible to give a
practical method of estimating machine time. When the data are appropriate,
considerable time may be saved by performing as many options as possible in
one machine pass. Also, complete but independent data decks may be stacked
to save program read-in time.
C. Machine Components Used
About 26,000 core storages
Standard FORTRAN input tape (NTAPEZ)
Standard FORTRAN output print tape (NTAPE3)
Four utility tapes (NTAPE4, NTAPE5, NTAPE6, NTAPE8)
-66-
SECTION VI
PROGRAM NOTES
A. Subroutines Used
In addition to the main program (page 77) the following subprograms
were used.
1. MPRINT, prints matrix in matrix format (page 91).
2. MMULTD, computes the product [C] of the two matrices [A] and
[B] (page 95).
3. MREAD, reads a matrix in either column binary or FORTRAN
format (page 98).
4. BINRD, reads column binary cards (page 103).
5. RDLN, reads and prints title card (page 113).
6. MNVRSX( computes the inverse of a complex matrix (page 115).
7. INVERS, computes the inverse of a real matrix (page 118).
8. SWEEPX, computes true mode and sweeps it from the related
matrix (page 123).
9. CENTER, computes the aerodynamic coefficients, centers of
pressure, and spanwise loadings (page 128).
10. LOADS, computes the three loads options and the structural loads
suboption (page 133).
11. DERIV, computes the flexible load coefficients used in the derivative
options (page 143).
12. MITER, matrix iteration (page 149).
13. NPNRMX, vector normalization (page 160).
14. DIVERG, divergent pressure calculations (page 165).
-67-
B. Generalized Tapes
Input, print, and utility tapes in this coding are defined as logical units
2, 3, 5, 6,7, and 8; however, these may be altered by pi" &,g the desired units
on symbolic cards HM030827 through HM030832 in the main program.
-68-
SECTION VII
FLOW DIAGRAMS
-( -
, -69-
FLOW DIAGRAM -MAIN PROGRA
TAPE REWINDSTAR DIENSONS QUIALECE CMMO FOMATSDEFNITONSTAPES
NO 98 99 50 10 101 102
50 &O'oYES
VFEIS NCNRC ~ IF >0 IF NONC g CMLE C2 NkNR 6(O READ: 0 -(i -I NO) PRINT: FLEXIBILITY [a][3-1NC= NxNC(N)NQ,Oi PRESENT
107 NO 108 NO 110 CALL NO 112
IF RED IF ES MCALLjE
IF YE RED F ES MRADREAD READ: IF YES[A [] NEXST NXSTN XT>O [A] CIOKR H M,L,IFORM,IROW IFORMzO[
>0 IFRRWNXST x NXST
121 12 235NO &126 &12 NO
NO CALL
STORE [A] IF s [w] [~i READ: IF YES MREAD READ ' _ FNTAPE 5 NEXST)O (NEXST xNEXST) NXST NXST>O [w] M, L, IFORM IFORM=ONXSTx NXST
119 133 137132 NO 136 PRINT
IF CLL CALL CALL THERM)JIF YES CALL THERMAL YE READ MRAD MREAD MMULTD MATRI)
DIVEG DTA MNIORMROWr rTEMPERATNQ: DVEG AT MNFOM IOW{,TlATTITplz[?Jtt/Tj MODE
PRESENT[a]
NO
-70-
GRAM - MAIN PROGRAM
90
TIT LI :REALTAPE REWIND TITL2:COMPLX CALL RDLN READ: READ: PRINT.
DEFINITIOjNS TAPES MAXR 50 TITLE CARD OPT IONS CARD ,f CONSTANT CARD OPTIONSIMAXQs2O CONSTANTS
101 10( 2 15106
YESIF NO PRINT:
FLXBIIY [a o STORE [a]J READ: CALL MREAD STORE []FLEXIBILITYPRESENILTY NTAPE 4 M, N, IFORM, IROW[a NTAPE 4 MATRIX
NO 112 10 1 2
READ: IF YES READ. READ. PRINT: CALLE[[A[CIFAEONAE4WEIGHTING MUT TR
MLIOM,,IFOR FORM ORM: [A] SIN BIAY [w] IN FORTRAN MATRIX RCOR [ 2][RESENTAP
[w]
1237N 1 34 12 393
17PRINT: 1419 10 PRINT,
CAULL MTERIAL YEF EXPERIMENTAL
TEMPERD ATURIX {ATTIT} -0 C~I RE AD: C FORCEL
{ATTIT}: [a]{t/T} MODE PRESENT f
L!J.1/TJNO
-70-
FLOW DIAGRAM - MAIN PROGRAI
143 144 147 148 15115
IF NO CALL PRINT: IF NO CALL PRINT:YE S INITIAL YES ADDITIONAL. IF Y
,MREAD DEFLECTION MREAD DEFLECTION MASS READ. M
MODE ,,-I MODE MATRIXNEDE L 0 J{h/H} NEEDED tJ{/} NEEDED
15159IS15 nIS? NO 158
ISNOPRINT: IF PRINT: IF NOPRINT:P
LOAD FACTOR READ: CHORDWISE YE READ: SPANWISE READMODE READj :IR CORDNAE COORDINATES NSTRPNCPT*
{X) Xi,-,,NRC i i 1,,,..C V co •-.-, NSTA
{/J NEEDED {XJ NEEDED {)NT
[so NO ITO
REWIND CALL STORE [( W] [Ch] DO CON:q K43/ IF A: [A] REALNTAPE4, MMULTD AND REWIND 204 [A] CON [A] COMPLEX
NTAPE- Ac [W] [Cwh] [CINTAPE6 + :IN[A] ["m
164 ias no
REA[] CLL: CAL MMLCTD A CN +: STORE IF EV
LJLJ~~TAE NDATA A:WLJ [] ~ ~RED
NO196 197 200 202 YES
PRINT:
IF AYEAS CALL MMULTD CORRECTED CALL CENT~ER READ: REWINDCOMPLEX A FORCE COEFF T'(6) [a] [w] [c TAPES20
.-- e] -' {Cc..zJz, NTAPE6 -
-71-
RAM -MAIN PROGRAM (CONTINUED)
151 152 153 154
YAS READ:. - READ: C2RADL PMwCA.MUTYAS E E DI IL i. L~ M ASSJ s MASS MATRIX MATRIX {CN i}
MATRIX LEISS
NONOD
164 166
PRINT: PITPANWISE Clio-READ: PRINT: REWIND READ: READ:)RDINATES NSTRP, MOPTi PODINT TE4Ca
i:I, NTPDATA AyE, Zi AY1 , Zi TAPES NTAPE 5 [W]NAP4Ci
fI q STRIP. NCPT, [Ch]
NO rio 172 NO 174 176 177lo
4[ARELCAL.L IF PRINT: IF SOE ACA] a. VRSX MGOOF*O INVERSE IFACOA]+M9PLEX A8 [A] MM[CMPLE NTAPES
COPLEX -1[A ERROR
18919ISO No 190 192
[A]RIN[A PRINT:
STORE [] IF EXPER 'YE TITLE IF [EA] REAL CALL MNVRSX IF YSINVERSE
NTAPE* DAIAE COMLE IS [A] IMAG. [A]-' MSOOF 0 0 ERROR
No
YES No 206 207 N 208 209 NO 210
REWIND IF YEOF DER. OPT YES IF DIVERS. YES EG04 LOADS OPT CALL LOADS CALL DERIV. OPTION CALL DIVE.NTAPE6 0 ND REQUES S REQUESTED REQUESTED
-71-
FLOW DIAGRAM - LOADS SUBROUTINE
424 a 4211426
DIMENSIONS IF DOIAD: So TOE TITEASTART EQUIVLENCE MUSS 'D 411111 SIVNT Lr-L MU GITSSEACOMMON TAPE$ Loan 1143 EFPICENl
PORINATS OfO143MATRIX
no3
429 431 4126ES 40
IFM]-A PRES TIFLE CALL o E I IFA CALL{cTORQUEL! 40] -LOD 2 CMIJ Z;AZ 003 h.II. UJT
PRESEN - {C ~ al -4 N {D}:[W432N{C}COFFCIN NO am
4572 46 0 462 46
IF ESU IF CALLO CATUNLO(a) YE 47659- LOADS 2 mm-Ul
noN~ l{C) NO NO72 IST
472 4?040-72-
DRAM -LOADS SUBROUTINE
44426 42? 426434 e2
00 11oEAI PRINT TITLE: [s.9Aatj-[A] PRINT TITLE:6WWT LHSH 0 TO $HEAR MOMENT
L49 1 COEFFICIENT N4VMSB COEFFICIENT
7n4 2 MATRIX NVMT MATRIX
460 45
+b T READ CALL DO{}Su C
412 470
D of CALL DOZZ3CAPZ LOADS 3 %/H. MIUJLTD J4-!I?,:LNj 0
4U40 48344 9
YES YES YES
READ IF TRIN IF TINNIF TURN ONNYAPE 4 LODSI SES LOANS 2 EFE LOADS 3 SENSE 416
a: [uJLIM? I GM?2 USTE UISHT 3
No Nono
-72-
FLOW DIAGRAM -LOADS SUBROUri
OFF
49490 492
46 OoF I}Q{Ir READ IF ONTURN ON CALL CALLRE554 NTAPE 8 SENSE SENSE MMULTD MMULTDNTP
00I,141 +N{CNIN} A -[A] L1H LI1 D4]F}Af
4968 500 02511 512 514
XFTU R N ON DTh - CA LLZ t LLI~SENSE SES 'IZ FFMULTO LIGHT MMULT
LIGHT LIHT .J Ii2HI 2 4a] f {F} +{H-TX f H4 4
2 2 - T fH)- D
1
X
OFF
LOADS FOR SENSE ON SENSE LOADS FOR LOADS FOR PRINT SENSE ON SENSE PRINT:ROANLIGHT LIGHT TRIMMED Sl LIGHT LIGHT Z, DT
ON 57 OFF 56 ON
CALLT ir CAULL NONT REWIN SENSE F SENSE ON SENSE OF
D-[MMULT {T MMLT PRNT NTAPE 6 LIGHT LIGHT LIGHT
0-TRU L {l L YES 1 2 3
-73-
)IAGRAM -LOADS SUBROUTINE
492 494 496
CALL CALL CALL FF CALL IlNCIMMMMULTDUT TP MMULTD SENSE M~LTD
{D}. f Fr} IH . [A] {D} A =[0] {F}I [B fF4 IHTo 0-u- {WCHx W. [ 64 E wc~xl
ON
512 514 516 518) 524
{4z K {Hf} SENSE CALL t'H.VK IHIl IF ON TURN ON
LIGHT MMULTO DxJ 1,IH SENSE SENSE
+{}OTX {}O]F JJJ F.{}{CNIN LIGHT LIGHT
L= D[.]OFF"
OFFF
4TN SENSE SENSE t RN: ff.Hl SRCUPITAP S LIGHT LIGHT LIGHT LIH LGT IH
-73-{F
FLOW DIAGRAM - DERIVATIVES SUBI
NO 23 24 25 26 27IF NO No
DIMENSIONS REIDTPS INITIAL TURN ON AERO TURN ON THRAL TUISTRTEQUIVALENCE TURN OFF COFIINS~ SENSE STABILITY 'U SENSE SAILITY S 91
COMMON SENSE LIGHTS REQSTED LIGHT DERIVATIVES LIGHT DERIVATIVES LFORMATS NX-O REQUESTED REQUESTED
OF 35 38 39 4243OF4 OFF 42
IF TURN ON PRINT TITLE: IF TURN ONPRINT TITLE: IF TURN ON P1SENSE ON SENSE LIGHT 2 AERODYNAMIC SENSE ON SENSE LIGHT 3 THERMAL SNSELIGHT f =STABILITY LIGHT /Dlr~~T* STABILITY LIGHT LIGHT D1
L ONOFF
**OFFjOF45 46 47-
II
READ CALL IF TURN ON EXP I-NX-U NTAPE 5 MMULTD SENSE ON SENSE MENTAL MM
[{W][ChI] £C = [W] [C@ D1 f LIGHT LIGHT FORCE+II COL.UMN I}
1JLJLMJ I I PRESEN
OFF 0
71 53 54 555+
+ 5I -. 0
IF _ FI CALL CALL READIF IF f-, fDICj IF MMULTD MMULTD NTAPE 8
0 N LLJNX{C} [A]{JE} {DI[wJ C~ TRASH
61 62 63 6
IF IFREWIND SENSE OFF SENSE ON NOP(6) IF + SENSE CALTAPES LIGHT LIGHT =NOP(6)-I NOP(6) LIGHT 2 cRA
1 2 ON D 2{h/H}
ON OFF -,0
I -74-
IRAM - DERIVATIVES SUBROUTINE
24 25 26 27 28 29 so 3
IF NO IF No NoAERO __TURN ON TRAL TURN ON IF TURN ONIF TURN ON PRINT
STABILITY *~SENSE STABILITY SENSE INERTIAL BENS SENSE ONSENSE LIGHT I TITLE:DERVATVES LIGT DRIVTIVS LGHT DERIVATIVE
REQUESTED 2 REQUESTED REUSLisH LIGHT IDj-H, fr/ INITAL2 3 1H COEFFICIENTS
OFF
43 4442
PRINT TITLE: IF TURN ON PRINT TITLE: REDCALL READTHERMAL SENSE ON SENSE INERTIAL RELTADUINAPSTABILITY LIGHT LIGHT DERITV EATIVS, NTAPE 45
"DERIVATIVES 4 4 A'[a] P[ 14]Nlt"t/N,
OFFL
47 4450
O IFO UN O EXPERI- CjCfDO READSENSE MENTAL NTAPE 60 NTAPE RLIGHT FORCE
COLUMN +{'•01 1.•c ,NQ A[.A]I PRESEN A IOL
0
5556 57 70
CALL READ READ CALL CALL YESMNULTD NTPEsNTAPE S MMULTD PRINT: CENTER 6
I [W][WIN{C} TRASH A = [] {o} =[B] {C} q.{D} {D} NO
63 n64 6?
LIGHTE MREAD ADDITIONAL SENSE OF RTRON D2{VH DEFLECTION LIGHT
-74-
SECTION VIII
SYMBOLIC LISTING
Some of the program FORTRAN symbols are defined below.
FORTRAN Symbols Definitions
A(I,J),B(I,3),C(I,J) Elements of matrix "working" arrays; also elementsof [a], [Chl, and [W]
ATTIT(I,J) Element of column matrix representing the product[aT] It/T}
CIOKR I/kr
CAPH H
CAPHO H0
CAPN N
CAPS S (reference area)
CAPT T
CAPXO x 0CAPZ Z
CBAR z
CNIN(I, J) Element of column matrix representing the product[M]In/Nj
CZRE(I,J) Element of experimental force column matrix
DELY(J) Ayj (width of strip j)
F(I,J) Element of total force distribution column
FA(I, J) Element of final aerodynamic force distributioncolumn
FLEXK K
FMOM(J, K) Element in load coefficient matrix [M/F]
FR(I,J) Element of rigid force distribution column
HADD(I, J) Element of additional deflection mode column
HR(I, J) Element of initial deflection mode column
-75-
FORTRAN Symbols Definitions
IFORM IFORM = 1 if matrix is in FORTRAN format;= 0 if matrix is in column binary format
IROW IROW = 1 if matrix is input by rows;= 0 if matrix is input by columns
L Number of strips (partitions) in the surfaceaerodynamic matrix
LH(K) Row number of last nonzero element in column k ofmass matrix
LHIGH(J) Column number of last nonzero element in row jof [V/F], [M/F], or [T/F]
LL(k) Row number of first nonzero element in column kof mass matrix
LOW(J) Column number of first nonzero element in row jof [V/F], [M/F],or [T/F]
NCPT(J) Number of control points for strip j
NEXST Number of external stores reserved
NOP(I) Option i, Field i of control card (data Item 2) -NQ Number of dynamic pressures read in
NRC Order of current case
NSTRP Number of surface strips used for derivatives options
NVMT Number of rows in [V/F], [M/F], or [T/F]
Q(I) (ith) dynamic pressure
SHEAR(J,K) Element in load coefficient matrix [V/F]
SMALS s
TCCP Chordwise center of pressure
TORQU(J, K) Element in load coefficient matrix [T/F]
TSCP Spanwise center of pressure
X(I) Element in chordwise coordinate matrix
Y(I) Element in spanwise coordinate matrixZI(IJ) Location of jth control point for strip i
The symbolic listing of the program is shown on the following pages.
-76-
-l 0
*~ .0 1. PO -O
a, - 4U 4-4 U.~t' "m 9- usOU
Inc * 2.. - 2 0z 0Z m U 0, M. 4- 01 -.. I% ~ 0 4 N 'D N4.; .0 we 'A .
M ; ui0iU. 1. - 24 .W
C; 4a aA A P. 04 -44 0 IL I soo x Z~n - :; - : N .. 0-- .. u W " QN% in woJu 4- -.. 422 2 " Z W% :"I Z N a4 * NN- ~ i r 0 0.. h W4 . .0
00.. - Kz Q.L a 4-s ;. .4 L - xwU.Wz
%A Iua -. . Z4U zn4.. z N .- ZN W
-- 04 0. .4 4 KK2 at. 04i x2 2 -UZ *U N6.. II 1KW
-2 -0.i- N~NI w ow b mw Wo -0204 - 1-l .UU -4 Z N .m -C2, z O U, .M 6l
0w~x Q0 #0 %IIWwlX4*~ 61-.. ii2 nn..- W UU WX
IL4 Z9- r"9 U .. 44 4 1- w ZU-4 aI4 0-.J W00. ;: - 4L L.-- .4 KK C U.%%- U - -=0 a I
in- .0 !A o- - -0 z 3-- O 0-.4 -Z *W " .j 4 z 4-04 4 1 0l a co-i9 NN0NN a KK t - 9 z UJ
I.ZlA -4- -4 1 in-- z aU 4 a I---
-. 0 . JZ4 j. W z z 0 :-L. a A- ý - Z &-. 4A 4c
U.. 0 so N 5;UNZ t4 0^-N O ZU W. 4..4-.*% W Z W X C 4ar0-C Z I- 0 - -- 4* P4 OC U. I-lU- A-4K .J
P4AC -i z 2 .4 . . .4- n . 0..i. .4-U..OX N- Ný l - nZ;ZZU. Ix OWWU.W--In
-i-CI a-.. .9. A-r 2 I @ %%U4W 00.
C a-% oo wu o 0 C .coU. 4W ZmrW40 1.-0 z z ; ;P &- ; NN09 O4244i mgmZNi22222
Qa IL in C- 0ie l - .I O 4 4 ~ * ~N 4 -QK 20C 4- I-4 -~3 -.-. 49 44- 4 449.-x .. 0- 0- W .; 4 40 x!0 0.4a z x-10
0.00.z(I --- -4wN -------0--
0 0 uj_4 - jAt In 4 2196 ---- 1 -. 99I1II
2 z >4494 4 444 ;;444444wj w -. 22 2222 I2 z z2222z x ac ac ccUU l cUc c a a a a
a- - l 000 00000 00000000 0 WL U UU. %, U..LL LU. LL. UL.U.U. U.bLU.IL &
ý4~~~~ ~ ~ ~ -4 -n' n01-C p 4 N4 m 4 It -%
-77-
cli)
In 0
0 L
a0 Z 0 L
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ce- -x- U)JI
- C4l .OLIf0'O4A aI-in4 - C~ 0 JL
Z-o-NZX WO-i 4
Coe *.Z- Wx LA 10 flr ' .wj" 0czO 6-l
C) l' M'~0 LL4 J CLi C m - I.-NWU.1
41-4r'd . 0-0 - I- CL- .A~2 - -
a u.-.uui 2-IL~ CL .* n
W :a - 0WWX0Lu =0 -I% W =0 00 ij zN rý A Wa 00.1l0Z - w z u.e .4 0
x M z .0 W N Ii -- WU * 0In-0
040. 000 Ouh 4d N
0x~ ~ rx .1 LiiZ
r ( l -4 0 M, ad L
x. I.-.* - UJ U W - t < I-X)()(..-1 OL?..* M. a.i 0 .i 'NIT
el4 N4~C-4 0 W- N M'j 0 .r4Wo.94 t-. I-. 0.- *- 0. 10 -tI' 0-.?
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-00. -I-I I- 'a0zt.- N In ItI Om Qaz0 oI nN -
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w c 0 t ( ccý (- 44 44 xz~ A x x Z-j.j4Oorn 0 0 00 ) 0N m4 A0a -0Q I-.l-I-Iý-Il- WWWW L L -- 44 W
'N CL 0. 0.0.%.0. 0.r.- 00 a, ýj -1 N In < .444444 4
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-78-
N,a
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4Ai z X6
U . IL A. 0
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W X 00 CL 2 -z 4
U0-4 2 -Z Uvi O . "i z I0.-
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0.I ao Z O A ULU4-U~~~ z-O -Z 0- a U
mii~ 2 a a., 9 -0, 0A
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g2Z4 * 0 -0 A- 9L ZO. a -k4 wL on 0-m I-aa ~ K a
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A .- 44 1.- - 0- zJ a P..a-2x LU LU 0,, =2 .C . .* =a w LU U
a. z 30 - . -.- U L w . A-z i
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