vortex to point on left panel being influenced; measured from left panel,. _' ...... 3d. IM=LM*2. DG3. 33. DO. 1. I=I,IM. DG3. 34. CDRAG(1)=O. DG3. 35. DO. 1.
NASA
TECHNICAL
NOTE
NASA TN D-8090
O
!
Z I--
Z
A VORTEX-LATTICE FOR OF
THE
TRIMMED
WITH
John
MEAN
METHOD CAMBER
NONCOPLANAR
MINIMUM
VORTEX
SHAPES PLANFORMS DRAG
E. Lamar
Langlo, HamDton,
Research Va.
Center
_.QO_'UT/O4z
23665 Z_7 S ._gl
NA1iONAL
AERONAUTICS
AND
SPACE ADMINISTRATION
-
WASHINGTON,
D.
C.
°
_
JUNE 1976
1. Report
No.
2.
NASA 4.
Title
TN
and
Government
Accession
No.
Subtitle
A VORTEX-LATTICE SHAPES
OF
MINIMUM
METHOD
TRIMMED
FOR
THE
NONCOPLANAR
VORTEX
MEAN
9.
E.
Performing
Sponsoring
Name
Va.
National
PLANFORMS
Abstract
Name
WITH
a vortex
This
method
mum
drag,
required
Code
8.
Performing
Orgamzation
Report
10.
Work
'11.
Contract
or
13.
Type
Report
505-06-II-05 Grant
of
No.
and
Technical
and
Space
method
lattice uses then
Administration
has
been
noncoplanar and
14.
solves
for
Sensitivity
analysis
mean
the
surface
wing
and
mean
minimum
Period
Covered
Note
Sponsoring
Agency
the
Code
chord
wing,
other
This
loading
which
can
be
method
specification.
span
theories,
a wing-winglet
surface
drag.
optimum
of the with
camber
vortex
with
determine
comparisons
a tandem
which
difficulties to
camber
studies,
include
by with
previous
plane the
developed planforms
overcomes
a Trefftz
which
(Suggested
camber
loading
will and
combination
for
provide
minithe
applications have
been
to made
by
Author(s))
18.
Distribution
Statement
surface Unclassified-
flow
Vortex-lattice
Unlimited
method
Interacting surfaces Optimization Security
Classif.
(of
No.
No.
presented.
Subsonic
19.
Unit
20546
loading.
are
Words
Organization
Address
trimmed
configurations
Mean
Performing
Address
Center
subsonic for
uses
Key
1976
6.
Notes
A new
17.
and
D.C.
determined
and
and
Aeronautics
16.
Date
June
23665
Washington, Supplementary
No.
DRAG
Research
Agency
15
Report
Catalog
L-10522
Langley
Hampton,
12.
5.
Lamar
Organization
NASA
Recipient's
CAMBER
7. Author(s)
John
3.
D-8090
this
Subject
reportl
20.
Unclassified
Security
Classif.
(of
this
page)
21.
sale by
the
National
Technical
of
Pages
22.
Category Price"
$7.00
185
Unclassified
For
No.
Information
Service,
Springfield,
Virginia
22161
02
A VORTEX-LATTICE
METHOD
OF TRIMMED
FOR
THE MEAN
NONCOPLANAR
WITH
MINIMUM John
Langley
CAMBER
SHAPES
PLANFORMS
VORTEX
DRAG
E. Lamar Research
Center
SUMMARY
A new subsonic be determined method
for
uses
This
for
the
Sensitivity and
are
The
versatility
canard
overcomes
method
uses
a Trefftz
then
solves
loading. at the
studies
of vortex-lattice
previous plane
for
the
design
minimum
analysis mean
camber vortex
difficulties
with
surface
drag.
chord
to determine
camber
loading
the
surface
can
This
optimum
of the
wing,
or root-bending-moment
arrangement
with
method
is demonstrated
(3) a tandem
the mean
span which
constraints
lift coefficient.
Comparisons
configurations,
with
Pitching-moment
as well
of the
by which
planforms
and
drag,
presented.
developed
noncoplanar
required
can be employed
been
lattice
minimum
will provide
has
trimmed
a vortex
specification. loading
method
other
wing,
have
theories
show
by applying and
been
made
generally
with good
method
agreement.
it to (1) isolated
(4) a wing-winglet
this
wings,
(2) wing-
configuration.
INTRODUCTION
Configuration which
the
body
design
and
located
by taking
highly
maneuverable
increased
together
specified
lift
Single better ref. sented also
called 1) and
in changing
the
Such
planform
design
local
at supersonic
presented
1 was
in reference
control
design
order
methods
speeds
developed
with
with
the tails
are
With
surfaces,
multiple
there
and
advent has
an
be
drag
that
of
been
could
induced
requires
after
sized
the
surfaces
minimum
approach
the wing,
at some
the mutual
inter-
initially. available for
example, from
1, by using
then
lifting
so that
design
are
surface, (for
and
configuration
a combined
begins
requirements.
coupled
be considered
elevation
usually
into account, and
closely
a trimmed
surfaces
in reference
taken
having
lifting
transports
stability
aircraft
to yield
the
are
account
coefficient.
of the
subsonic
effects
into
interest
designed
ference
its
for
the
to optimize
wings
flying
refs.
2 and
an established same
the
mean
at subsonic 3). analysis
mathematical
The
camber
speeds design
method model,
(for
surface, example,
method
pre-
(Multhopp
but the
design
type),
methodsolves for the local meanslope rather than the lifting pressures. In the usual implementation of reference 1, the design lifting pressures are taken to be linear chordwise, but must be represented in this solution by a sine series which oscillates about them. An example presentedherein demonstrates that corresponding oscillations may appear in pressure distributions measured on wings which havebeen designedby the methodof reference 1. The methoddevelopedherein overcomes this oscillatory lifting pressure behavior by specifying linear chord loadings at the outset. The developmentapproachused in the two-planform design problem will be similar to that used for a single planform. The analytic methodemployed, selected because of its geometric versatility, is the noncoplanartwo-planform vortex-lattice methodof reference 4. The design procedure is essentially an optimization or extremization problem. Subsonicmethods(for example, see refs. 5 and6) are available for determining the span load distributions on bent lifting lines in the Trefftz plane, but they do not describe the necessary local elevation surface. This is one of the objectives of the present method which will utilize the Lagrange multiplier technique (also employedin refs. 2 and 3). The methodof reference 4 is usedto provide the neededgeometrical relationships betweenthe circulation andinducednormal flow for complex planforms, as well as to computethe lift, drag, andpitching moment. This paper also presents the results of precision studies and comparisons with other methodsand data. Several examples of solutions for configurations of recent interest are also presented. The FORTRANcomputer program written to perform the computation is described (appendixA), along with details of the program input data (appendixB) andoutput data (appendixC). Listings andtypical running times of example configurations are given (appendixD), anda FORTRANprogram listing is provided (appendixE). AppendixF provides details concerning the changesneededto substitute a root-bendingmomentconstraint for the basic constraint on configuration pitching-moment balance. SYMBOLS The geometric description of planforms is basedon the body-axis system. (See fig. 1 for positive directions.) For computational purposes the planform is replaced by a vortex lattice which is in a wind-axis system. Both the bodyaxes andthe wind axes have their origins in the planform plane of symmetry. (Seesketch (a) for details.) The axis system of a particular horseshoevortex is wind oriented and referred to the origin of that horseshoevortex (fig. 1). For the purpose of the computer program, the length dimension is arbitrary for a given case; angles associated with the planform are always in degrees. (The variable namesusedfor input data in the computer program are described in appendixB.) 2
m
element
A/,n
of influence
tains
induced
strength;
AR
aspect
linearly
i
normal
flow
matrix
A,
at lth point
total
number
of elements
is
chord
location
where
chord
value
toward
zero
scaling
polynomial
Fw,/,
n - Fv,l, n tan
4n due to nth horseshoe
_bl
, which
vortex
con-
of unit
N x N 2 2
ratio
fractional
ai,bi,c
function
varying
coefficients
in spanwise
b
wing
CB
root-bending-moment
CD
drag
coefficient,
CD, o
drag
coefficient
CL
lift coefficient,
load
changes
at trailing
from
constant
value
edge
span
o
coefficient
about
X-axis,
Root
bending q_Sref(b/2)
Drag q_Sref
at
CL = 0
Lift q_Sref
C m
pitching-moment
CN
normal-force
ACp
lifting
c
chord
cl
section
Cref
reference
coefficient
coefficient,
pressure
lift
coefficient
coefficient
chord
about
Y-axis,
Normal force q_Sref
Pitching
moment
q_SrefCref
moment
to
influence function which geometrically relates inducedeffect of nth horseshoevortex to quantity which is proportional to induceddownwash or sidewashat slope point l (see sketch (a) and also eqs. (5)
Fw,l,n' Fv,/,n
and
Fw,l,n, Fv,l,n
sum
(6))
of influence
planform left
wing
panel
denoted
G
function
_- Nca ;
maximum
L
lift
N+
about
of span
maximum mum
denoted
1 - n
(see
(see
eq.
indicate
of spanwise
Mach
reference
N
vortex
(brackets
moment
free-stream
number
i
number
pitching
n]
by
Fw,/, n
or
Fv,/, n
by two symmetrically
to be extremized
+ 0.75
K
function
caused
by fig.
n
and right
point
l
on
horseshoe
vortices,
wing
vortex
panel
1)
(19))
"take
the
scaling
coordinate
at slope
located
greatest
terms
integer")
(see
eqs.
(25) to (27))
origin
number
stations
where
pressure
modes
are
defined
as used
in
1
number number
of elemental of chordal
in reference
panels
on both
control
points
from
leading
rows
in spanwise
sides
at each
of configuration;
of
m
span
stations
maxias used
1
m
Nc
number
Ns
total
of elemental
number
panels
of (chordwise)
on configuration
qo_
free-stream
Sref
reference
area
S
horseshoe
vortex
4
to trailing
edge
direction
in chordwise
of elemental
semispan
dynamic
pressure
semiwidth
in plane
of horseshoe
(see
fig.
2)
row
panels
U
free-stream
velocity
X,Y,Z
axis
of given
system
horseshoe
vortex
(see fig.
body-axis
system
for
planform
(see
fig.
wind-axis
system
for
planform
(see
sketch
distance
1)
1)
(a))
along
X-,
Y-,
and
Z-axis,
respectively
along
X-,
Y-,
and Z-axis,
respectively
= distance
incremental
movement
of
X-Y
coordinate
c/4
midspan
x-location
of quarter-chord
X3c/4
midspan
_-location
of three-quarter-chord
y*,z*
y
and
z
distances
symmetry,
canard
local
as viewed
height
with
elevation
height,
positive
Prandtl-Glauert
_n
vortex
independent
to wing
by local
slope
in vector
located
to points
plane,
chord,
panel
on right
on left
positive
half
of plane
of
panel
down
referenced
to local
trailing-edge
_z/_x)
of
N/2
elements
(see
eq.
(1))
deg correction
flow,
strength
behind,
panel
of elemental
vortices
direction
down
local
of attack,
subsonic
respect
in streamwise
of elemental
image
from
normalized
/th elemental
angle
from
origin
factor
to account
for
effect
of
N/2
of compressibility
_1 - Moo 2
of nth element
variable
in vector
in extremization
(F)
process
elements
in
incidence
77
A
leading
edge
spanwise
coordinates,
nondimensional
spanwise
coordinate
planform
leading-edge
Lagrange
multiplier
along
fractional
dihedral
(see angle
constraint in
quarter-chord
in
= tan- 1/_--_
left
(see plane
also X-Y
where
vortex
eqs.
used
plane,
_'
wing
angle
(21)); panel,
on right
of elemental
as sweep
angle
deg
)
canard
design
i,j,k
indices
to vary
le
leading
edge
6
over
semispan
deg
chord
mean
camber
on left
panel
measured
(20) and
on left
angle
to point
Subscripts:
C
planform
plane,
by local
of point
panel,
dihedral
sweep
assumption,
X-Y
normalized
trailing
from
vortex
in
on local
height
is to be
(28))
from
Y-Z
b/-"-2
(19))
location
eq.
up, deg
based
angle
eq.
chord
function
horseshoe
ment
(see
local
measured
angle
sweep
chordwise
computed
0", 0 "_
positive
nondimensional
distance
4'
angle,
the
range
indicated
also
from
being
right
horseshoe
influenced;
panel
vortex
dihedral
deg wing
panel;
panel,
because
of spanwise
qS' = -_b, deg
of small horseshoe
angle vortex
fila-
associated
l_n
with
1 to L
left
R
right
w
wing
Matrix
and horseshoe
vortex,
reslSectively,
ranging
from
leg
trailing
root-chord
vortex
point
N/2
trailing
v
slope
leg location
notation:
()
column
vector
[]
square
matrix
r-\
X
Flow
angle at each
of attack determined slope point
/
"w-., _t
g
/ v- Typical /_
spanwise
vortex
filament
U
Vortex-lattice angle
Sketch THEORETICAL
This
section
camber-surface have
dihedral.
respect ble;
to their
however,
trailing
filame
of attack
presents design
For chord vertical
the
application
of two lifting
a given lines
displacements
DEVE LOPMENT
of vortex-lattice
pIanforms
planform, in the
(a)
local
wing
axis
of the
which vertical (see
methodology
may
be separated
displacements sketch
solution
(a)) are
surfaces
of the assumed
to the vertically surfaces
meanand with
to be negligi-
due to planform
separation 7
or dihedral are included. The wakes of these bent lifting planforms are assumedto lie in their respective extendedbent chord planes with no roll up. For a two-planform configuration the resulting local elevation surface solutions are those for which both the vortex drag is minimized at the design lift coefficient and the pitching moment is constrained to be zero aboutthe origin. For an isolated planform no pitching-moment constraint is imposed. Thus, the solution is the local elevation surface yielding the minimum vortex drag at the designlift coefficient. Lagrange multipliers together with suitable interpolating andintegrating procedures are used to obtain the solutions. The details of the solution are given in the following five subsections. Relationship BetweenLocal Slope and Circulation From reference 4, the distributed circulation over a lifting system is related to the local slope by
r
where technique
the matrix
"1
LAJ is the aerodynamic
described
in reference
4.
influence This
matrix
coefficient has
matrix
elements
based
on
the paneling
of
m
Al,n
which,
because
=lIFw,l,n(X',Y,Z,S,_P',dP)
of the
Fw,l,n(X
'
assumed
,y,z,s,_
-
spanwise
'
,_b) -
F
v,l,n(X
symmetry
w,/,n(X
-I- F
F
'
of loading,
'
,y,z,s,_
w,l, N+l-n(x
,Y,Z,S,g2,dp)
_
tan
leads
c_ll
(2)
to
,4_)lef t panel
"
'Y'Z'S'_"_b)right
panel
(3)
and
Fv,t,n- (x',y,z,s,_P',_b)
- Fv,/,n(X',y,z,s,gT,_))lef
+ Fv,l,N+
8
l_n(x',
t panel
y, z, s, gJ', qS)right
panel
(4)
where
(y tan Fw(x',y,z,s,@',O
¢,' - x')
cos
) = (x') 2 + (y sin
4))2 + cos2
do(y2 tan2
$'
+ z2 sec2
_'
_'
+ (y + s cos
- 2yx'
tan
_')
- 2z cos
_ sin
4_(Y + x' tan
_')
(x J(x'
+ s cos
, u.l
e_ i 0 ,..I
).
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APPENDIX
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106
D
APPENDIX
FORTRAN
This Data
program
Corporation
library
Minor
program
the
program
the
matrix
The
is
1120008
and
is
seven
to
on
for
an
the
computer
73 with
program
to
geometry
Each by
of
a three-digit
four
with
in
basic
77
to
features
and
79.
The
depending
on
CIRCULI
uses
uses
place
layout.
steps,
three
SUBROUTINE
following
CIRCUL3 automatically
vortex-lattice
each
and
PROGRAM
PROGRAM
takes
addition,
allow
characteristics.
dihedral
the
and
computers.
PROGRAM
PROGRAM
In
other
words,
selection
OVERLAY,
columns
on
Control
system
These
and
and
abbreviation.
number
use
uses
the
operating
ll20008
dihedral
The
for
aerodynamic
matrix
PROGRAM,
a three-letter
to
configuration
consists
3.0 to
the
configurations
the
2.3
stepping.
for
matrix.
of
program
75
for
SCOPE
510008
a well-conditioned
ill-conditioned
SUBROUTINES.
columns
the
for
from
without
technique
words
version
prior
technique
configurations
solution
words
This
sequenced
the
for
with
PROGRAM
vary
solution
language,
required
and
to
the
IV
be
UPDATE
and
LISTING
system may
requirements
630008
dependent
computer
using
technique
and
and
in
written
PROGRAM
FORTRAN
modifications
storage
words;
CIRCUL2
in
6000
conditioning
solution
510008
written
series
tape.
The
and
was
E
of
OVERLAYS is
these table
identified parts is
is an
index
listing: Name PROGRAM OVERLAY
Abbreviation
of part
Page
GEO
GEOMTRY
109
0 (WINGTL)
PROGRAM
}
WINGAL
119
DGO
SUBROUTINE
FTLUP
TLU
120
SUBROUTINE
SIMEQ
SEQ
122
SUBROUTINE
DRAGSUB
DGS
124
DGI
125
DG2
130
DG3
135
GIA
140
ZOC
148
OVERLAY
1 (WINGTL)
PROGRAM OVERLAY
CIRCUL1
A
)
I (WINGTL)
PROGRAM OVERLAY
CIRCUL2 1 (WINGTL)
PROGRAM
CIRCUL3
SUBROUTINE OVERLAY
GIASOS
2 (WINGTL)
PROGRAM
ZOCDETM
SUBROUTINE
INFSUB
INF
150
SUBROUTINE
SPLINE
SPL
151
SUBROUTINE
TRIMAT
TRI
153
DUM
153
PROGRAM aThe PROGRAM
DUMMY
PROGRAM
a DUMMY
is for
default
purposes
of
GEOMTRY. 107
APPENDIX JOS,I,lO00,063000,1000. JSER.LAMAR, JO"N
A4062 E
NURFL. JPDATE(F,I,N,C,L=O} RE*IND(NEWPL) JPDATE(Q,P=NEWPL,C,L=O) 4UN(S,,,COMPILt) 5ETINDF. _GO. RE,IND(NEWPL) REWIND(TAPE50) JPDArE(Q,I=TAPE50,P=NEWPL,L=O) RU_(S,,,COMPILE,,GLO) SET INDF. GLO. EXIT.
108
E R43|0
000503_00N
IO0110 38510
_1212
RIOI
APPENDIX
*DECK
C C C C C C
E
VLMCGEOM PROGRAM GEOMTRY(INPUT,OUTPUT,TAPE5:INPJT,TAPEB=OUTPUT,TAPE?5,TAPE5GEO I0) DIMENSION XREF(25), YREF(25), SAR(25), A(25), RSAR(25), X(2S), 15)9 BOTSV(2}, SA(2)9 VBORD(51}9 SPY(5092}9 KFX(2)9 IYL(50,2}9 25092) COMMON /ALL/ BOToMt8ETA,PTEST,QTEST,TBLSCW(50),O(_OO),PN(_OO),PV(_GEO IO0)oS(400),PSI(WOO},PHI(50},ZH(50),NSSW COMMON /MAINONE/ ICODEOFoTOTAL,AAN(2),XS(2),YS(2),KFCTS(2},XREG(25GEO 1,2),YREG(25,2),AREG(2592)9OIH(25,2}oMCD(25,2),XX(2592)oYY(2592),ASGEO 2(2592),TTWD(25,Z},MMCD(25,2)*A_(2),ZZ(?b,2),IFLAG COMMON /DNETHRE/ TWIST(2),CREF,SREF,CAVE,CLDES,STRUE,AR,ARTRUE,RTCGEO IDHT(2)oCONFIG,NSSWSV(2),MSV(?),KBOToPLAN, IPLAN,MACHoSSWWA(50) COMMON /CCRRDD/ CHORD(50),XTE(50),KBIT,TSPAN,TSPANA REAL MACH REWIND 50 PART
ONE
- GEOMETRy SECTION
COMPUTATIO_ OWE
-
I_PJT
OF
REFERENCE
WING
POSITION
ICODEOF=O TOTAL=PTEST=OTEST=TWIST(1)=TWIST(2)=O. IF (TOTAL.EQ.O.) RTCDHT(1)=RTCDHT(?)=O.O YTOL=I.E-IO AZY=I.E÷I3 PIT=I.5707963 RAD=57.2_578 IF (TOTAL.GT.O.) GO TO 7 C C C C C C C C C
! C C C C C C C C C
SET PLAN VARIABLE SET PLAN
EQUAL SWEEP EQUAL
SET TOTAL OF GROUP
EQUAL TWO
TO ]. FOR WING TO ?. FOR
A WING
ALONE
A WING
- TAIL
TO THE NUMBER DATA PROVIDED
OF
COMPuTAION
-
EVEN
FOP
A
COM_INATION
SETS
READ (5,98) PLAN,TOTAL,CREF,SREF IF(ENDFILE 5) 93,! !PLAN=PLAN
SET AAN(IT) DEFINE THE
EQUAL TO THE MAXIMUM PLANFOR_ PERIMETER OF
_UMHER OF CURVES THE (IT) PLANFORM.
SET RTCD_T(IT) EQUAL TO THE ROOT CHORD HEIGHT SURFACE (IT),WHOSE PERIMETER POINTS ARE BEING RESPECT TO THE WING ROOT CHORD HEIGHT WRITE (6,96) DO 6 IT=!,!PLAN READ (5,98) AAN(IT),XS(IT),YS(TT),RTCDHT(IT) N=AAN(IT) NI=N*I MAK=O IF (IPLA_.EQ.I} PRTCON=IOH IF (IPLA_.EQ.2.AND.IT.EO.I) oRTCDN=IOH IF (IPLA_.EQ.2.AND.IT.FO.2) PRTCDN=IOH
FIRST SECONO
REQUIRED
OF THE LIFTING READ I_, WITH
TO
GEO Y(2GEO IYT(GEO GEO GEO
GEO GEO GEO GEO GEO GEO GEO GEO GEO GEO GEO GEO GEO GEO GEO GEO GEO GEO GEO GEO GEO GEO GEO GEO GEO GEO GEO GEO GEO GEO GEO GEO GEO GEO GEO GEO GEO GEO GEO GEO GEO GEO GEO GEO GEO GEO GEO GEO GEO
I 2 3 5 6 7 8 9 I0 II 12 13 14 15 !7 18 19 2O ?I 22 ?3 2W 25 _6 27 ?B 29 3O 3! 32 _3 35 3b ]7 ]8 39 W0 42
,,5 _6 _8 w9 5O 51 52 53 5(* 55 56 57 58 59 b0
109
APPEND_
2
3
% C C C
5
b C C C C C C C C 7 C
B
I0
II
12
110
E
WRITE (6,97) PRTCON,N,RTCDHT(ITI,XS(IT),YS(IT) W_ITE (6,|091 DO 5 I=l,Nl READ (5,98) XREG(I,ITI,YREG(I,IT),DIH(I,IT),AMCD MCD(I,IT)=A_CD IF (I.EO.I) GO TO 5 IF (MAK._E.O.OR.MCD(I-I,IT).NE.21 GO T9 2 MAK:I-I IF (ASS(YREG(I-I,IT)-YREG(I,IT)).LT.YTDL) GO TO 3 AREG(I-I,IT)=(XREb(I-I,ITI-XREG(I,IT))/(YREG(I-I,ITI-YREG(I,ITI) ASWP:ATA_(AREG(I-I,IT))*RAD GO TO 4 YREG(I,II)=YPEG(I-I,ITI AREG(I-I,IT)=AZY ASWP=90. J:I-I WRITE
PLANFORM
PERIMETER
POINTS
AND
ANGLES
WRITE (6,106) i,XREG(J,|T),YREG(J, iT;,ASWW,DI,(J,II),MCU(J, DIH(J,ITI:TAN(OIH(J,IT)/RAD) CONTINUE KFCTS(IT):MAK WRITE (6,1061 NI,XREG(NI,IT),Y_EG(NI,IT) CONTINUE
READ
GROUP
SET SA(|),SA(2) CURVE(S) THAT READ(5,1051
2
PART l - SECTION 2 DATA AND COMPUTE DESIRED
EQJAL TO CAN CHANGE
THE SWEEP SWEEP FOR
wING
POSITION
ANGLE,IN DEGREES, EACH PLANFURM
FOR
CONFIG,SCW,VIC,MACH,CLOES,SA(1),SA(21
WRITE (6,99) CONFIG IF(ENOFILE 51 g3,8 IF (PTEST.NE.O..AND.OTEST.NE.O.I GO TO 95 IF (SCW.EQ.O.) GO TO I0 DO 9 I=l,50 TBLSCW(1)=SCW GO TO II READ (5,98) STA NSTA:STA READ (5,981 (TBLSCW(1),TBLSCW(I*I),TBLSCW(Io2),TSLSCW(I*3),TBLSCW(GEO IIo4),TBLSCW(I*5),TBLSCW(I*6),TSLSCW(I*7),I=I,_STA,8) DO 37 IT=I,IPLaN N=AAN(IT) Nl=N*l DO 12 I=l,y XREF(II=_REG(I,IT) YREF(1):YREG(I,IT) A(1)=AREG(I,IT) RSAR(1):ATAN(A(1)) IF (A(1).EQ.AZY} RSAR(1)=PIT CONTINUE XREF(NI)=XREG(_I,IT) YREF(NI)=YREG(NI,IT) IF (KFCTS(IT).GT.O) GO TO 13 K:I
GEO GEO GEO GEO GEO GEO GEO GEO GEO GEO GEO GEO GEO GEO GEO GEO GEO GEO .... GEO IT) G£.0 GEO GEO GEO GEO GEO GEO GEO GEO GE0 GE0 THE FIRSTGEO GEO GEO GEO GEO GEO GEO GEO GEO GEO GEO GEO GEO GEO
bl 62 63 64 65 b6 67 68 b9 70 71 72 13 14 75 76 77 78 79 HO 8l 82 83 84 85 H6 _7 88 89 90 91 92 93 94 95 96 97 98 99 I00 lO1 102 I03 104 I05 GEO I06 GEO I07 GEO I08 GEO I09 GEO II0 GEO III GEO I12 GEO I13 GEO I14 GEO I15 GEO ll6 GEO I17 GEO 118 GEO ]19 GEO 120
APPENDIX
13
15 16
C C C 17
IB C C
C 19
20 C C 21 22
23
2_
SA(IT)=RSAR(1)*RAO GO TO 14 K=KFCTS(IT) WRITE (6,102) K,SA(IT),IT S8=SA(IT)/RAD IF (ABS(SB-RSA_(K)).GT.(.I/RAD)} REFERENCE PLANFORM COORDINATES WITHOUT CHANGE IN SWEEP O0 16 I=I,N X(1)=XREF(I} Y(1)=YPEF(I} IF (RSAR(1).EQ.PIT) GO TO 15 A(I)=TAN(RSAR(1)) GO TO I6 A(I)=AZY SAR(1)=RSAR(1) X(NI)=XREF(NI) Y(NI):YREF(NI) GO TO 35 CHANGES
IN
WING
SWEEP
ARE
MADE
E
GO TO II ARE STORED
UNCHANGED
FOR
WINGS
HERE
IF (MCD(K,IT).NE.2) GO TO 94 KA=K-I DO 18 I=I,KA X(1)=XREF(1) Y(1)=YREF(1) SAR(I)=RSAR(1) DETERMINE LEADING EDGE INTERSECTION BETWEEN FIXED AND VARIABLE SWEEP WING SECTIONS SAR(K)=Sd A(K)=TAN(Sd) SAI=SB-RSAR(K) X(K*I)=XS(IT)÷(XREF(K,I}-XS(IT))*COS(SAI)*(YREF(K*I)-YS(IT))*SIN(SGEO IAI) y(K.I)=YS(IT)÷(yRE_(K÷I)-YS(ITj)*COS(SAI)-(XREF(K*I)-XS(ITJ)*SIN(&GEO IAI) IF (ABS(SH-SAR(K-I))oLT.(.I/RAD)) GO TO 19 y(K)=X(K÷I)-X(K-I}-A(K)*Y(K*I)*A(K-I)*Y(K-I) Y(K)=Y(K)I(A(K-I)-A{K)) X(K)=A(K)*X(K-I)-A(K-I)*X(K*I}*A(K-I)*A(K)*(Y(K,I)-Y(K-I)) X(K)=X(K)/(A(K)-A(K-I)) GO TO 20 ELIMINATE EXTRANEOJS BREAKPOINTS X(K)=XREF(K-I) Y(K)=YREF(K-I) SAR(K)=SAH(K-I) K=K*I SWEEP THE 8REAKPOINTS ON THE VARIABLE SWEEP PANEL (IT ALSO KEEPS SWEEP ANGLES IN FIRST OR FOURTH OIJADRANTS) K=K÷I SAR(K-I):SAI*RSAR(18 219
DO 29 NV=I,NSSW NSCW=TBLSCW(NV) NP=NR*I
OG3 DG3 DG3
720 221 P22
NR:NR÷NSCW PHIPR=ATAN(PHI(NV))_RAO
OG3 DG3
2_3 2_4
22
WNII=WNII*CORAG(J)_A(I,J)_PI_S_EF/(SNN_CSS_CRPHI) CONTINUE
23
WRITE (6,47) Y(1),CO_AG(1),WNII IF (I.EQ.NMA(1).AND.IPLAN.EQ.2) CONTINUE
_PlTE
(6,4W)
DO 26 I=I,IPLAN IUZ=NMA(1) DO
24
J=I,IUZ
JJ=J*(I-I)_NMA(1) ZZH(J)=Y(JJ) XTT(J)=CORAG(JJ) 24
25 2_
CONTINUE IUU=NSSWSV(1) DO 25 J=I,IUU JJ=J*(I-I)*NSSWSV(I) CALL FTLUP (YC(JJ),PPP(JJ),*I,IUZ,ZZH,XTT) CONTINUE
IF DO
(I.EQ.2) 27 J=_A,K8
D=XCFT
NSCW=TBLSCW(J) AI=NSCW_O*O.75 IMAX=INT(AI) DO 27 K=I,NSCW JK=JK÷| E=I. IF (K.GT.IMAX) 27 2U
E=(I.-(K-.75)/NSCW)/(I.-D)
CIR(JK)=PPP(J)_E CONTINUE CONTINUE WRITE NR=O
IF DO
(6,37)
CLDES
(NV.EQ.(NSSWSV(1)*I)) 29 [:NP,NR
WRITE
(5,38)
PNPR=PN(1)_BETA PVPR=PV(1)_HETA PSIPR=ATAN(BETA_TAN(PSI(1)))_RAO 29
WRITE CONTINUE WRITE WRITE
DG3
229
(6,39)
PNPR,_VPR,Q(1),ZH(NV),S(1),PSIPR,PHIPR,CIR(1)
(6,35) (6,3b)
DG3 OG3 DG3
230 231 2112
CREF,CAVE,STRUE,SREF,BOT,AR,ARTRUE,MACH
OG3 DG3 DG3
2 ]3 234 235
CLTOT=CMTOT=O. DO 31 I=I,NSSW IF (I.EQ.I) WRITE IF (I.EQ.(NSSWSV(1)*I)) SPANLD=O. DO 30 IJ=I,NSCWMIN IK=(I-I)*NSCwMIN÷IJ
138
OG3 225 DG3 226 OG3'227 DG3 228
(6,42) WRITE
(6,4])
DG3
236
DG3 DG3
237 238
OG3 DG3
239 240
APPENDIX
E
SPANLO=SPANLD*2.*CIR(IK)*COS(ATA_(PHI(1)))
DG3
74!
CLTOT=CLTOT*B.*S(I_)*CIR(IK)/SREF*COS(ATAN(PHI(1)))
DG3
?42
CMTOT=CMTOT+B.*S(I_)*CIR(IK)*P_(IK)*BEIA*COS(ATAN(PHI(1)})/(SREF*CDG3 IREF) 30
CONTINUE WRITE
(6,45)
(I.EQ.NSSWSV(1)) (I.EQ°NSSWSV(1)I
IF IF
(I,EQ.NSSWSV(1)) WRITE (I.EQ.NSSW.AND.IPLAN.EQ.2)
WRITE(6,40) C 3_
FORMAT FORMAT
35
38 39 (*0
28ER/) FORMAT
REF.
CHORO,GK,25HC
CLA,CM2
AVERAGE
AREA,gX,3H_/2,SX,THREF,
TRUE
AR,8X,7HTRUE
AREA
FORMAT IIAHCM
%2
IA O I FORMAT I0 A D FORMAT I
PLANFORM DESIGN
G//6OX,IHY,IIX,AHCL*C) (/IIIAOX,58HS E C 0 I _ G//6OX,I,Y,IIX,AHCL*C)
VORTEX
OESCR|PTIONS/)
=,_]O.6,SX,I?HCL V=,FIO.6)
N
D
DEVELOPED DEVELOPED
DG3 DO3
746 P41
DG3 DG3
?AIA 248
DG3 063 DG]
?49 25O P_I
DG3 DG3 0G3
252 253 254
DG3 DG3
?55 756
L
0
P A
A L
N
PLANFORM) PLANFORM)
W
I W
THIS THIS S
A
L
E S
H
ALONG
A N
F
0
R
M
S
P
A
N
L
PLANFORM=,FIO.6/ PLANFORM=,FIO.6) S C A ) /( U
L *
E C
0
F S
A I
C N
T E
0 (
R D
S I
257 258 259 260 961 262
DG3 063 063
?63 ?64 265
DG3
?66 _7
0G3 DG3
?60 ?69 770
0G3 DG3 DG3
771 ?72 PI3
DG3 0G3
274 ?lAA
DG3 DG3 OG3
275 216 777
DG3 DG3
2/8 279 2_0
DG3 DG3 063
281 ?82 783-
COMPUTED=,FIO.6,SX,DG3
COMPUTED=,FIO.6,SX,?gHNODG3 V=,FIO.6) A _ F 0 R M S P A N
P ON ON
2H E D R A L ) )//30X,23HDISTANCE 3HWN/(U*COS(PHI)) ) FORMAT (36XFIO.5,10XFIO.5,3XFIO._) (IOX,I_HFI_ST (IOX,IS_SEC3NO
PW4 745 745A
SWEDG3
Y
FORMAT (55XFIO.5,3XFIO.5) FORMAT(/////AX, 127HS I A N D ( N 0 R M
FORMAT FORMAT tNO
HORSESHOE
(/////15X,7HCL DES=,FIO.6,SX,12HCL MOMENT CDNSTRAINT,5X,SHCD (////40X,SGHF I R S T = L
(//50X,3OHCL 50X,3OHCM
DG3 DG3 OG3
063 DG3
(/////15X,IIHCL COMRUTED=,FIO.6,SX,SHCD
FORMAT PITCHINO FORwAT
?AZH 743
,AX,IDG3 NUMDG3
AR,WX,IIHMACH
(_F15.5)
27X,SHANGLE,AX,6HCLDES=,F7.4/) FORMAT (/W5X,45HSECONO FORMAT (17X,BF]2.5)
I
(+5 %6
CMA=CMTOT-CM! WRITE (6,44)
FORMAT (IHI,///25X,IHXIIX,IHX,IIX,IHY,IIX,IHZ,12X,IHS,SX,gHC/4 IEP,AX,SHOIHEDRAL,3X,IOHGAMMA/U AI/24X,3HC/4,9_,_H3C/W,WAX,SHANGLE.DG3
41
%3
(6,44) CLI,CWI CLA=CLTOT-CLI
CLDES,CLTOT,CMTOT,CD
(IOXlIO,IOXIIO) (////AX,IIH
14HREFERENCE
37
CLI=CLTOT CMI=CMTOT
CONTINUE CONTINUE RETURN
33
36
Q(IK),SPANLD
IF IF
IF (I.EQ.NSSW.AND°IPLAN.EQ.2) IF (I°EQ.NSSW.AND-IPLAN.EO.2) CONTINUE
31 C
2wAA OG3 DO3
PLANFORM,5X,THFACTORS,Sx,150G3
139
APPENDIX SUBROUTINE _________
C__ C C C C
PURPOSE
C C
M_TRIx
SINGULAR PERFUP_ING
PPOHLEM
VALUE THE
DECOhPOSITION OF A A:UOV (T) FACTO_IZATION,
RANK,TH_ SINGUI.AP VALUtS, IHE MOMU_ENOU_ SOLUTION , At_O A LFAST SOI.,A_ES SOLUTION
CALL
GIAO005 GIAO006
AN AND
THE FOR
GIAo007 GIAO008 GIAO009
THE
AX=H.
lOP
OPTION IOP=l
PANK
lOP:?
IN
WILL
bE
ThE
RFTIIPNEO
O_DEPtD
AI_I_ITION
HASIS FOP THF LAST
T(_
C C C C C
TO
I_F
ThE
AS
IQP:2.
WILL
BE
PETUHNEL)
IN
IN
WILL
IOP=I V
HE
RFTURNEU
_D[;ITI()N _AT_IX
TH_
IN
lOP=2, IN IN APLUS.
ADDITION
THF
PSEUDO
lOP=5
SAP_
AS
lOP=w,
ADDITION
T_E
LEAST
WILL
BE
RETURNED
IN
MATRIX
A,
ROW
DIMENSION
FOR
V
GIAO035 BIAO036
Ih_TEGER
SPECIFING
TMF
MAXIMU_
C C
M
INPUT
INTEGER
SPECIFING
THE
NUMBER
OF
ROWS
C C
N
INPUT
INTEGER
SPECIFING
TPE
NUMBER
OF
COLUMNS
A
AN
A
CONTAINS
NOS
NUMBER
C C
B
AN
OR
RIGHT
HA_D
INPUT
VALU_,
SIDES
TO
ON
INPUT,B
I_rTEGER
SPECIFING
IN T_E l'SEi; T(,
ELEMENTS DETERMINE
IAC.(_T.I3
H BE
TWO-DIMENSIONAL
lOP=5.
THI_5
PEAL _T LEAST w_ICH IS
PATRIX
C_N_INING TrE
7£Hn
IN
ARRAY
GIAO037 GIAOO3B _IA0039
A. IN
WITH
N. ON UESTROYEU.
EXCEPT
A. ROW
INPUT, ON
WHeN
f)IMEN-
GIAO0_O 6IAO0_I GIAO0_?
A OUTPUI
GIAO043 _IAO0_
IOP=I,
BIAO0_5 bIAO0_6
SOLVEO.
bIAO0_7 GIAO0_8
AR_AY(_D
CONTAINS
SYSTEM OF EQUATIONS THE LEAST SQUARES NOT BE DIMENSIONED
ACC_)PACY VALUE IS
IF
ISOMETRIC
INPUT/OUTPUT
IOP:3
AN
THE OF
FOR THE CONTAINS B NEED IAC
TWO-DIMENSIONAL
_
FOR
INPUT
C C
X
THE
NOS)
RIGHT
USED
FOR
GIAOO_g
_AND
SIDES
_IAO050
B
_IAO051 _IAO05? GIAO053
TO BE SOLVED, UN OU[PUT, SOLUTIONS FOR T_E EQUATIONS, FOR OTHER OPTIONS, THE
NUMBER
OF
OF THE INPUT THE TEST FOR
A
DECIMAL
DIGITS
MATRIX. [_IS ZERO SIngULAR
PANK, TEST
_IAO030 bIAO031
SOLUTIONS
DIMENSION
ND
A
_IA0028 _IA0029
ROW
MAXIMU
C C
HE
GIAO032 _IA0033 GIAO03_
THE
DIMENSION MATRIX
wILL
H.
SPECIFI_G
COLUWN INPUT
61A0023 _IAO02_ _IAO025
SOLUTIONS
SwUAPES
INTEGER
AND THE
(_IAO02I 61A0022
A.
INVF_SE
INPUT
INPUT/OUTPUT
IN
MATHIX
S_UAhES
MD
SION MD CONTAINS
_ETUHNEO THE U
_ILL
QGIAO018 GIAO019 _IAOO?O
GIAO026 GIAO027
SA_E AS RETURNED
.C C
IN
_.
IOP=_
IN
HETUHNE[)
URTMO_UNAL
WILL _E r_ATRIx.
LEAST
IN
_E
AN
SOLUTIO_ OF THE WILL
IN
P_U_HA_
VALUFS
OPTI()NS
.AIRIx
SAMF
CALLING
SID_GULAR
TPE HOmOGEnOUS N-IRANK COLH_N5
TPANSFEPMATION lOP:3
_IAO014 _IAO015 bIAo016 _IAO017
CODE
IRAh_,
C C
_IAO012 GIAO013
GIASOS(IOP,_D,NO,M,N,A,£,OS,N,IAC,O,V,IRAN_,APLUS,IERR)
C C C
140
D_ X
GIA0001 _IA00U_ GIAO003 GIAO00_
GIAO010 GIAOO]I
C C C C
C C C
HEAL
USE
C C
C C C
TME A:BY
WITH OPTIONS FOR THE ORTHOGOh'AL HASIS FOR PSEUDO INVERSE OF A
C C C C C
GIASOS(IOP,MD,ND,_tN,A,NOS,B,IAC,Q,V,IRAN_,APLUS,IERR)
TO COMPUTE N MATRIx
C C
E
OF
BIAO05_ @IAO055 @IAO056 GIAO057 GIAOOSB 6IA0059
HE
COMPUTED
USINb
THE
6IAOOBO
APPENDIX
E-NOPP
C C C C
IF
IAC,LT.13
OF
E
A
_ULTIPLIED
BY
THE ZERO TEST WILL RE E-NOkM OF A MULTIPLIED
2-*(-4_)
,
GIAOOBI
COMPUTEO USING BY IO**(-IAC),
GIAO062 GIAO063 GIAO064
THE
GIAO065 BIAO066
C C C C
A ONE ORDERED
C C
AN OUTPUT ORTHOGONAL
C C
UPON RETURN FROM THE SURROUTINE WILL CONTAIN AN ORTHOGONAL RASIS FOR THE HOPOGENOUS SOLUTIONS IN THE LAST N-IRANK COLUMNS FOR ALL OPTIONS EXCEPT i ,
GIAOO7| GIAOOT2 _IA0073
IRANK
RANK
GIAOO74 GIAO075 GIAO076
APLUS
AN
C C C C C C
IERR
THE
OUTPUT
ERROR
ARRAY VALUES,
TWO
WHICH
WILL
CONTAIN
THE
GIAO067 GIAO068 ARRAy WHEN
(ND
X N) lOP=I,
WHICH CONTAINS THE V MATRIX
IMPLIES
C C
K.GT.O K=-I
I_PLIES IMPLIES _E_T),
C C
LOWE_
C C C
(OUTPUT) APRAY
UF
MATRIx
(ND
A,
IF
X
M)
IOP
WHICH
UOES
CONTAINS
NOT
EQUAL A
UUMMy
WITHU,WITHV A(MD,_,)
NORI_AL KTH THAT THIS RANK T_E
,
_(MD,h!flS)
GIAO080 GIAO081
PETtlRN
SIKGUEAR t SIN(_ _DTPlX T_AN
GIAO084 6IA0085
VALU_ _OT FOUNI) AFTER 30 ITER, THE (_IVEN IAC(ACCUPACY REQUIREIS CLOSE TO A MATHIX wHICH IS OF
IPANK
RArJK
OF
V(Kfl,N)
ANU Thh
,(%(N)
IF
MATRIX
THE
ACCIJRACY MAY
ALSO
IS _E
REDUCED,
.E (?Sh)
THE
E-NnRM
OF
MATRI_
C
510
GO TO 7TEST
510 = ZTEST
ZTEST
=SQPT(SU_)*2,O_*(-48)
IF
(IOP,NE.I
El5
WITHU=,TRUE,
AS
ZERO
TEqT
FOR
SINGULAR
VALUES
GIAOI02 GIAOI03 GIAOI04
GIAOI07 GIAOIO8 GIA0109
J:I,N SU M + A(I,J)**2 = SQRT(SU_)
WITHU=.FALSE. WITHV=,FALSE, GO TO 520
A
GIA0105 GIA0106
l=l,M
IF (IAC.GT,13) GO TO ZTEST = ZTEST_IO,_(-IAC) 505
GIAOO90 GIAOO91 GIAOO92
GIAO099 GIA0100 GIAOIOI
COMPUTE
I)O 500 SLJI_ = 7TEST
GIAOOR9
_IA0097 GIAO098
,APLUS(NU,_')
SIZE=O,O NPI=N÷I
500
GIA0086 GIAOO87 _IA0088
GIAO094 GIAO095 GIAO096
TOL=I,OE-60
SUM: 0,0 r)O 5On
GIAO077 61A0078 GIAO079
GIAOO82 GIAOO83
REDUCED
C C
THEGIAO069 GIAO070
INDICATOR
K=O
DIMENSION
A
N
ARRAY NEE[) NOT NE DIMENSIONEU _UT MUST APPEA& I_ T_E CALLIr_6 SEQUENCE,
C C C
LOGICAL DIMENSION
SIZE
UI_ENSIO_AL
INVERSE
TPIS
OF
DIMENSIONAL MATRIX EXCEPT
MATRIX
PSEUDO
4 OR S PARAMETER
C C
TWO V
OF
THE
C C
DIMENSIONAL SINGULAR
_
)
50S
2,0_*(-_8)
GO
TO
515
GIAOllO GIAOlll GIAOll2 GIAOll3 GIAOII_ GIAOI15 GIAOll6 _IAOI17 GIA0118 GIAOll9 GIA0120
141
APPENDIX
E
WITHV=,TRUE, 520
GIAOI21 6IA0122 GIA0123
CONTINUE G 0,0 X -- 0,0 DO 30 I
=
HOUSEHOLDER
REDUCTION
3
J
THE =
I-TH
S + A(J,I)**? 0.0 .LT. TOL) SQRT(S)
F = IF(F H :
A(i,|) .GE. 0.0) F*G -S
=
F DO
S/H S K
=
A(K,J) CONTINUE Q(I) IF(I
: G .FQ.
S = 0.0 DO II J S = S *
GO
=
IF(F H =
16 17 19 20 30
142
DIA@ONAL.
GO
TO
GIA0137 GIAOI38 6IA0139
10
-G
GIAOI40 GIAOI4I GIAOI42
TO
10
GIAOI43 GIAO144 GIAOI45 GIAOI46 GIAOI_7 bIAO148 GIAOI49
I.M
N)
÷
Gb
THE
61A0150 bIA0151
F*A(_,I)
TO
I-TH
GIA0152 6IA0153 GIA0154
20 POw
TO
QIGHT
bO
TO
20
(_ =
.GE. 0.0) F*G -S F
-
- G
=
00 16 S = S DO ]7
= L,N A(J.K) = L,N
K + K
CONTINUE IF(.NOT.
wiThY)
GIAOI61 GIAOI6?
bIAO169
_IAOI70
L,M
A(J.K) = A(J.K) CONTINUE Y = AHS(Q(1)) ÷ IF(Y .GT. SIZE)
bIAO155 bIAO156 61A0157
GIAO166 bIAO167 GIAOI68
G
DO 15 J = L,_' E(J) : A(I,J)/,
DO ig d S = 0.0
SUPEP-r)IAG.
6IA0163 61A016_ _IA0165
A(I,I÷I)
=
OF
GIAO1S8 GIAOI59 GIA0160
= L,N A(I,J)**2
A(I.I÷]) 15
HELOW
GIAO131 GIAO132 GIAOI33 GIAO134
=
G
G = 0.0 IF (S .LT. TOt) G = SO_T(S) F
GIAOIB7 6IA0128
GIAO135 GIAO136
:A(_,J)
ANNIHILATE
II
COLUMN
DO 7 K = I,M S = S ÷A(K,I)*A(K,J)
7
10
FORM.
I,M
S = G = IF(S G =
A(I,I) = F-G IF(I .EQ. N) DO 9 J = L,N S = 0.0
8 9
BIDIAGONAL
GIAOI29 GIAOI30
ANNIHILATE DO
TO
= G 0.0 I.I
E(1) S = L =
3
GIA0124 GIA0125 GIAOI26
1,N
*
(_IAO[71 _IA0172 61A0173
A(I,K) ÷
A_
