Rochester Institute of Technology
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2000
Mechanical supported adaptable wing
Cory Pike
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MECHANICALLY SUPPORTED ADAPTABLE WING
by
Cory
J.
Pike
A Thesis Submitted in
Partial Fulfillment of the
Requirement for the
MASTER OF SCIENCE
IN
MECHANICAL ENGINEERING
Approved by:
Dr. Mark H. Kempski
Department of Mechanical Engineering
Dr. P. Venkataraman
Department of Mechanical Engineering
Dr. Hany Ghoneim
Department of Mechanical Engineering
Dr. Satish Kandlikar
Head, Department of Mechanical Engineering
(Thesis advisor)
DEPARTMENT OF MECHANICAL ENGINEERING
ROCHESTER INSTITUTE OF TECHNOLOGY
Permission granted
MECHANICALLY SUPPORTED ADAPTABLE WING
I, Cory J. Pike, hereby grant permission to the Wallace Library of the Rochester Institute of
Technology to reproduce my thesis
in whole or in part. Any reproduction will not be for
commercial use or profit.
Dedication
This
thesisis
dedicated
to_{my}
_{loving}
parents,Wes
and_{Cheryl,}
andto_{my}
ABSTRACT
There
are several advantages to theincorporation
ofan adaptable_{wing}
on an aircraft.Among
these areincreased
fuel
_{efficiency,} greater _{maneuverability,} and animproved ability
tonegotiate adverse conditions.
This
thesis explorestheconcept of a_{mechanically}
supportedadaptable_{wing,} wherethe supports are
in
theform
ofvirtual spars_{running lengthwise along}
a
_{wing}
of uniform crosssection.With
no crosssectional support to the wing, the skinspanning
theareabetween
thevirtual spars experiences a certain amount ofdeflection
due
tothe various aerodynamic
forces acting
on the wing.This
thesisdevelops
a processfor
optimizing
(minimizing)
the number of virtual spars required to support the_{wing}
whileTABLE OF CONTENTS
ABSTRACT i
TABLE OFCONTENTS ii
LIST OFTABLES iv
LIST OFFIGURES v
CHAPTER1  INTRODUCTION 1
1.1BACKGROUND 1
1.2BRIEFOVERVIEW 2
CHAPTER 2 CHOOSINGTHEAIRFOILS
5
2.1DETERMINING AIRFOIL SHAPES 5
2.2AERODYNAMIC PROPERTIESAND APPROXIMATIONS 7
2.2.1 Physical CharacteristicsandSpecifications 7
2.2.2 Aerodynamic PropertiesandValues 7
2.2.3
_{Wing}
Skin Properties 12CHAPTER 3 DATA SETUPUSING
MICROSOFT
EXCEL 97 AND VBA 14
3.1THE NEED FOR MACROS 14
3.2PREPARING THE DATA FORANALYSIS 14
CHAPTER 4 XFOIL 16
4.1GENERAL DESCRIPTIONOFTHE PROGRAM 16
4.2USINGXFOILTO OBTAIN THE PRESSURE COEFFICIENT DISTRIBUTION 17
CHAPTER 5
MICROSOFT
EXCEL97 AND VBA REVISITED 22
5.1 GROUPCPXMACRO 22
5.2EXPLANATIONOF GROUPCPX MACRO 23
CHAPTER6
MATLAB
ANALYSIS 24
6.1BRIEF INTRODUCTIONTO
MATLAB
24
6.2OVERVIEWOFTHE ANALYSIS PROCESS
_{(MASTER.M)}
256.3DESCRIPTION OF INDIVIDUAL FUNCTIONS 27
6.3.1
_{Setting}
theRelevant Parameters (Parameters.m) 276.3.2
_{Approximating}
theAirfoilPerimeter(ArcApprox.m)
286.3.3
_{Dividing}
theAirfoil Perimeter(MapToP.m)
396.3.4
_{Determining}
Element Orientation(GetAlpha.m)
456.3.5
_{Establishing}
Pressure ValuesatNodes(Loading.m)
476.3.6Finite Element Analysis
(FEA.m)
536.3.7
Optimizing
Each Nodal Support_{Spacing}
_{(Opt.m)}
686.3.8
Controlling
theOverall FEAandOptimizationProcess(FEAJDpt.m)
71CHAPTER 7 RESULTS 74
CHAPTER8 DISCUSSIONOF RESULTS AND CONCLUSION 80
APPENDIX A C5 AIRFOIL COORDINATES DATA
83
APPENDIXB
MICROSOFT
B.l CONVMACRO 86
B.l.l Conv Variables 86
B.1.2ConvCode 86
B.2CONVALL MACRO 87
B.2.1 ConvAll Variables 87
B.2.2 ConvAll Code 87
B.3GROUPCPX MACRO 89
B.3.1 GroupCpx Variables 89
B.3.2GroupCpxCode 89
APPENDIXC
XFOIL ADDITIONAL INFORMATION 91
C.l THEORY REFERENCES 91
C.2PRIMARY PROBLEM FORMULATIONS 92
C.3CPXS SUBROUTINE(FORTRAN
_{77)}
94C.4PRESSURECOEFFICIENT VSX DATA 95
C.5PRESSURE COEFFICIENTSVS.XPLOTS 100
APPENDIX D
MATLAB
MFILES VARIABLESAND CODE
103
D.lMASTER.M 103
D.l.l Master.m Variables 103
D.1.2Master.m Code 105
D.2 Parameters.m 108
D.2.1 Parameters.m Variables 108
D.2.2 Parameters.m Code 709
D.3 Arcapprox.m 110
D.3.1ArcApprox.m Variables 110
D.3.2 ArcApprox.m Code Ill
D.4MAPTOP.M 112
D.4.1 MapToP.m Variables 112
D.4.2 MapToP.mCode 113
D.5 GetAlpha.m 114
D.5.1 GetAlpha.mVariables 114
D.5.2 GetAlpha.mCode 115
D.6LOADING.M 116
D.6.1 Loading.m Variables 116
D.6.2 Loading.m Code 117
D.7FEA_OPT.M 119
D.7.1 FEAjOpt.m Variables 119
D.7.2FEAJDpt.mCode 120
D.8 0PT.M 122
D.8.1 Opt.m Variables 122
D.8.2Opt.m Code 123
D.9FEA.M 124
D.9.1FEA.m Variables 724
D.9.2 FEA.m Code 725
APPENDIX E MAPLEVFINITE ELEMENT ANALYSIS 127
LIST OF TABLES
Table
1.1 List
of softwarepackagesutilized in this thesis4
Table 2.1 Some
generalcharacteristicsof theLockheedGeorgia C5B
7
Table 2.2
Properties
of the standard atmosphere for an altitude of40,000
ft(12,192 m)
10
Table 2.3
Initial
properties ofAL 7075T6
skin onC5
wing12
Table
6.1_{Sample}
_{results of optimal nodal support spacings for one}_{iteration}_{on}all
5
airfoils71
Table 6.2
Uniform
assignment of kout value for all5
airfoilsfromTable 6.1
based on a minimum kout value (c5d,kout=
_{1022)}
72
Table B.l
List
of variables used in
Conv
macro86
Table B.2
List
of variables used m
ConvAll
macro87
Table B.3
List
of variables used in
GroupCpx
macro89
Table D.l
List
of variables used inMaster.m
103
Table D.2
Listof variables used inParameters.m
108
Table D.3
List
of variables used inArcApprox.m
110
Table
D.4_{List}
_{of variables used in}_{MapToP.m}
_{112}
Table D.5
List
of variables used inGetAlpha.m
114
Table
D.6_{List}
_{of variables used in}_{Loading.m}
_{116}
Table D.7
List
of variables used inFEA_Opt.m
119
Table
D.8_{List}
_{of variables used}_{in}_{Opt.m}
_{122}
LIST OF
FIGURES
Figure 1.1
Example
of a nodal support spacing around an airfoil3
Figure
1.2_{Example}
_{of deflected skin material between supports}_{(tradling}
_{edge of}airfoil in
Figure
_{1.1)}
3
Figure 2.1
Plot
of allLockheedGeorgia
C5A
airfoil crosssections6
Figure 2.2
Vertically
_{rescaled}C5A
airfoils(for
variance_{emphasis)}
6
Figure 2.3
Example
of a pressure coefficient distribution
8
Figure 2.4
Example
of a pressure distribution at high altitude
9
Figure 2.5
Division
of a wing into sections and individualelements
13
Figure 3.1
Flowchart for theConvAll
macro15
Figure 4.1
PlotofPressure Coefficient
vs.X
xc5a.ps20
Figure
5.1
Flow
_{chart for the}GroupCpx
macro22
Figure 5.2
Sampleof an optimal nodal support distribution23
Figure 6.1
Flow
chart fortheoverall analysis process
_{(Master.m)}
25
Figure
6.2
Flowchart forParameters.m
27
Figure 6.3
Plot
of c5a airfoil coordinates
28
Figure 6.4
Arcs
defined by3
points(left)
and a center and2
points_{(right)}
3 1
Figure
6.5
Determining
_{the}center of an arc defined by
3
poestts on the arc3 1
Figure 6.6
Initial
arc approximation using the first
3
datapoints33
Figure
6.7
Addition
_{of a second arc approximation}35
Figure
6.8
Closer
view of thefirsttwo arc approximations36
Figure
6.9
Occurence
_{of concave arcs}(A &
D)
on the c5a airfoil37
Figure 6.10
Plot
of all arcs used toapproximate the c5a airfoil37
Figure 6.1 1
Flow
chart for arcapproximation function_{(ArcApprox.m)}
38
Figure
6.12
Mapping
apoe^jt alongan arc into global coordfnates39
Figure 6. 13
Howtodeterminewhichgoverning arc to follow41
Figure
6. 14
ResultsofMapToPfunctionforthec5a airfoil42
Figure 6.15
Closeup
view of c5a airfoilofFigure
6. 14
showing the leading nodeindicated by the open square
43
Figure 6. 16
Flowchart forMapToX
function44
Figure
6.17
Variable
and sign convention for theGetAlpha
function45
Figure 6.18
FlowchartforGetAlpha
function46
Figure
6. 19
Plotofpressurecoefficientdata fromXFOIL (c5a)
47
Figure
6.20
Four
pointsto be fit usestg a cubic polynomial48
Figure
6.21
Cubic
polynomialfit to original four points ...49Figure
6.22_{Pressure}
coefficientresults of theLoading
function51
Figure
6.23
Flow
_{chart}for theLoading
function52
Figure 6.24
Generalized
displacementsand forces for theEulerBernoulli
frame element(width
=1)
54
Figure
6.25
Various
loadingconfigurationsfor anEulerBernoulli
frameelement
56
Figure
6.26
Applying
pressures(from Loading
Figure 6.27
Twoelement
global assembly of applied distributedloadingon abeam
59
Figure
6.28
Distinguishing
between local and global degrees offreedom
64
Figure 6.29
Flowchart forFEA
function67
Figure
6.30
Example
of the nodal support spacing optevuzationprocess
69
Figure 6.31
Flow
chart forOpt
function!
70
Figure 6.32
Flowchart forFEAJDpt
function73
Figure
7.1_{Trial #1}
[THICKNESS
=0.00079375
_{M}(1/32 IN),
_{ALLOWABLE DEFLECTION}=_{0.001}
M]
74
Figure 7.2
Trial#2
[THICKNESS
=_{0.00079375}
_{M}_{(1/32 IN),}
_{ALLOWABLE DEFLECTION}=0.005 M]
75
Figure 7.3
Trial#3
[thickness
=0.003175
_{m}_{(1/8 in),}
_{allowable deflection}=0.001 m]
76
Figure 7.4
Trial#4
[thickness
=0.003175
_{m}_{(1/8 in),}
_{allowable deflection}=0.005 m]
77
Figure 7.5
Magnified
view of the trailing edge of the results from
Trial #2
(Figure
_{7.2)}
78
Figure 7.6
Magnified
_{vmw of the leadentg edge of the results from}Trial
#2
(Figure
_{7.2)}
79
Figure
C.l
Plot
ofPressure Coefficient
vs.X
xc5a.ps100
Figure C.2
Plot
_{of}Pressure Coefficient
_{vs.}X
_{xc5b.ps}100
Figure
C.3_{Plot}
_{of}_{Pressure Coefficient}
_{vs.}_{X}
_{xc5d.ps}_{101}
Figure
C.
4_{Plot}
_{of}_{Pressure Coefficient}
_{vs.}_{X}
_{xc5c.ps}_{101}
CHAPTER
1
INTRODUCTION
1.1 BACKGROUND
Since
thedawn
of _{civilization,} manhas looked
upward to the_{sky}
and observed nature's airborne _{creatures;}from
the smooth, effortless_{soaring}
of the majestic eagle, to thehighly
energeticflights
ofthehummingbird.
_{Early}
civilizations most_{likely}
looked
on thesecreatures with awe, and
_{in many}
cases even reveredthem.These
animals seemedto_{defy}
theconstraints of
_{gravity}
and were able to_{enjoy}
suchfreedom
of _{movement,} as toleave
mansimply
envious.From
thetime that mankindfirst
tookhold
ofthe "airfoil" concept, to thehighspeed
computer airfoil
design
and analysis programs of_{today,}
aircraftinvention
has
been
traditionally
based
on a singleframe
of mind... a_{constant,}rigid
airfoildesign.
In
all ofthe advancesin
speed, altitude, etc.,oftoday's aircraft, andin
the_{ability}
to_{fly}
faster
andhigher
than_{any}
otheranimal, therestill exists onemajorskillthatmost_{flying}
animalshave
thatour aircraft_{simply}
cannot compare with... maneuverability.Manmade
designs
ofrigid
wings providethe_{necessary}
_{lift,}
andtheadditional control surfaces_{(rudders, flaps,}
_{stabilizers, etc.)} provide somebasic
maneuverability,but
none can provide the aircraft with the_{agility}
of a creature that can change the actual shape ofits wing
at_{any}
time.Airfoil
_{adaptability}
is
acapability
that almost all_{flying}
animals possess, and use to their advantage.Yet
noknown
aircraft at this time canduplicate
thisfeature.
Flaps
provide a certain amount of change of shape, and some _{aircraft,} such as theLockheedGeorgia
C5
_{Galaxy}
(mentioned later in
thispaper)
usedifferent length
sections of afew different
airfoilsfrom
theroot ofthe_{wing}
to thetip.
However,
none can_{actively physically}
change the shape ofthe airfoil crosssection ofthe
_{wing in flight. Even today,}
thisremainstobe
onetraitof nature's_{flying}
animalsthatmanhas
notbeen
abletomimic.The
goal of an adaptable airfoil crosssection need not achieve adrastic
changein
mind: _{payload, speed, weight, altitude, purpose,} _{etc.}
_{The}
airfoilis
optimizedfor
theseparameters,
andthe_{wing}
is
designed.
_{However,}
oncethat_{wing}
is
constructed, inefficienciesinherently
arise.Fuel
optimization at_{cruising}
speed and altitudeis lost
the entire time theaircraft
is
_{climbing}
ordescending.
_{Maneuverability}
atlower
speeds and altitudesis lost
athigher
speeds and altitudes.Even
to stretch theimagination
andlook into
thefuture,
anoptimal
design in Earth's
atmosphere,may
not work as wellin
the atmosphere of anotherplanet.
Hence
valid reasons existfor exploring
the realm of_{actively}
controlled airfoilshapes.
The issue
of active shape control of wingsfor
moreefficientflight has been discussed
for
several years(see References [15]). The
studieshave been
centeredabout_{using}
variablecamber, twist or surface shape control to adapt the
_{geometry}
ofthe_{wing}
to_{changing flight}
conditions.
1.2
BRIEF
OVERVIEW
This
thesis explores the_{attainability}
of several_{wing}
crosssection shapes through themanipulation of anumberof
"nodal
supports"
firmly
clamped to theskin at"optimal" points
around the airfoil.
Due
to thelarge
number of variables that enterinto
thedesign
of such acomplicatedsystem, several simplifications
have been
employedin
orderto obtain aninitial
approximation ofthenumber and
location
ofthenodal control points aroundthe airfoil.The
goal of this thesis
is
toinvestigate
the_{feasibility}
of_{using}
a"nodal
support"airfoil/wing
design (See Figure 1.1).
Each
nodal supportin Figure 1.1 is defined
as a virtual spar_{running}
_{along}
thelength
of the wing, which _{completely} clamps the
_{wing}
skinin
place.In
thisthesis,
these nodalsupports are the
_{only}
means of support and manipulation of the wing._{By}
_{allowing only}
spanwise
_{wing}
skin _{support,} the skinis
expected todeflect in
the gapsbetween
nodal?> 0'
OriginalMapped Airfoil Deflected Airfoil
o Supports
e e e~Q o o e
44Supports Required A\rage_{Spacing}=0.39105m
2 3 4 5 6
ActualChord,m.a.c. =8.4818m
Figure 1.1 Example
ofa nodal support_{spacing}around an airfoil
0.2
0.1
0.15
Original MappedAirfoil
Deflected Airfoil
O Supports
(.05 8.1 8.15 8.2 8.25 8.3 8.35 8.4 8.45
ActualChord,m.a.c.=8.4818m
This deflection
resultsin
a_{deviation}
from
thedesired
airfoil shape.The
goal ofthis thesisis
to
define
and maintain various airfoil shapes within a specifieddeflection
tolerance whileminimizing
the number of nodal supports required.This
support minimization objectiveis
accomplished_{according}
tothe_{following}
analysis process:Step
1.
Select five different
airfoils.Step
2.
Determine
theaerodynamic_{loading}
on thefive
airfoilsbased
on specifiedparameters.Step
3.
Divide
each airfoilinto
a specified number offinite
elements.Step
4.
_{Apply}
the_{loading}
found in
_{Step}
2
to thefinite
elementsestablishedin
_{Step}
3.
Step
5. Calculate
optimal supportlocations based
on skindeflections due
to_{loading}
appliedin
_{Step}
_{4,}
wherethedeflections
aredetermined
throughfinite
elementanalysis.The
scopeofthis thesisinvolves
abasic understanding
of_{aerodynamics,} a_{slightly}
more
involved
knowledge
offinite
element _{analysis,} and a_{basic understanding}
ofdesign
optimization.
The
thesis alsoinvolves
the use of severaldifferent
software packages andassociated
_{programming}
syntaxes/languageslisted in Table 1.1.
Software
Language/Syntax Usedfor
AcknowledgementsMicrosoft
Excel 97
Visual Basicfor
Applications
Preparing
datafiles
for import into
MATLAB
Copyright 19851996
Microsoft Corporation.
Allrightsreserved.
XFOIL 6.8 FORTRAN
_{Obtaining}
pressurecoefficient
distributionforeach airfoil
Created
_{by}
MarkDrela, Ph.D,
Professor,
MIT Departmentof AeronauticalandAstronauticalEngineering
MATLAB5.2
Builtin (similarto
C)
Bulkof
programming
(mapping,
FEA,
optimization,
plotting)
Copyright
19841998The
MathWorks,
Inc. Allrights reserved.MAPLE V
Release4.00
Builtin Symbolic
Computations
Copyright 19811996 Waterloo
Maple Inc.
Allrights reserved.
Table 1.1 Listof software packages utilizedinthis thesis.
Due
to thefact
that a gooddeal
of this thesisis in
theform
of computer _{code, the}various
functions
andsubroutineswillbe
represented_{mainly}
_{by}
conceptualflow
charts so as to avoidthedetails
ofthesyntaxitself.
The
actual algorithms employed areincluded in
theirentirety
andpresentedin
several appendices.Each
respective algorithmis
accompanied_{by}
aCHAPTER 2
CHOOSING
THE AIRFOILS
2.1
DETERMINING
AIRFOIL
SHAPES
Once
designers
makethedecision
togo with an adaptable_{wing,}_{they}
would nolonger
be
_{designing}
a single airfoil to meet alldesign
constraints._{Instead,}
the_{wing design}
wouldinvolve
severaldifferent
airfoil shapes to allowfor
optimal performance under a_{variety}
ofcircumstances.
Design for
speed,lift,
maneuverability, and overall_{efficiency}
would nolonger be limited
to one solid airfoil._{Instead,}
the goal wouldbe
such that a pilot of anaircraft could
_{ideally}
just dial in
an airfoil shape ona_{computer,} andthen gettheperformancerequired
from
thatparticular airfoil shape.An
even moreideal
situation wouldbe
one where the shape ofthe_{wing is}
controlled_{dynamically by}
a computer that_{constantly}
monitors thecurrent
_{flying}
conditions.Concerning
the shape of the airfoil, shape coordinates canbe
obtained throughpublished
_{books,}
through the aircraft_{manufacturers,} or evendownloaded from
theInternet.
Since
the goal ofthisthesis_{lay}
in
the analysis of several airfoils,any five
woulddo.
Upon
exploration of the
UIUC Airfoil Coordinates
Database1 on the_{Internet,}
a_{group}
offive
airfoils seemedto stand out as
far
as_{catering}
to thegoals ofthis thesis.These
airfoil shapesbelong
to the LockheedGeorgiaC5A Galaxy.
These
airfoil shapes weredesigned
as"boxed"
sets that were connected
from
the root ofthe_{wing}
to the tip.This design
wasfirst
implemented
as a redesign to theC5A
model's wingsin
1981,
and thenit
was carried overand used onthecurrent
C5B
model aswell.2
The
C5A
airfoil_{geometry files}
chosenfor
thisthesis were c5a.dat, c5b.dat, c5c.dat, c5d.dat, and c5e.dat, and these
five
geometries are1_{http://www.uiuc.edu/ph/www/mselip/ads/coord} _{database.html} A
web site maintained
_{by}
Michael_{Selig,}
DepartmentofAeronauticalandAstronautical
Engineering,
University
ofIllinoisat_{UrbanaChampaign,}
Urbana, Illinois,
61801. Informationisavailableforuse undertheGNU General PublicLicense: http://amber.aae.uiuc.edu/~mselig/pd/gpl.html.2
displayed
_{graphically in Figure 2.1}
andFigure
2.2. A
printoutofthedownloaded data files is
given
in Appendix A.
0.2
0.2
0.3
X c5a o c5b f c5c ? c5d 0 c5e
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 xcfim,(%chord)
Figure 2.1 Plot
of allLockheedGeorgiaC5Aairfoil crosssections
0.08
0.06
"'
X c5a 0 c5b
+ c5c n c5d 0 c5e
O 6 0
8&&88i*.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 xdim,(%chord)
Figure2.2
These
airfoils werechosenbased
onthe assumption that anadaptable_{geometry wing}
would utilize each shape while subject to similar performance requirements.
The
adaptablecross sections would then
be
varied enough to alter the aerodynamic capabilities of aprototype
_{wing}
as neededin
flight.
_{So,}
instead
of_{using}
allfive
airfoils_{simultaneously}
atspecified positions
_{along}
_{the wing, this thesis} explores the_{ability}
to change the entire_{wing}
from
one airfoil shapeto_{any}
oftheotherfive
shapes.2.2
AERODYNAMIC
PROPERTIES AND APPROXIMATIONS
2.2.1 Physical Characteristics
andSpecifications
The
most current version of theLockheedGeorgia C5 is
the_{C5B,}
hence
theaerodynamic
data
usedherein
was collectedbased
on thedesign
specifications.3Again,
thesame airfoil cross sections were utilizedonthe
C5B
asin
the_{C5A,}
sothe application ofC5B
data
to theC5A
airfoilsis
appropriate.The design
parameters relevant to this thesis aregiven
in Table 2.1.
General
Characteristics
Value (English
_{Units)}
Value (SI
_{Units)}
Wing
span_{(b)}
222.8 ft
67.91
mWing
area_{(S)}
6,200
ft2
576.0
m2Long
range cruise speed_{(M)}
0.77 Mach
0.77 Mach
Maximum
gross weight_{(W)}
840,000
lb
381,018
kg
Table2.1 Somegeneral characteristics oftheLockheedGeorgiaC5B.
2.2.2
Aerodynamic
Properties
andValues
Note
that thevaluesfor
speed andweight characteristics notedin Table 2.1
are_{fairly}
high,
nearmaximumvalues.This
wasdone
as a means of_{testing}
theXFOIL
programtosee3
Generalcharacteristicsfor theLockheedGeorgiaC5
_{Galaxy}
obtainedfromtheLockheedMartin web site at:if it
would convergefor
a_{relatively high Mach}
_{number.}_{However,}
_{a reasonable} _{test} ofthefinite
_{element analysis routine was also}_{desired;}
_{one}_{in}
_{which at some point on} _{the} _{airfoil}there would exist a negative
local
pressure_{(pL)}
that would cause the skin of the_{wing}
todeflect
outwards.In
order to establish the pressuredistribution
on _{the airfoil/wing, the}pressure coefficient
_{(Cp)}
_{distribution}
mustbe
_{determined first.}
An
example of a_{Cp}
distribution
is
shownin
Figure
2.3.
0.2 2
1 ~ ^~
^^^~~*i "
i 1 1 1 1
\
Lower SurfaceCL ^"*w *"*
o o
^\
^
"V^ ^~
^
^^
<D
'o
_{^}
it CD 1 _
O ?" '
O
3
J
UpperSurface05 S
0) o

J
i. CL
3
V
4
i i i i i
0.2 0.4 0.6
xdim,
(%chord)
0.8 1.2
Figure 2.3 Example_{of a pressure}_{coefficient}distribution
This
pressure coefficientdistribution is
convertedtolocal
pressure values_{using}
Pl
=In
Equation
(2.1
_{)}
, the pressure coefficient(Cp)
is usually
notless
than 4(see
Figure 2.3).
Hence
the_{best way}
to achieve a negativelocal
pressure_{(pL)}
with ahigh
freestream
_{velocity}
_{(V0)}
as notedin
Table
_{2.1,}
is
tohave
alow
freestream
pressure(p0). This
can
_{be easily}
achieved_{by}
_{selecting}
_{freestream}
_{values} _{at} _{a}_{high}
_{altitude.}_{The resulting}
pressure
distribution
wouldbe
similarto thepressuredistribution depicted
in Figure 2.4.
x 10
2.5
7a 1.5 a.
3 05 01
0.5
0.5
0.2
UpperSurface
0.2 0.4 0.6
xdim,
_{(%chord)}
0.8 1.2
Figure 2.4Example_{of a pressure}distribution
athighaltitude
In Figure
_{2.4,}
note that even though thepressure valuesfor
theupper surface arelower
thanthepressure values
for
thelower
surface_{(creating}
_{lift),}
theupper surface pressure values areprimarily
positive_{(pushing}
in
on the airfoil)._{However,}
Figure 2.4
shows a small area of negative pressure_{(pulling}
outwards on the _{airfoil)}near the_{leading}
edge ofthe airfoil ontheupper surface.
This
negativelocal
pressure_{(p)}
areais desired
as a means of_{testing}
thefinite
element analysis.4
So,
for
the purpose of thisthesis,
an altitude of40,000 ft (12,192
m) was used todetermine
theatmospheric propertieslisted
in Table 2.2.
Atmospheric
_{Property}
Value (SI
_{Units)}
Pressure
_{(po)}
_{1.882748 E4}
N/m2Density (p)
0.3027496
kg/maKinematic
_{Viscosity}
_{(v)}
4.698136 E5m2/s
Speed
ofSound
_{(a)}
295.07
m/sTable2.2 Propertiesofthestandard atmosphereforanaltitude of40,000 ft (12,192m).5
XFOIL has
the_{capability}
of_{performing}
either aninviscid flow
analysis ora viscousflow
analysis.For
theinviscid
analysis,XFOIL
requires specific valuesfor
angle of attack(ALFA),
Mach
number_{(MACH),}
andReynolds
number(RE).
_{However,}
in
ordertoperformthe more realistic viscous
flow
_{analysis,}XFOIL
also requires a valuefor
thelift
coefficient(CL).
Each
ofthese parameters required_{by}
XFOrL
canbe determined from
the valuesin
Table 2.1
andTable 2.2.
The Reynolds
numbercanbe
calculated as6RE
= _{general}(/
=length)
(
2.2
_{)}
v
Vc
i \RE
= _{airfoils}(c
=chord)
(
2.3
)
v
RE
= taperedwings(c
=mean aerodynamic_{chord)}
_{(}
_{2.4}
_{)}
v5
Dataobtainedfrom: Aerodynamicsfor Engineers.
3rd
Ed.,
JohnJ. Bertin &MichaelL._{Smith,}
_{1998 Prentice}Hall, Inc.,
pp.650651 Table 1.2b.6
depending
on the application.Of
these_{equations,}
the one thatbest
applies to this thesisis
Equation
_{(}
2.4 ).
_{However,}
it
is based
upon two yet undetermined values, thefree
streamvelocity
(V)
andthe mean aerodynamic chord_{(m.a.c.)}
.The
free
streamvelocity
(V)
canbe
determined
_{easily}
asV
=Ma
=0.77(295.07 m/s)
V
=227.20
_{m/s}(508.24
mi/hr)
(2'5)
However,
determination
of the mean aerodynamic chord_{(m.a.c.)}
of the airfoils_{is slightly}
more complicated.
The
m.a.c,_{by}
definition,
is
the chord of an_{imaginary}
_{wing,} withconstant_{chord,}
_{having}
thesame aerodynamic characteristics as theactualwing.7
The
m.a.c.is
given_{by}
2
m.a.c.=3
(
CrCt
^
V R'^T
j
c
+c.root chc
CT
=tip
chordCR
=_{exposed root chord at side of}_{fuselage}
In
other _{words,}it is
a_{way}
of_{accounting for}
the_{tapering}
of a_{wing from}
root to tip.Since
exact valuesfor
thelength
of the chord at the root and_{tip}
were _{unavailable,} anapproximation of the m.a.c. was made
_{by dividing}
the_{wing}
area_{(S)}
_{by}
the_{wing}
span_{(b),}
which
is
calculatedtobe
__S__
576.0
m2b
67.91m
_{(2.7)}
c=
_{8.4818}
m(27.83
_{ft)}
From this,
theReynoldsnumber canbe
calculatedasVc
(227.20
_{m/s)(8.4818m)}
~~v~4.6981365m2/s
_{(2.8)}
RE
=_{41. 0E6}
'
Definition from: FundamentalsofFlight.
2nd
The lift
coefficient canbe
determined
from
Equations
_{(}
2.9
_{)}
&
_{(}
2.10
_{)}
based
on thefact
_{that,}
under_{cruising}
_{conditions,}
_{the total}_{lift}
_{(L)}
_{of}_{the}_{plane equals}theweight
(W).
CL
=pV2S
(
2.9
y
cL
=w
3723161.5
N
lpV2S
^(0.3027496
kg/m3)(227.20
m/s)2(576_{m2)}
2'
2
CL
=0.8274
(2.10)
2.2.3
_{Wing}
Skin Properties
The only
remaining
parameters requiredfor
_{succeeding}
analyses arethose_{pertaining}
to theactual skinthatmakes_{up}
theairfoil shape.Due
tothefact
that theactual material composition oftheC5
skin was_{unavailable,}AL
_{7075T6,}
a common aluminum_{alloy}
usedin
aircraft_{applications,} was usedherein. The
modulus of_{elasticity}
andinitial
thickness(it
willbe
altered_{later)}
are givenin Table 2.3.
Property
Value (English
Units)
Value (SI
_{Units)}
Modulus
of_{Elasticity}
_{(E)}
10.4 E6
lb/in2 71.7E9N/m2Thickness
_{(r)}
1/32 in
7.9375 E4
mTable2.3 Initialproperties ofAL 7075T6skinonC5wing.
This
thesis assumes aconstant airfoil size_{along}
the_{wing}
with a chord equal to the m.a.c._{Hence,}
the pressuredistribution
remains constant_{along}
the wing.The
pressure coefficient_{(Cp)}
distribution
calculated_{by}
XFOIL
is
based
on a_{wing}
of unit _{width,} and an analysis ofa oneunit width section willbe
usedto representtheresults_{along}
therest ofthewing.
This
unitywidth sectionis broken up into many
smaller sections_{(elements),}
which8
Equationobtainedfrom: FundamentalsofFlight,
2nd
Ed,
Richard S.Shevell,
1989_{PrenticeHall,}
_{Inc.,}
pp.57,
Equation_{(3.9)}
9
AL7075T6properties obtainedfrom: MechanicsofMaterials.2nd
Ed.. Ferdinand Beer &E.Russel
are utilized
later
in
thefinite
element analysis.Figure 2.5
depicts
thedivision
of the_{wing}
ndividuol
Element
Analyzed Section
into
sections as well asthedivision
ofthesesectionsinto individual
elements.Figure2.5
Divisionof a_{wing into}sections andindividualelements
An underlying
assumptionin
thisthesisis
that the_{wing}
section willbe divided into
alarge
number of elements._{Hence,}
each skin elementdepicted
in Figure 2.5
canbe
approximatedasflat.
_{Using}
theflat
approximation and awidth_{(vv)}
equal toone unit allowsfor
thecalculationofthearea moment ofinertia
_{(I)}
/
= =(l
_{m)(7.9375}4
12
12
V A_{(2.11)}
Z=4.16745llm4
andthecrosssectional area
(A)
A
=_{wt} =(l
_{m)(7.9375}4_{m)}
A
=(7.93754m2)
(2.12)
for
each skinelement.Now
that theairfoilshave been
chosen, the coordinatedata
obtained, and all relevantparameters,
design
specifications, and approximationsdefined,
the next_{step is}
tobegin
theCHAPTER
3
DATA
SETUP
USING
MICROSOFTEXCEL 97 AND
VBA
3.1 THE NEED FOR MA CROS
Unfortunately,
a smallhurdle is
encountered evenbefore
_{beginning}
analysis.This
arises
from
thefact
thatthedownloaded
airfoil coordinatedata files (see Appendix
_{A)}
arenotin
aformat
that_{is readily}
usable_{by}
XFOIL (coordinates from
_{trailing}
edge to_{leading}
edgealong
the upper surface andfrom
_{leading}
edge to_{trailing}
edge_{along}
thelower
surface).Thinking
ahead tofuture
applications, not_{knowing}
if different
airfoils_{may be}
neededinstead,
and theinherent
tediousness of_{rearranging}
thedata
spurred the_{writing}
of afew
short
VBA
macrosin Excel
thatwouldimport
the_{data,}
sortthrough_{it,}
and preparefiles for
use
in XFOIL
andMATLAB.
The
macros are storedin
afile appropriately
namedworkhorse.xls, andthe
first
macrobegins
withthe_{clicking}
ofthisbutton
ImportandConvert
AllC5 Airfoils
3.2 PREPARING THE DATA FOR
ANALYSIS
The
macro thatis
runis
called_{ConvAll,}
andit
makes a call to another subroutinesimply
calledConv.
This
smaller subroutine sorts through the two columns ofdata for
each airfoil andreorganizesit into
the_{trailing}
edgeleading
edgetrailing
edgeformat.
The
processfollowed
_{by}
theConvAll
macrois
presentedin
Figure 3.1.
The
codefor Conv
Ask for Total Numberof
Airfoils
Save Compilation File
as"allfoils.txt"
Ask for Airfoil "j Coordinates File Name
Open File
Rearrange Coordinate Data
(ConvMacro)
Add Data toa
Separate Compilation File
Save Individual Data File for
XFOIL
Next Airfoil
Figure 3.1 Flow
chartfortheConvAllmacro
Upon
exitfrom ConvAll
the airfoil coordinatedata is found
reorganizedinto
5
separate
files
_{(xc5a.dat,}
xc5b.dat,xc5c.dat, xc5d.dat, xc5e.dat) that are_{easily imported into}
XFOIL,
and1 large
coordinatefile
_{(allfoils.txt)}
that_{is easily loaded}
_{by}
MATLAB.
At
thistime,
it
shouldbe
notedthatthiscompilationis only
usefulif
thesamenumberof coordinatesdescribes
all of the airfoils.This is due
toMATLAB's
_{inability}
toimport
matrices withdifferent length
columns.Files
withdifferent length
columns couldbe incorporated into
theMATLAB
functions in
this thesis with some alterationsin
the codeitself.
However,
thevariable number of airfoil coordinates
between files
wouldnotimpact
thebasis
ofthis_{thesis,}
because
each airfoilis
redividedinto
an equal number of equallength
skin _{elements,} atwhich pointtheoriginal airfoil coordinates areno
longer
needed.The
next_{step is}
to_{dig}
into
XFOIL
to obtain the pressure coefficientdistribution based
on the parametersdefined
CHAPTER
4
XFOIL
4.1
_{GENERAL DESCRIPTION}
_{OF THE PROGRAM}
Given below
is
abrief
overview of theXFOIL
program as wasincluded in
theXFOIL 6.8 User Primer
written_{by}
the program's creatorMark
_{Drela,}
_{Professor,}
MIT
Department
ofAeronautical
andAstronautical
Engineering.
The
_{theory}
references andprimary
problemformulations
are givenin Appendices C.l
andC.2.
General Description
XFOIL is an interactive program for the design and analysis of subsonic
isolated airfoils. It consists of a collection of menudriven routines
which perform various useful functions such as:
Viscous (or
inviscid)
analysis of an _{existing} airfoil, allowing* _{forced}
or free transition
*
transitional separation bubble(s)
* _{limited}
trailing edge separation
* _{lift}
and
_{drag}
predictions just beyond CLmax*
KarmanTsien compressibility correction
 _{Airfoil} _{design} _{and} _{redesign}
by
interactive specification ofa surface speed distribution via screen cursor or mouse. Two
such facilities are implemented.
*
FullInverse, based on a complexmapping formulation
* Mixed_{Inverse,} _{an} extension of XFOIL 's basic panel method Fullinverse allows multipoint design, while Mixedinverse allows
relatively strict geometry control over parts of the airfoil.
Airfoil redesign
_{by}
interactive specification ofnew geometric parameters such as *
new max thickness and/or camber
* _{new} _{LE} _{radius} *
new TE thickness
*
new camber line via geometry specification *
new camber line via
loading
change specification*
flap
deflection*
explicit contour geometry (via screen cursor)
Drag polar calculation with fixed or varying Reynolds and/or Mach numbers.
Writing and reading of airfoil geometry and polar save files
(Versaplotderivative plot package used)
XFOIL is best suited for use on a good workstation. A highend PC is also
effective, but must run Unix to support the XWindows graphics.The source
code of XFOIL is
_{Fortran,}
77. The plotlibrary
also uses a few C routines for the XWindows interface.4.2
_{USING XFOIL TO OBTAIN}
THE PRESSURE COEFFICIENT
DISTRIBUTION
As
mentionedin
thelast
paragraph oftheprogram_{description,}
XFOIL
is designed for
a
Unixbased
workstation.The
files
were_{easily}
copied andcompiledon such a_{system,} andupon
_{running}
_{the program,}theuser seesthe_{following}
_{startup}
menu:XFOIL Version 6.8
QUIT Exit program
.OPER Direct operating point(s)
.MDES Complex mapping design routine
.QDES Surface speed design routine
.GDES Geometry design routine
SAVE f Write airfoil to labeled coordinate file
PSAV f Write airfoil to plain coordinate file
ISAV f Write airfoil to ISES coordinate file MSAV f Write airfoil to MSES coordinate file
REVE Reverse writtenairfoil node ordering
LOAD f Read buffer airfoil from coordinate file
NORM Buffer airfoil normalization toggle
NACA i Set NACA 4,5digit airfoil and buffer airfoil
PANE Regenerate paneled airfoil from buffer airfoil .PPAR Show/change paneling
.PLOP Plotting options
WDEF Write xfoil.def file with current settings
RDEF Read xfoil.def file
NAME s Change airfoil name
For
each _{airfoil,} thefirst
_{thing}
that needs tobe done is
toload
thatdata file into
XFOIL.
Once
theLOAD
commandis
given and thefilename
specified,XFOIL
reads thedata
and performs severalinitial
calculationsbased
onthosecoordinates as seenbelow.
XFOIL c> load xc5a.dat
Labeled airfoil file. Name: c5a
Number of input coordinate points: 61 Counterclockwise ordering
LE _{x,y} = _{0.00012} _{0.00166}

Chord = _{1.00012} TE _{x,y} = _{1.00000} _{0.00000}j
Blunt _{trailing} edge. _{Gap} = _{0.00258}
Paneling parameters used. . . Number of panel nodes 140 Panel
_{bunching}
parameter 1.000 TE/LE paneldensity
ratio 0.150Refinedarea/LE panel
density
ratio 0.200 Top side refined area x/c limits 1.000 1.000 Bottom side refined area x/c limits 1.000 1.000XFOIL c>
Now
that the airfoildata has been
_{loaded,}
the next_{step is}
to switch to the_{operating}
menu
for
analysis.XFOIL C> OPER
<cr> Return to TOP LEVEL
! Redo last ALFA,CLI,CL, ASEQ, CSEQ,VELS
PACC Auto polar accumulation enable/disable toggle PADD Add point to polar save file and/or
dump
file PFIL f Specify new polar save and/ordump
filenamesVISC r Viscous/Inviscid toggle
.VPAR Change BL parameter(s) RE r Change Reynolds number MACH r Change Mach number
TYPE i Change type of Mach,Re variation with CL INIT BL initialization
_{flag}
toggleITER Change viscoussolution iteration limit
ALFA r Prescribe alpha
CLI r Prescribe inviscid CL
CL r Prescribe CL
CSEQ rrr Prescribe a sequence of CLs
GRID
_{Cp}
vs x grid _{overlay} togglePREF Reference
_{Cp}
_{overlay} enable/disable toggleFREF Reference CL,CD..
_{display}
enable/disable toggleCPX Plot
Cp
vs xCPXS Save _{Cp} vs x to a file
CPV Plot airfoil with pressure vectors (gee wiz)
VPLO BL variable plots
.ANNO Annotate plot
HARD _{Hardcopy} current plot
SIZE r Change plotobject size
CPMI r Change minimum
_{Cp}
axis annotationFMOM Calculate
_{flap}
hinge moment and forcesFNEW rr Set new
_{flap}
hinge pointVELS rr Calculate velocity components at a point
DUMP f Output Ue,Dstar,Theta,Cf vs _{s,x,y} to file
CPMN
CAVI
.OPERi c>
Report minimum _{Cp}
Auto cavitation accumulation enable/disable
Once
withinthismenu, thesequenceof commandsis
asfollows:
r)
. OPERi2:
) .OPERi3;
)
. OPERi4;
)
. OPERi5;
) .OPERi6]
I .OPERiz
1 . OPERi8:
) .OPERic> alfa
(Give
aninitial
angle of_{attack;} a= 2 _{was}chosen)
c> re
(Enter Reynolds Number from Chapter
_{2)}
c> mach
(Enter
Mach Number from Chapter
2)
o vise
(Switch
from inviscid
toviscous_{analysis)}
c> cl
(Enter lift
coefficientfrom Chapter
_{2)}
o cpx
(Plot
_{Cp}
vs._{x)}0 hard
(Create
a_{hardcopy}
PostScript
file
ofthe_{Cp}
vs.x_{plot)}c> cpxs
(Save
theactual_{Cp}
vs. xdata
toa_{file)}
2.D
1.5
P
1.0
0.5
0.0
0. 5
1. D
XFOIL
V6.B c5b
Ma 0.7700
Re  _{41.}
0010
a 3.8291 c, = _{0.8274}
CM
D.098
Cn O.DD790 L/O= _{1 10.33} Nrr _{9. DO}
Figure 4.1 Plot_{of}PressureCoefficient_{vs.}X _{xc5a.ps}
The final
command was notincluded
in
the original software.However
it
wasnecessary in
ordertoobtain a_{working data}
setthatcouldbe easily imported into MATLAB.
By
collecting bits
and pieces ofother subroutinesin
severallocations in
thesource _{code, this}additional
CPXS
subroutine was created_{by}
the author ofthis thesis to write the appropriateCp
vs. xdata
to afile.
The
actual codefor
this subroutineis
givenin Appendix C.3.
This
additionalcommand would
be
enteredasfollows
.OPERi c> CPXS xc5a.cpx
where xc5a.cpx
is
thefilename
where thedata
is
stored.Once
all ofthe airfoilshave
been
analyzed, the end result should
be five data files
(xc5a.cpx,
xcSb.cpx, xc5c.cpx, xc5d.cpx,xc5e.cpx),
displayed in Appendix
C.4,
andfive PostScript files
(xc5a.ps,
xc5b.ps, xc5c.ps,Note
that throughout the entireXFOIL
analysis, no_{scaling has been}
done
to theoriginal airfoil
data. All
calculationshave been
_{done using}
an airfoil ofthecorrectshape,but
with a unit chord.
In
terms of the pressure coefficientdistribution though,
the results arebased entirely
on the shape of_{the airfoil,} not the size. Therefore these_{Cp}
values_{apply}
toany scaling
ofthechord(xvalues).
Once
theXFOIL
analysisis
_{complete,} another quick macroin
Excel will prepare theCHAPTER
5
MICROSOFT
EXCEL 97 AND
VBA
REVISITED
5.1
GROUPCPX MACRO
At
this point of the variable_{wing geometry}
_{analysis,} there exists onelarge
_{file,}
allfoils.txt, which contains the airfoil coordinate
information
for
all _{the airfoils,} andfive
separate
files
that contain pressure coefficientdistributions.
_{Combining}
thesefive files into
one
large file
_{(allcpx.txt)}
proves much more efficientfor
file
management purposes._{Again,}
this
_{is providing}
allfiles
arethesame_{length,}
which_{they}
arein
this casebecause XFOIL
useda
default 140
panels around each airfoil.The
bulk
of theVBA
code_{necessary for}
the currentdata
consolidation taskhad
already been
createdin
theConvAll
macro mentioned earlier(see Chapter 3).
After
suitableperturbation, the new
GroupCpx
macrois
createdin
the workhorse.xlsfile.
GroupCpx is
invoked
with a clickofthebutton
Importand
_{Group}
AllPressure
Distributions
The Visual Basic
codefor
this macro and the variables usedin
that code canbe found in
Appendix
B. The basic
processthismacro worksthroughis
givenin Figure 5.1.
Define
Variables
AskforTotal Numberof
Airfoils
AskforCp
* DistributionFile 1
Name
OpenFile
Add Data toa Separate CompilationFile
Next Airfoil "
Save
CompilationFile as"allcpx.txt"
5.2
EXPLANATION
OF GROUPCPX MACRO
The
structure andfunction
ofthismacrois
_{practically identical}
to that oftheConvAll
macro
from Chapter
_{3. The only}
majordifferences
are:Instead
of_{importing}
airfoil_{coordinates, this}macroimports
_{Cp}
distributions
There is
no callto thesmallerConv
macroThe
singlefile
generatedfrom
thisis
calledallcpx.txtUpon
exitfrom
theGroupCpx
macro, allinformation regarding
the pressurecoefficient
_{(Cp)}
distribution is found
condensedin
to the allcpx.txtfile.
This
allcpx.txtfile,
andthe allfoils.txt airfoil coordinates
file
are nextimported into
MATLAB,
where_{they}
are usedin
a sequence ofMATLAB
R routines which work towards the end goal of_{determining}
an optimal nodal support
distribution
around the airfoils.A
sample of such a supportdistribution is depicted in Figure 2.1.
w
1
3Original Mapped Airfoil
Displacements
O Supports
e o o e o o
44 Supports Required
Avsrage
_{Spacing}
=0.39105mooo e
62 3 4 5 6
Actual
Chord,
m.a.c. =8.4818mCHAPTER
6
MATLAB
ANALYSIS
6.1 BRIEF
_{INTRODUCTION}
_{TO}
MATLABAs
mentioned_{briefly}
in Chapter
_{1,}
thebulk
of the_{programming}
for
this thesisis
accomplished
here.
MATLABallowsthreebasic
means of_{entering information:}
Command
line
Script
mfileFunction
mfileThe
commandline is
the most_{direct way}
of_{working in MATLAB.}
_{However,}
if
along
series of commands or_{many iterations}
are _{required,} then_{working solely}
at thecommand
line
will not suffice.This
thesis makes use ofthe command_{line only}
_{briefly}
tostarttheanalysis process andtomonitor
its
progress.The
script mfile_{is,}
_{by}
_{far,}
a moreefficient_{way}
of_{managing information. An}
mfileis simply
atextfile
writtenin
aformat
thatis
understood andrun_{by}
MATLAB.
A
scriptmfile
is
_{basically}
theequivalent of_{typing}
allthecommandsthatwouldbe
enteredin
orderatthecommand
line.
This
caters to therapiddetermination
of_{results,} and a greatdeal
ofeasein making
alterations._{Also,}
the script mfile providesfor
the use of certain commonprogramming
tools:for
loops,
while_{loops,}
if
statements, etc.,which when usedproperly,canbe
quite powerful.The
current analysis utilizes one main script mfile_{(Master.m)}
to controltheoverall process.
The
function
mfileis very
similarto thescript mfile exceptfor
thefact
thatfunction
mfiles are
_{typically}
genericin
_{design,}
and_{usually}
require specifictypes ofinput
toproducespecific outputs.
Another
majordifference between
the script mfile andthefunction
mfileis
the_{handling}
of variables.A
variable assignedin
ascriptmfileis
_{automatically}
assigned ablock
of_{memory}
andis
placedin
the active workspace, whereas the variables usedin
function
mfiles are clearedas soon asthefunction
is
complete.This
thesis employs severalfunction
mfiles_{(Parameters.m,}
ArcApprox.m,
MapToP.m,
GetAlpha.m,
Loading.m,
6.2
OVERVIEW
OF THE
ANALYSIS
PROCESS
_{(MASTER.M)}
The
entire analysis processis
governed_{by}
thescript mfileMaster.m. This is
themaincontroller of all the
input
and output to andfrom
theindividual functions.
Master.m it
requires no
input
and produces no official "output"._{However,}
sinceit is
a script_{file,}
thevariables that are assigned
here
are storedin
the active _{workspace,} which aidsin debugging.
Master.m
also serves as a goodlocation
to place_{plotting}
commands to_{graphically}
representthe results
from
each operation.For
the sake of_{simplicity,} the_{only plotting}
commands thatare
included in
thecode arethose atthe_{very}
endthat_{display}
thefinal
nodal support_{spacing}
aroundtheairfoil.
The
actual codefor
_{Master.m,}
andthevariablesutilizedherein
are givenin Appendix
D.l. In its
mostsimple_{form,}
theanalysis processtakes theform displayed in Figure 6.1.
Clear Previous
Results
Set Global Variables
SetParameters (Parameters.m)
Load Airfoil Coordinates (allfoils.txt)
Calculate Perimeter (ArcApprox.m)
Determine Perimeter Scale
Factor
LoadCp Distribution
(allcpx.txt)
Scale Coordinates and_{Cp}Distribution Accordingly
Redetermine Perimeter (ArcApprox.m)
Calculate ElementLength
MapAirfoil into Elements (MapToP.m)
DetermineLeading EdgeNode
Determine Global ElementOrientation
(GetAlpha.m)
EvaluateNodal Loading (Loading.m)
LowerSurface
Store Nodal Coordinateand
LoadingData
The
flow
chart givenin
_{Figure 6.1 is}
synonymous with the_{following}
_{process,} whichis
explained
in detail in
Section 6.3:
1)
Type
"Master"at the command
line.
This
_{immediately}
calls_{up}
andbegins
to run theMaster.m
scriptfile.
2)
The
Parameters
function,
and allthepertinent variables are assigned values.3)
The
allfoils.txtfile
is
_{loaded,}
andthetotalnumber of airfoilsis determined.
4)
A
quickdetermination
of the perimeter of the airfoilis determined
_{by}
the ArcApproxfunction. From
this_{function,}
a scalefactor
is
calculatedfor
each airfoil so that allhave
thesame perimeter.
5)
The
allcpx.fileis
_{loaded,}
and each pressure coefficientdistribution
and each airfoil arescaled.
6)
Another
pass around the airfoils with theArcApprox
function
creates several variablesused
_{in mapping}
_{(meshing)}
theairfoil.7)
Each
airfoilis
mappedinto
a set number of equallength
elements around theperimeterusing
theMapToP
function.
8)
The
orientation ofeach ofthese elements with respect to a global coordinate systemis
determined using
theGetAlpha
function.
9)
The
pressure coefficientdistribution is
fit
with a series of cubic polynomialsin
theLoading
function
as ameansof_{determining}
apressure valueateachelement node.10)
The
airfoils are scaledtoactual sizefor finite
elementanalysis.11)
The
finite
element analysisis
performed_{using}
theFEAJDpt, Opt,
andFEA
functions
starting from
theleading
edge ofthe airfoil and_{traveling}
toward the_{trailing}
edge,first
along
thelower
surface, then theupper.The FEAJDpt function
controls the optimization of support_{spacing}
across allfive
airfoils,suchthatall airfoilsaretaken
into
accountin
theanalysis.The Opt function
employs a zeroorder optimization method to maximize eachsupport