TECHNICAL
NOTE2751
A SIMPLEAPPROXIMATEMETHOD
FOR CALCULATING
SPANWISE
LIFT DISTRIBUTIONS
ANDAERODYNAMIC
INFLUENCE
COEFFICIENTS
AT SUBSONIC
SPEEDS
By FranklinW. Diederich
LangleyAeronauticalLaboratory
LangleyField, Va.
T
Washington
August1952
n<
NATIONAL
ADVISORY
COMMITTEE
FORAERONAU
TECHLIBiARY
KAFB,
NM
-
orlb5m
—
Illllllll
-
TECHNICAL
NOTE2751
A SIMPLEAPPROXIMATE
METHOD
FORCALCULATING
SPANWISE
IJ3?T
DISTRIBUTIONS
ANDAERODYNAMIC
INFLUEIK
E COEFFICIENTS
AT SUBS3NIC
SPEEDS
By Franklin
W. Diederich
SUMMARY
.
.
Several
approximate
e methodsforcalculating
liftdistributions
at
subsonic
speedsarecombined
andextended
to forma simplestep-by-step
procedure
forcalculating
symmetric
andantisymmetric
liftdistributions
forarbitrary
angle-of-attack
conditions
on sweptandunswept
wings.
Methods
of estimating
therequired
aerodynamic
characteristics
are
included,
butanyavailable
theoretical
or experimental
results
maybe
usedin several
stepsof theanalysis
to shorten
theworkandincrease
theaccuracy.Theextension
of themethodto thecalculation
of aerodynamicinfluence
coefficients
sndof spanwise
momentdistributions
is
indicated.
INTRODUCTION
~ empirical
method(ref.1) forcalculating
spanwise
liftdistributions
on unswept
wingshasbeenusedextensively
in thepast. In
modified
form(ref.2) ithasbeenapplied
to thecalculation
of spsnan improved
wiseliftdistributions
on sweptwings. Morerecently,
methodof calculating
theliftdistributions
dueto twisthasbeen
published
(ref.3).
.
In thepresent
papera limitation
of themethodof reference
3 as
applied
to antlsymmetric
twistsis indicated
andremoved.Themethods
of references
1 to 3 srethencombined
witheachotherandwiththe
procedure
results
of snother
investigation
(ref.4) intoa step-by-step
whichstartswiththelift-curve
slopeandtheadditional
liftdistributionandproceeds
withthecalculation
of symmetric
basicliftdistributions,
theliftdistribution
dueto roll,therolling-moment
coeffiThe
cientdueto roll,and,finally,
antisymnetric
liftdistributions.
-. —
NACATN 2731
2’
additional
lifl-distribution
ishereindefined
as theliftdistribution
forconstant
angleof attackacrossthespanwitha liftcoefficient
equalto 1. Thebasicliftdistribution
is defined
as theliftdistributionof a twisted
wingatizero(total)
lift. Meansof calculating
theaerodynamic
characteristics
requti~in themethodarecontained
in
thispaper,butif anyof thesecharacteristics
areknownfromother
sources
theymaybe incorporated
in theprocedure
witha result-t
savingin timeandimprovement
of accuracy.
Themethodof thispaperis derived
in appendix
A. It is outlined
andsomeexamples
of itsapplication
to various planformsandangle-of’=
attackconditions
arepresented
anddiscussed
in thebodyof thepaper.
Formulation
of themethodin matrixnotation
forthepurpose
of obtaining
aerodynamic
influence
coefficients
suitable
foraeroelastic
analyses
is
accomplished
in appendix
B of thepresent
papsgin them-er employed
as themethodof thepresent
papersupersedes
in reference
5. Inasnmch
thatof reference
2, on whichreference
5 is%ased,appendix
B ofithe
present
papersupersedes
reference
5.
.. .
.
r
.
.
Meansforestimating
thespanwise
distributions
of pitching
moments
(orlocalcenters
of pressure)
required
in an aeroelastic
analysis
are
indicated
in appendix
C.
SYMBOLS
A
aspectratio,b2/S
a
distance
of localaerodynamic
centerfromleading
edge,
fraction
of chord
an
coefficient
inFourierseriesfor 7
a
angleof attack,
radians
unlessspecified
otherwise
=
average
angleof attack,
radians
‘e
effective
angleof attack,
radians
ai
induced
angleof attack,
radians
dczdcz
control-effectiveness
parameter,
— —
db/ da
b
wingspan
..
.
3
NACATN 27’51
h~
coefficient
inFourierseriesfor a sin6’
c1,2,3
coefficients
foradditional
liftdistribution
cL
ILft/qs
wingliftcoefficient,
c&
winglift-tuneslope,perradianunlessspecified
otherwise
Rolling
moment/qSb
rolling-moment
coefficient,
damping-in-roll
coefficient,
-CZP
rolling-moment
coefficient
dueto rolling
perunithelix
angle(radians)
at tip
to planeof symmetry
chord,measured
parallel
.
average
chord, S/b
section
liftcoefficient
section
lift-curve
slope,perradianunlessspecified
otherwise
section
lift-curve
slopein incompressible
flow,per
radian
distance
fromleading
edgeto centerof pressure
dueto
aileron
deflection,
fraction
of chord
Y
Ccl/r
loading
coefficient,
liftdistribution
loading
coefficient
foradditional
?’b
loading
coefficient
for%asicliftdistribution
yd
liftdistribution
loading
coefficient
forunit-rolling
7r
liftdistribution
loading
coefficient
forresidual
71
loading
coefficient
forunitangleof attack
72
twist
loading
coefficient
forunitlinearsntisymmetric
.—
NACATN2751
control
deflection
in planeperpendicular
to hingeline,
radians
section
lift-curve-slope
ratio,CZ 2fi
a/
dimensionless
lateral
distance
fromwingroot,
Lateral
distance
bw
--22
...=
— ..n
-.
?.
.
F
A/v COS..A
plan-form
parameter,
f
addit-ional-lift-distribution
componentiue
to sweep
I
momentdueto
contribution
of functionf .to rolling
1
fy+2dy++
rolling,4
Jo
m“
.—
J
functionf
abscissa
of centroid
of srea.of
P
K
center
ccrcrection
factorforeffectmftaperon lateral
of pressure
insteadyroll
‘o
kO‘
k2
slope,
finite-span
correction
forwinglift-curve
equation
(7)
slopeaccording
finite-span
correction
forwinglift-curve
to slender-wing
theory
——
finite-span
correction
forbasicliftdistribution,
equation
(12)
..
..-
finite-span
correction
forliftdistribution
in roll,
.
equation
(15)
.
k3
finite-span
correction
forresidual
liftdistribution,
equation
(22)
kk
finite=span
correction
forTolling-moment
coefficient
due
to roll,equation
(17) ““
2
section
Mft
A
angleof sweepback
at quarter-chord
line
—
-—
—
.
●
5
N.ACA
TN 27’.51
f%
effective
angleof sweepback
in compressible
flow,
tanA
tan-l
m
taperratio,Tipchord/Root
chord
free-stream
Machnumber
4?
pressure
difference
between
upperandlowersurfaces
dynamic
pressure
wingarea
t
wingthickness
widthof fuselage
e
.
trigonometric
variable
corresponding
to N,
co+++
dimensionless
lateral
ordinate,Lateral
ordinate/Semispan
dimensionless
lateral
ordinate
of wingcenterof pressure
effective
lateral
center-of-pressure
location
of resultant.
loadcausing
rolling
momentdueto rolling
Subscripts:
II
two-dimensional
flow
III
three-dimensional
flow
DESCRIPTION
OF THEMETHOD
Symmetric
LiftDistributions
.
.
The liftdistribution
foranysymmetric
angle-of-attack
distributionmaybe considered
to consist
of twopartsja basicliftdistributionwithzerototalliftandan additional
liftdistribution.
Thebasic
liftdistribution
fora giventwistis defined
as thedistribution
for
thegivenwingwiththeangleof attackreduced
equally
at everypoint
untilthetotalliftis zero. Theadditional
liftdistribution
is defined
as thedistribution
whichthewingwouldcarry if it wereuntwisted
and
6
NACATN2751
theliftcoefficient
wereequalto 1.0. If theadditional
andbasic
liftdistributions
aredefined
in termsof theirloading
coefficients,
then
.
?
(1)
andtheliftdistribution
foranyvariation
of angleof attackor twist
acrossthespanmaybe writtenin theform
7 = Cj#i7a+ 7b
.
(2)
Thissection
Is concerned
withthedetermination
of thequantities
course,
be usedinstead
of thevaluesgivenherein.
Thewinglift-curve
slopemaybe obtained
froIu
Lift-curve
slope.thesection
lift-curve
slopeanda fin~te-sp.~
correction
as
=CzaCOSA ~-C%t
where c1~ and ~
(3)
—
aredetermined
as follows:
Thesection
lift-curve
slopeis takenfor thesection
perpendicular
to thequarter-chord
lineat a Machnumberequaldo M cosA; itmaybe
estimated
fromtherelation
(4)
.
.
7
NACATN 2751
cl
wheretheratio — a is givenin figure1 fromtheaverage
of the
c‘%
theoretical
dataof reference
6 forairfoils
of theNACA63A,64A,and
65Aaswellas the 63, 64, 65, and66 (withsubscripts)
seriesas-a
function
of theeffective
Machnuniber
M cosA forseveral
airfoil
thickness
ratios(perpendicular
to thequarter-chord
line).Forall
commonly
usedairfoil
sections
thelift-curve
slopein incompressible
flow Cz
is bown or maybe calculated.
%.
,,
—
—
——
Fromthevalueof Cla a ratio
maybe calculated
and,hence,a plan-form
parameter,
defined
in reference4 as
According
to reference
4, thevalueof ~
of thisplan-form
parameter
as
%=
‘4
r
maythen.hegivenin terms
(7)
F1+—+2
F2
.=
_—
ThefactorkO is plottedin figure2 as a function
of theplan-form
another
factor~’,
psrsmeterF. Forverylargeanglesof sweepback
alsoshownin figure2, shouldbe usedinstead
of ~.
*
.
Theadditional
liftdistribution
Additional
liftdistribution.maybe obtained
fromexperimental
dataor theoretical
calculations
(ref.7, forinst=ce)or maybe estimated
fromtherelation
derived
in
appendix
A,
—
8
NACATN 2751
7a = cl:+
c
C2:P+c3f
.
wherethesweep-correction
functionf is givenin figure3 andwhere
thecoefficients
Cl, C2,and C3 aregivegin figure4. Thefunction f depends
on an effective
angleof sweepback
Ae defined
by
Theelliptic
distribution
~
‘~~
is als~shom-infigure3 as the ““ ““‘“-””~
j
valueof thefunctionf for Ae.= O.
Unlikethevalueof ~a givenin reference
2, thatgivenby equation(8)reduces
to thecorrect
value(that-given
by low-aspect-ratio
theoryor slender-wing
theory)
whentheaspectratioisverylowandto
thecorrect
(strip-theory)
valuewhenthea=~ect”rat-io”is
veryhigh.
In general,
equation
(8)maybe expected
to-apply
to mostpractical
plan
formswithnearlystraight
quarter-chord
lines,exceptthosewithvery
largeeffective
anglesof sweepback,
saygreater
th”hn
about60°. For
suchwingsthe liftdistribution
maybe estimated
fromtheresults
of
morefullyin
slender-wing
theory(refs.8 and9),as explained
to slender-wing
theory,
theliftdistribution
i~”
appendix
A. According
likelyto be almost-elliptic
fora wingwitha plan-form
parameter
less
thanabout-2.5.
A wingwitha plfi-form
p-eter greater
thanabout2.5
In thetwo-dimensional
regionthe
canbe divided
intothree-regions.
liftdistribution
is proportional
to thechord,in thetipregionthe
liftdropsoffwithinfinite
slope,andin therootregiontheliftdrops
to about 2/n ofthe valueitwouldhaveif itwereproportional
to the chord.(Seefig.5.)
.
—
—
-—
—
—
--.
—
=
-—
,..
Thelateral
ordinate
of thecenterof pressure~ of theaddit-ional
liftdistribut-ion
or of anylifi-distribution
for-aconstant
angleof attackacross
‘thespan*S numerically
equalto themomentabout
theoriginof thefunction7a,sincetheareaunderthefunction?’a
is 1. Therefore,
.
.
y+=c~
1+2X +C2;
3(1+ k)
+CJ3
(9)
*
iM
.
.
9
I?ACA
TN 2751
where J is theabscissa
of thecentroid
of areaof thefunctionf
andis givenin figure6 as a function
of theeffective
angleof sweep.
back Ae. E@atio~(9)applies
to a linearly
tapered
wing. Fbrawing
whichis notlinearly
tapered,
theterm l+2L
of equation
(9)may
-
,,
,
3(1 + k)
be replaced
by
..-:
.-
Basicliftdistribution.As described
in appendix
A, thebasic
liftdistribution
maybe obtained
fromthelift-curve
slopeandthe
additional
liftdistribution
as follows:
Yb =klCk(a - =)7a
(lo)
__.—
where ~ is evaluated
fromtheintegral
.—
(11)
andthecoefficient
kl is obtained
fromtherelation
1
4
F1+—+2
F2
kl =
whichis plottedin figure2. Theintegration
indicated
in equation
(11)
maybe performed
graphically
or numerically
(bymeansof Simpson’s
rule,
forinstance).
10
NACATN 2’751
If therearediscontinuities
in thean@e-o&-attack
distribution,
theyshouldbe fairedbeforetheangle-of-at~ack
distribution
is usec-.
..
in equations
(10)and(11). Apparent-ly
thebestresults
areobtained,
whenthefairing
extends
about0.3semispan
on either
on theaverage,
sideof thediscontinuity
andpassesthrough
themidpoint
of thediscontinuity;
thefairedcurveshould
havethesameareaas theunfai=d
.
one.
*
*
Antisynmetric
LiftDistributions
Theliftdistribution
for_any
antisymmetric
twistmaybe considered
to consist
ofltwopart-s:
a rolling-type--distribution,
whichis thedistribution
forthegivenwingwitha linear‘Zihti.symmetric
twistof sufficientmagnitude
to havethesamerolling
momentas thetwistdistributionof interest,
anda residual
distributi~n,
whichisthedifference
between
therolling-type
andthetruedistribution
andwhich,by definition,haano rolling
moment.If theUnit-iolling
typeof distribution
is defined
by itsl;ading
coefficient
:
.
+ .—
.
cc1
~d~—
zc~d
where Cld is therolling-moment
coefficient
of thedistribution
fora
linearantisymmetric
twistwithunitangleattheti~(’and
is,hence,
coefficient
thenegative
of theconventionally
defined
rolling-moment
dueto rollingCz , andif theresidual
distribution
is defined
by its
P)
loading
coefficient
—
....
thenanyantisymmetric
lift-distribution
maybe writtenas
—
7 = CZd~7d+ 7r
(13)
:q
.
—.
11
NACATN 2751
where ~ is thetipanglerequired
fora linearantisymmetric
distributionwiththesamerolling
momentas thedistribution
oi’”
interest.
Thissectionis concerned
withthecalculation
of’ ?’d
,
, Cld) cLe
and Yr. If thevaluesof 7d and CZd areknowntheymay,of course,
be usedinstead
of thevaluespresented
herein.
.—
Unit-rolling
distribution.The liftdistribution
72 fora linear
antisymmetric
twistwithunitangleat thetipmaybe obtained
from
reference
10 or 11 or from”the
relation
,
‘–
—
where
k2
4
F1+—+2
~2
{
(15)
—
distriis derivedin appendix
A sndshownin figure2. Theunit-rolling
by thedamping-in-roll
coeffibution 7d is equalto 72 divided
cient C?,d.
._
—
Damping-in-roll
coefficient.Thedamp’ing-in-roll
coefficient
may
be obtained
fromreference
10 or 11 or,as in reference
4, fromthe
relation
. .-.
(16)
12
NACATN 2751
where k4 is a factordefined.
by
k4=
, F
...
I-6
Fl+—
F2+4
i
(17)
“
u
.
,
.
whichis-plotted
in figure2. ThefactorK is a correction
fortaper
effects
on thelateral
centerof pressure
introduced
in reference
12 and
.
~L1
is equalto twicethefactor P usediri””that
paper.Another
expresb/2
-...
sionfor..-C2maybe obtained
&om themethodof thepresent
paper:
d
(18)
where Cl) C2,and C3 srethefactors
usedin equation
(8)and I is
themomentof inertia
of thefunctionf CQefined
as
—
.
-— .—-.
(19)
Thevalueof I IS givenin figure6. If thewingdoesnothavea
mustbe substituteti
-far-.
lineartapertheexpression
4.~l& y$+2dy+
Jo c
.
2 —.
1 +3A
theterm 31+X
A comparison
of equations
(16) and(18)reveals
thatthefactorK
21+3kc
is equalto theexpression—
/
31+X
~ -t-C2 +–IC3.
.
13
NACATN 2751
?L,
Thetwotapercorrections
K and ~
b/2
—
y~ ,
●
.
arecompared
in figureT(a)
forunswept
wings sincethecorrection
— P doesnot applyto swept
(
b/2
wings andexcellent
agreement
is seento exist,exceptforplan-form
)
~L!
parameters
lessthan6, forwhichthevaluesof — J? givenin referb/2
ence12 aresomewhat
uncertain.FigureT(b)showstheeffectof sweep
on thetapercorrection
factorusedin equation
(18).Thisfigure
serves to explain
thestatement
madein reference
12 thatexperimental
—
—
FL1
.
.
evidence
indicates
thattheeffectof sweepon thefactor— P is
b/2
likelyto be small,a statement
whichis difficult
to reconcile
with
thegreatdeviation
of theadditional
liftdistributions
of swept
(particulsrly
swept
forwsrd)
wingsfromelliptical
andalsoof their
linearantisymnetric
liftdistributions
fromthatof an elliptic
wing.
As maybe seenfromfigureT(b),however,
thefactorK, whichis twice
..
thefactor— P andis obtained
fromspanwise
liftdistributions,
is
b/2
indeedsubstantially
unaffected
by sweepforanglesof sweepback
between
0° andabout350, whicharetheonesforwhichmostof theaforementioned
experimental
evidence
wasobtained;
however,
forsweptforward
wingsand
forhighlysweptback
wings,sweepdoeshavetheexpected
effecton K.
...:——
—
Inasmuch
as thecorrection
of equation
(18)permitssweepto be
tslcen
intoaccount,
whereasthatof equation
(16)doesnot,theuseof
equation
(18)appears
to be preferable
forsweptwings. Forunswept
wingsbothcorrections
givealmostidentical
results.
Residual
liftdistribution.Theresidual
liftdistribution
maybe
obtained,
as shownin appendix
A, fromtherelation
yr = ‘Sc,d($
- ae)yd
wheretheeffective
tipangle ~
.
.
is defined
by
(20)
.. ..— —
14
NACATN 2751
●
(21)
or
✎
andwherethefactork~ is definedby ‘
k3 .
r
Fl+—+64
9
❞
(22)
✎
~
✍✍✍✍
,Anydiscontinuities
in theangle-of-attack
distribution
mustbe
fairedbeforethedistribution
is usedinequation
(20)or (21).A
convenient
procedure
forfairing
thisdistribution
is to plotrthe
ratio u/fi overthespanandto fairit-&ssuggested
fordiscontinuous
Thefaireddistribution
maythenbe used
symmetric
distributions.
directly
in thesecondformof’equation
(21).
ILLUSTRATIVE
EXAMPLES
Themethodpresented
in thepreceding
sections
hasbeenapplied
to
thecalculation
of liftdistributions
fora variety
of planformsin
orderto illustrate
certain
trendsaswell_as
.
to
compare
theresults
.-.
withthoseof othermethods.
Additional
liftdistributions
werecalculated
forsixwingsof
different
planformsin incompressible
flo~; Theresults
areshownin
anglesof.-sweepback,
taperratios)
ad
figure8. Theaspectratios,
plan-form
factors
of thesewingsaregiven.in
table1. In thecalculationof theplan-form
factors
a section
lift-curve
slopeof 21rwas
constantsCl, C2,and C3>
assumed.Alsogivenin tablel.are.the
calculated
by
whichweretakenfromfigure4. Theliftdistributions
themethodof reference
13 (with15pointsonthespan)and,intwo
calculated
by themethodof reference
l!t
cases,thelift-distributions
(with126lifting
pointson thespan)area@o shownin figure8. There
is goodagreement
between
theapproximate
liftdistributions
given‘by
themethodof thispaperandthosecalculated
by thetheoretical
methods.
✎✍
✎✍
✝
✎
✎
w
-i
NACATN2751
.
.
.
.
15
Thelift-curve
slopesandlateral
centers
of pressure
of theadditionalliftdistributions
of thesixwingplanformsconsidered
in figure8 aregivenin table1, as calculated
fromequations
(3)and(9)of
thepresent
paperandas calculated
by themethodof reference
13 (with
15 pointson thespan).Thevaluesof ~ required
in equation
(3)
forcalculating
C~ wereobtained
fromfigure2 andarealsoincluded
in table1. Again,thereis goodagreement
of theapproximate
values
of C% and ~ obtained
by themethodof thispaperwiththetheoretical
values.
Alsocalculated
forplanforms1 and2 werebasicliftdistributions
dueto a unit arabolic
twistanddueto a unitdeflection
of a halfsemispau
flapr%5 cosA = 1). Theresults
of thesecalculations
are
givenin figures
9 and10. Fortheparabolic
twist G = y+2,equation(11)yields,
in conjunction
withthefunctions7aSW
taken
FCL
fromfigure8, a valueof Z Thefunctiona - ?i is shownby the
firstcurvein figures
9(a)and10(a).Thevaluesof kl weretaken
fromfigure2 andare0.57and0.66forplanforms1 end2, respectively.
Thesevaluesof ~ and kl,theapproximate
valuesof C% givenin
table1, andtheadditional
liftdistributions
givenin figure8 were
usedin equation
(10)to calculate
thebasicliftdistributions
ccl
(shownby thesecondcurvein figs.9(a)and10(a)).
7b=—
~b
()
Thebasicdistributions
givenby thesecondcurveof figures9(a)
and10(a)andtheadditional
distributions
givenin figure8 maybe
combined
withthecalculated
valuesof ZZ ad C~ t; givetheloading
forthecasewherethewinghas zero angle of attackat therootandis
twisted
parabolically
so as to havea unitangleat thetip. Thiscase
is illustrated
by thelastcurveof figures
9(a)and10(a).Theseparticular
results
arecompared
withtheliftdistributions
whichwouldbe
obtained
fromthemethodof reference
13. Theagreement
isbetterin
thiscaseforph” form1 thanforplanform2.
.
.
Fortheeffective
twistdueto flaps,theangle-of-attack
distributions
areshownin thefirstpartof figures
9(b)and10(b).The
valuesof 75 forthefaireddistribution
werecalculated
fromequation(11)andare0.26forplanform1 and0.2Tforplanform2. The
(10)to calculate-the
basic
Cme %’aired- ~ wasusedin equation
liftdistributions
(shownin thecenterpartof figs.9(b)and10(b))
andthetotalMft distributions
(shownin therightpartof figs.9(b)
and10(b))as fortheparabolic-twist
case. Thevaluesof C~ and kl
. .—
.
16
NACATN 2751
—
usedto calculate
thebasicliftdueto parabolic
twistwerealsoused
in thiscase. Theliftdistribution
calcuiat-ed
bjthe methodof reference13with15pointson thespan(a correction
forthediscontinuity
in thesngle-of-attack
distribution
was included)
is compared
withthe
liftdistribution
calculated
by themethodof t-his
paper. Again,
betteragreement
of theapproximate
withthetheoretical
distribution
‘;
is obtained
forplanform1 thanforplanform2.
.-.
:-.
.-.
In arderto illustrate
thetreatment
of antisymmetric
angle-of–attackdistributions,
calculations
havebeenmadefor-p-lan
form2 of
antisymmetric
liftdistributions
dueto a linearsntisymnetrti_twist
(damping
in r;ll)anddueto-aunitdeflection
of half-semispan
outboardailerons
8
COSA
=
1).
Forthelinear-twist
casethedistri(’%
bution72 was calculated
fromequation
(1~),with ~ =0.80 obtained
of damping
in roll C2 = 0.33 wasthen
fromfigure2. Thecoefficient
d
calculated
fromequation
(18),inwhichthevalues
k4
= 0.63 obtained
-.fromfigure2 snd I = 1.15 obtained
fromfigure”6
wereused. Theunitrolling
distribut-imn
wasthenobtained
by dividing72 by thisvalue
of Cld. Thedistribution
is shownin figuren(a). Alsoshownin figure.ll(a)
is theunit-rolling
distribution
calculated
by themethodof
reference
13. T’he
coefficient
of damping
in roll~alculated
by that
methodis Cld= 0.38. Theunit-rolling
distribution
andthecoefficient
..of damping
in rollobtained
by theapproximate
methodof thepresent
paperandby thetheoretical
methodof re-ce
13“arein goodagreement”:
—
-.
-Theftmctiona/y+ fortheaileron-type
angle-of-attack
distributionis shownat thetopof figuren(b);——
thefairing
extends
0.3semispanon eithersideof thediscontinuity.
_Thevalueof ~ “was
obtained
-.
fromthesecondformof equation
(21),withthe-approximately
calculated
..
cc~
f~ction ~d ~—
givenin figuren(a), andis Ue = 1,12.The
ZcZd
M ““valueof ks w= obtained
fromfigure2 andis k3 = 0.68. Withthese
valuesfor k3 and ~, thefunctionYd givenIn figuren(a), the
previously
calculated
valueof Cl=,andthefairedfunctiona/~
was
givenin figuren(b), theresidua~
liftdistribution
Yrs &
Cr
calculated
fromequation
(20) andis showqinthemiddlepofiion
of figuren(b). The liftdistribut-ion
dueto titleron
deflection
wasthen
calculated
fromequation
(13)andis sho~—”in
pafiof fig.— thebotto’m
ure11(%).Alsogivenis theliftdistribution
calculated
by themethod
of reference
13with15pointson thespa~andwitha correction
forthe
Th~~-wo
liftdistributions
arein
discontinuity
in thesngleof attack.”
goodagreement.
()
. —-_
=
—
-i
,s
:
—
.—
“
17
NACATN 2751
DISCUSSION
Inasmuch
as experimental
determination
of pressure
distributions
is a verytedious
process,
complete
information
concerning
thespsnwise
liftandmomentdistributions
corresponding
to allangle-of-attack
conHence,analytical
methgdsforcalculating
ditions
is rarelyavailable.
thesedistributions
areusedalmostuniversally
in designing
airplanes.
Mostof thesemethodsaretheoretical
(forinstance,
forsubsonic
speeds
seerefs.9 and13 tO 16). Thereliability
of thesemethodsiswell
established;
forwingswithoutfuselage,
nacelles,
tiptanks,or
external
stores,
theygenerally
furnish
verygoodapproximations
to the
truedistributions,
provided
theangleof attack,
airfoil
thickness,
mostof themarerelatively
andMachnumberarenottoolarge.However,
timeconsuming.In orderto overcome
thisdeficiency,
calculations
for
manyplanformshavebeenmadeby someof thesetheoretical
methods
(refs.7, 10,and11,forinstance),
so thatno further
calculations
needbe madefortheangle-of-attack
conditions
considered
in thesecalAnother
wayof avoiding
thetedious
calculations
required
culations.
forthetheoretical
methodsis to use empirical
methodssuchas those
of references
1 to 5, aswellas thatof thepresentpaper,whichconsistsin a combination
of thesemethods.
.
.
Compared
withthetheoretical
methods,
an empirical
methodfor
calculating
liftdistributions
hasthedisadvantage
of beingless
accurate
on theaverage,
although
theaccuracy
of theresultsof the
disadvantage
of empirical
empirical
methodis oftenadequate.Another
methods
as compared
withtheoretical
methodsis thefollowing:Once
theaccuracy
of a theoretical
methodhasbeenestablished
%y comparison
withknownresults
in a fewcases,confidence
canbe placedin the
results
of thismethodforwidelydifferent
cases. Whenan empirical
methodisused,thedegreeof confidence
thatcanbe placedin its
results
for cases widelydifferent
fromthoseforwhichitsaccuracy
hasbeenestablished
is notnearlyso high. As faras themethodOr
thepresent
paperis concerned,
thisli@tationimplies
thatforthe
purpose
of calculating
lift-curve
slopesandadditional
liftdistributionsthemethodis restricted
to planformswithnearlystraight
quarter-chord
linesandeffective
anglesof sweepback
whicharenot
if theseaerodynamic
characteristics
greater
than,say600. However,
havebeenobtained
experimentally
or by meansof accurate
theoretical
methods,
otherliftdistributions
andaerodynamic
parameters
canbe
calculated
by thismethodfora muchwidervarietyof planformsby
usingtheknownlift-curve
slopesndadditional
liftdistribution;
also,
theaccuracy
of thefinalresultscanbe improved
in thismannerover
thatattainable
by starting
outwithvaluesof thelift-curve
slope
andadditional
liftdistribution
obtained
by themethodof thispaper.
.-
-.
—
-.
—
—.
18
NACATN 2751
—
ThematrixschemeOF appendix
B andthemethodforcalculating
the
chordwise
centerof pressure
givenin appendix
C makethemethodof’this
paperreadily
applicable
to theaeroelasti~_~alysis
18.
— of reference
Forwingswithstraight
quarter-chordMnes,
moderate
anglesof
sweepback,
andmoderate
aspectratiostheliftdistributions
calculated
by themethodof thepresent
paperhavebeencompared
withthosecalculated
by theoretical
methods
,incofiecti&i
withtheillustrative
examples.Theagreement
between
theliftdistributions
andassociated
aerodynamic
para&terscalculated
by themethodof thepresent–paper
andthosecalculated
by theoretical
methods
hasbeennotedto be good.
Inasmuch
as theapproximate
methodof-this.paper
isbasedon liftinglinereasoning,
andinasmuch
as lift-ing-lirie
theoryis validonlyfor
wingsof highaspectratio,a comparison
of theresultEoftheapproximatemethodwithresults
obtained
by morerigorous
theoryforwingsof
verylowaspectratiomaybe of interest.The liftdistributions
for
fromreference
15, provided
thewingsdo not
suchwingscanbe obtained
havereentrant
trailing
edges.
Fora wingof verylowaspectratioandwitha parabolic
twist
~ . @2; thus 7a = ~~~z,
C~ = ~ A~”-andkl“=3 (seefig.2)*
>
.
.
.-
_-
‘
also, ~, as obtained
fromequation
(11),is 1/4. Hence,fromequations(10)and(2),
7b =
2+*2
-;)F=
and
=A ~sine+ ~ stn39
)
6.
(
-.
intablelof ref(wherecos9 = y+),whichis alsotheresultgiven,
erence15. Thatthisperfect
agreement
of thetworesults
is fortuitous
maybe seenfromthefactthat-for
a linem symmetric
twist u = IY*],
.
*
19
NACATN 2751
and
=:++;)-==
whereas,
according
to reference
15,
7=:
co.% log tan(; + :) + sin Ej
[
m.
.
“
“forthiscase. Thetwo difitributions
areshe%.infigure12(a)andare
,in goodagreement.
Thecoefficient
of damping
— — in rollof a wingof verylowaspect
(18).‘Fromfi-&e 2
ratiocanbe obtained
approximately
fromequation
andequation
(17)the factork4 maybe seento be asymptotically
equalto F/8 as F approaches
O; for F= O thefactorsCl and C3
areO andthefactor C2 is 1 (seefig.4),so thatas a resultof the
definition
of F,
Cld=$A
15. Thefactork2 = 0.5 for F = O, so that
as givenin reference
_. —
as givenin reference
15.
NACATN 2TT1
20
—
=
Theunit-rol.limg
distribution
of-awingof verylowaspectratio_
is then
.
72
7~=—
Cld
—
—,=
..
antisynmetric
twist;
andthefactork3 iS 2/2. Fora parabolic
on
theright
wingand
a
=
-y~
on
theleftwing.
Theeffeca=F2
32
tivetipangleis ae = —, andtheresidual
liftdistribution
on the
15X
rightwingis
—.
—.
>1
.——
.“
so that–the
liftdistribution
on therightwingis
+ —8 sin26 + ~ sin 3e
=A ~ sin.tl
1511
)
(
whereas,
according
tu reference
15,
1
43
+2
7 =A— coselogtan~+~
4 2
3fl‘in20
()
[ 3Yt
.
forthiscase. Thesetwodistributions
areshownin figure12(b)and
arein goodagreement.
.
N.Ac~
m 2731
21
Forthecubicantisymmetric
twist a = y+3,theapproximate
method
of thepresent
paperhappens
to givethesameresultas reference
15:
7 =A1T sin2@+~sin4@
16
(
)
In general,
therefore,
themethodof thepresent
papergivesresults
forwingsof verylowaspectratiowhich.are
in excellent
agreement
with
theresults
furnished
by low-aspect-ratio
theory.
.
.
.
Themethodof thispaperhascertain
advantages
overtheoretical
methodsapartfromthefactthatthetimerequired
foran analysis
by
theempirical
methodis onlya fraction
of thetimerequired
foran
analysis
by meansof oneof thetheoretical
methods
andis indeedcomparableto thetimerequired
to obtaina desired
liftdistribution
by
interpolating
betweenthosefurnished
in references
7, 10,and11. The
methodof thepresent
paperis ratherflexible,
so thatit maybe used
in somecaseswherepresenttheoretical
methods
areinapplicable.
For
instance,
in thecaseof a wingwitha fuselage,
nacelle,
or tiptank,
neither
themethodof thispapernoranygenerally
available
theoretical
methodcangivethelift-curve
slopeor theadditional
liftdistribution.
However,
if thisinformation
hasbeenobtained
experimentally,
themethod
of thispapercangivetheliftdistribution
foramysymmetric
or antisymmetric
twistwithan accuracy
sufficient
formanypurposes.(Ifthe
winglift-curve
slopeis different
fromthelift-curve
slopef,or
thewing
forthevalue
alone,thefactorskl) k2j k3,and k4 canbe obtained
of theplan-form
parameter
whichcorresponds
to thegivenlift-curve
sloperatherthanforthevalueof thetrueplan-form
parameter.)
Anotheradvantage
of theempirical
methodoversometheoretical
methodsis theeasewithwhichit lendsitselfto thecalculation
of
aerodynamic
influence
coefficients,
whicharerequired
in an aeroelastic
analysis.(Seeappendix
B.) Also,fora wingin transonic
flowthe
theoretical
methodsusedwithlinearized-theory
corrections
forcompressibility
effects
tendto giveunsatisfactory
results,
as pointed
outin reference
6. Theempirical
methodof thepresent
paper,however,
is capable
of furnishing
usefulresultsin certain
casesof transonic
flowbecauseit is basedon an interpretation
of thethree-dimensional
Glauert-Prandtl
rulewhichpermits
theuseof knawnsection
characteristicsandreconciles
thatrulewithsimplesweeptheory.(Seeref.4.)
Forunswept
wingsof lowaspectratioandforsweptback
wingswhichare
slenderin thesenseof reference
9, thatis,wingswhichhavea small
I
valueof Ail - Mz,equation
3 haabeenfoundto yielda goodapproximationto theobserved
valuesof thelift-curve
slopein someinstances
22
—
NACATN 2751
of trans~nic
flowup to and%eyonda Machnumberof 1, provided
the
wingshavethinsections
andar>a-tlowanglesof attack.
— -—_
..-_
.—
CONCLUDING
REMARKS
—
.
A relatively
simple
methcdof calculating
spanwise
liftdistributionsforanyangle-of-attack
condition
hasbeenderived
by combining
.—Themethod
is such
andextending
several
previous
theoret-ical
methods.
thattheworkcanbe shortened
andtheresu-lts
improved
atvariousstages
of thecomputations
by introducing
experimental
or theoretical
valuesof
certain
aerodynamic
parameters
whenever
the-y
areavailable.In addition,
themethodlendsitselfreadily
to formulation
of theload-distribution
problem”ln
termsof aerodynamic
influence
coefficients. “
Theresults
obtained
by thepresent
~thod compare
favorably
with
thoseobtained
by moretime-consuming
thed?ies.
LangleyAeronautical
Laboratory
National
Adviso~Committee
forAerotiutics
Langley
Fieid,Va.j~Y l) 195~-
___
.
----
23
NACATN 2751
APPENDIX
A
DERIVATION
OF THEMETHOD
Additional
LiftDistribution
In themethodof reference
1 theadditional
liftdistribution
78
forunswept
wingsis givenby theaverage
of thewingplanformandan
ellipse
of equalarea,so that
(Al)
relation
forsweptwingsis givenin reference
2 as
The corresponding
.
()
lcmo+f
Ya‘52=.=
(A2)
—
—
of reference
2, thesection
liftwhere ~ and ‘~ are,in thenotation
curveslopeandtheaverage
section
lift-curv=
slope,respectively,
and
foreachsweep.This
where f is a function
of y* whichis different
functionf fulfills
a purposesimilar
to thatof thefunctionf used
in thepresent
paperbutdiffers
fromit numerically.
Thatbothof theserelations
areinapplicable
to wingsof either
veryhighor verylowaspectratiomaybe seenfromthefactthatfor
veryhighaspectratiostheadditional
liftdistribution
is equalto
thechorddistribution
c/F,whereasforverylowaspectratiosit is
the
defined
by theellipse$~G2
(seerefs.8 and15),provided
to cover
wingdoesnothavea reentrant
trailing
edge. Consequently,
theentirerangeof aspectratiostheequation
fortheadditional
lift
distribution
shouldhavetermswhichinvolve
thechordandtheellipse
anda function
whichvarieswithsweep;theextentto whicheachof these
threefunctions
entersintotheliftdistribution
shouldvarywithaspect
ratio.Consequently,
thefollowing
relation
suggests
itself:
.
.
Ya =C1:+C2$
c
R+
c,’
(A3)
.—
—
—
24
NACATN 2751
.
.
where f is a newfunct.ion
whichvarieswithsweepandis givenin fig.ure 3, andwherethe Cls varywithaspectratiojin
sucha waythat..—
Cl
.is 1 forinfinite
aspectratio, !2 is 1 kforzeroaspectratio,andthe—”
sumof allthree C’s is always1. A set..of-C1s
obtained
by analyzing
‘- ‘“an
—-+
=
theresults
of thecalculations
by themet—hod
of reference
3 fora large
variety
of plan.fo~ ispresented
in figfie
Theplan-foim
.
-.-.
—4.
....G__
. ..-__ parameter
4) waschosenas theabscissa
””rather
@an =
F ‘~
(proposed
ipref.--”
.
.
—
theaspectrati~proper
in orderto takes~eepandcompressibility
effects-intoaccount
in a reamersimilar
to that~hployed
irireference
4 for-the —
winglift-curve
slope.
.
In general,
equation
(A3)maybe expe~ted
to applyto mostpractical
pla~formswithnearlystraight
quarter-chord
lines,exceptthosewith
verylargeeffective
anglesof-sweepback
.Ae,and-tofurnish
moreaccurate
‘
results
thando theapproximat=formulas
of references
1 and2, inasmuch‘as equation
(A3),unlikethoseformulas,
takesaspectratiointoaccount;
Wingswithverylargeanglesof sweepback
canbe analyzed
by slenderwingtheory.(Seerefs.9 and16.) The~ypeof wingsanalyzed
in reference9 is characterized
by a constant
chordwhichis filleted
generously
at thetrailing
edgeto theextentthat-the
rootchordis st/2timesthe
chordfarawayfromtheroot. As maybe‘deduced,
fromreference
9, this
wingcanbe divided
intothreeregions
in themannerindicated
in figure5. Thepressure
distribution
in theQartof therootregionforward
of theapexof-thetrailing
edgeis thes.&eas “itwouldbe ifthewing
consisted
of thatpartof therootregion_
:only,
andtherefore
it canbe
obtained
fromthedelta-w@gtheory(ref.8). The”
spa-riwise
liftdistributionis constant
andequalin msgnitud&-”to
tha&inthetwo-dimensional
region.In thetwo-dimensional
regionthechordwise
pressure
distribu--tionisthesameas in two-dimensionaQow;
t~espanwise
liftdistributionis constmt-ndis givenby
‘-
“-:
n_=
J---
.
—
—.
—..
._
—.
~=
In thetipregionthepressure
is zerobehindtheleading
edgeof the
tipchord,andthespanwise
liftdistribution
dropsoffto zeroifian
wingtheliftne~r
approximately
elliptic
fashion.Foran unfilleted
he proportional
to thechord,
therootmustbe less;if it is assumed””to
theliftat therootis 2/fitimesthe~~fttithe
two-dimensional
——
region.
-.
--
——
.
---—
As shownin reference
9, thelift-curve
slopedepends
on theparameter A/cotA, whichforlargeanglesof..~weep
iS approximately
equal _ ~“ ‘~-.
—
M
WA @l/,2,1gfL
..
.
25
to A/cosA. Tw.%YR.lW%z’@Jl
&UrLrL
m w3xm@’aw ‘a@aI”+tfi
Viak!I’
@&TF:-)
formparameterF, becausefora wing
— witha largeangleof sweepback
theflowinplanesperpendicular
to theleading
edgeis almostincompressible
at allsubsonic
speeds,
sathatthesection
lift-curve
slope
,t$j
’approximately
2fi.Theva@&@n Of=~
with A/cotA givenin
figure7 of reference
9 is therefore
‘&roducedin figure2 of the
present
paperas a variation
of ~’ with F, where ~’ represents
,:tihe:ratio:zfl~r>
a~bfilfi~s~arsle~~fi:ql%n~
“fi~.~fi~d
~fl~flo~sg~
‘*”T f
-;:.~n
~~s~:ts-lo-.is[%
yngm’tnoitw~irstaib
tlilaf~YYCMY miJ-@ti211OJ i10~tEj131
9tf3@ r19vl~ai n.::hdiz~
BasicLiftDistribution
In reference
l.thebasiclif<~istribution
Yb iS givenas One- ..
ha~ of thestrip-theory
vak@,-c@~nSo= y
.
—
.
——
.--F.E
.
=
___
—
(A5)
_
—
.._—
(A6)
Bothresults
wer&bb%Ve5fhk&”~~%’~&e~~
e’m!&&%!@&syi~~
%+e?>~i+
‘*
ence17,however,
indicates
thepossibility
of a morerational
approach,
in thatit provesby lifting-line
theorythatan average
angleof
attackZ defined
by
-.,.,- , ,-J3
.,,
..
— .=
—
.
(A7)
.
1
26
NACATN 2751
canbe obtained
foranysymmetric
twistdistribution
from
.-
(A8)
Thereasoning
behindthisequation
is somewhat
as follows:According
to lifting-line
theorytheliftdistribution
foranyangle-of-attack
distribution
is givenby therelation
Y=
%+(a
-d
.
—
(A9)
andthedistribution
fora unitangleof attackacrossthespanby
.
(A1O)
where ai and ail aretheinduced
anglesof attackappropriate
to
theliftdistributions
Y and 71,respectively.
- Theseequations
may
be combined
to yield
(
7 = 71a+ 7ail- Ylai
)
so thattheliftcoefficient
forthegiventwistis
.
.
1
=
1
o
1
ylady*+
J(
o
Yail- Ylaidy+
)
(All)
27
NACATN 2751
Thesecondintegral
maybe shownto be zeroby setting
-.
Y=
?i-
an sinnL3
theory,
so that,by lifting-line
(where(3s Cos-ly+)
.
z
ai=—
1 &l
sinne
nanl
sine
Thenboththeintegral
of yail andthatof 2’laiareequalto
2-(
E
angle-of-attack
distributions,
so
‘ananl forallsymmetric
xl
thatthekcdifference
is zero. Consequently,
(A12)
by dividing71
andsincetheadditional
liftdistribution
may be obtained
additional
a
=
1
forthe
by itsliftcoefficient
(whichis C&, since
liftdistribution)
equation
(A8)follows
fromequation
(A12).
Thepreceding
development
suggests
an equation
forthebasiclift
distribution
in theformof equation
(A6),exceptthatthefactor c/5
mustbe replaced
with CC71FCLin orderthatthetotalliftof the
(A13)
““-
.
“-
——
where kl is a constant
whichdepends
on t~eplan-form
parameterF
.
.
andtakestheplaceof thef@ctQu,
j...”<n”~uationion
.—(Ji6)-.
.--.i
—Thevalueof kl hasbeenderived
in.reference
~ by meansof series
..—.
expansions.
In thederivation
of reference
3 it wasrequired
thatthe
‘“-liftsandrolltig
m~-a,.of!ti&imffifi%tiY~bbti%w
&tv~.@re
_q&tiw:(.A13)
be equalto thevaluesgivenby lifting-line
theory.In thefoll~wing
paragraphs
thes~e resultis obtained
in a slightly
different
manner
whichfacilitates
theext nsionof theme rodto antisyninetric
lift-. .-_.,
$.nff~a
1 --distributions.
“
—
#fII
-,~ m“= ~a
Q flia
.
Inasmuch
as thevaluesof thelift-curve
slopeanddamping
in roll
.*
as givenin references
h-and12 arebasedQh lifting-lfie
results
for
elliptic
wings,a reasonable
valuefor kl”shouldbe obtainable
through
analysis
of elliptic
win~f?~%b~~h~in$s~ lk&ting-line
theorypre_ .~ A-_ LL
dietsa liftdistributions
flia
——
-
-’*
.-
(SIA)
29
NACATN 2751
Fora symmetric
angle-of-attack
distribution,
onlyoddvaluesof n
is thengivenby
areheededin equation
(Alk).Theliftcoefficient
(A16)
thefirst
Sincetheliftis to be zeroforthebasicliftdistribution,
thesymmetric
casethe
termof theseriesmustbe omitted,
so thatfor
Forthecaseof
firsttermof theseriesis thatforwhich n = 3.
constant
angleof attack,
however,bl =a and allother b’s are
zero,so that
yl =
8TIA
—asine
A+2q
.
(A17)
(A18
)
...
and
4
Ya =Xsine
(A19)
*
Consequently,
substituting
equations
(A15),(A18),and (A19)int~
equation
(A13)gives
. .-..—.——
(A20)
.
.
since Z = bl. Thisdistribution
shouldbe thesameas thatgivenby
equation
(A14)(starting
withtheterm n = 3). Clearly,
no value
“ of kl cansatisfy
thiscondition
exactly.If theftisttermsin the
twoseriesaresetequal,however,
theotherswillnotdiffermuch;
furthermore,
sincethefirsttermis usually
muchlargerthantheothers,
theliftdistributions
willgenerally
be almostidentical
if thefirst
.-
30
NACATN 2751
.
termsof theseriesareequal. In orderto satisfy
thatcondition,
kl mustbe takenas
/.
kl = A+2?l
Ai-67L”
(A21)
-—
.
whichis identical
to theresultobtained
in.reference
3, exceptthat
—
there q wasassumed
to be 1.
.
A lifting-surface
andsweepcorrection
for kl maybe deduced
in
themanneremployed
forthedamping-in-roll
.coefficientin
references
4
and12.-If’theproductklC% thatoccursin equation
(A13)is obtained
fromequattins
(A18)and(A21)andwritten
in theform
..-
A/3
klc~.=A—
cja
-+q
3
.
a correction
identical
to thatfor C& butbasedon one-third
the
aspect–ratio
suggests
itself.Consequentlyl
uponintroducing
thesweep
andlifting-surface
corrections
inthisequ6tion
anddividing
by C~,
thefoltiwing
relation
is obtained:
kl=
‘i’
+4(%-Y+
27
Cos’
“,,
‘
r
-—
(A22)
Thisfactoris plotted
in figure2 against
theplan-form
parameter
A
.
F~—
TIcosA
>—.
Lift–Distribution
in Roll
If an equation
similar
‘coequatim(A13)is applied
to a linear
antisynmnetric
angle-of=attack
distribution-”-with
unitvalueat the
tip (a= p), therelation
.
.
31
.-.–
NACATN 2751
72
= ‘Zcbpya
(A23)
—
-.
.
of thedif- ‘
is obtained.A factork2 is usedratherthan kl because
ferentnatureof the induction
effects
forthesymmetric
andantisymmetriccsses.,Thedesired
valueof k2 maybe obtained
fromthelift
distribution
of elliptic
wings,as follows:
—
Fortheangle-of-attack
distribution
in thiscase,equation
(A15)
yieldsb2 = $ andallother b’s arezero. Hence,equation
(A14)
yields
72 =
.
aftersubstitution
of equations
(A18)
whereas
equation
(A23)yields,
and(A19),
A
72 = k2Cz
—aA+2q~
4 sin0 cose
..
-——
Thedesired
valuesof k2 ~y be ~btained
by equating
thesetwoexpres{
sionsfor 72:
k2=—A+~
A+47
(A24)
Thisvalue,also,is identical
to thatobtained
in reference
3 for TI= 1.
andsweepeffects
in thesame
Itmaybe corrected
for lifting-surface
manneras klj so that
k2 =
.
(A25)
Thisfactoris alsoplottedin figure2 against
theplan-form
parameterF.
32
NACATN2751
Damping-in-Roll
C~efficient
By calculating
the rolling
momentof theliftdistribution
givenby
equation
(A23),a valueforthedamping-in-roll
coefficient
Cld (=“lp)
maybe o%tained:
“
.
.
1
Cld S:C2U
COS
1’
A k$o
0
-,
Theproduct’of
the’
$actorsk2 and ~
in thefollowing
derivation.
YaP*w
(J@6)
i~referred
to as thefactork~
-..
--
If equation
(A3)is substituted
intoequation
(A26),thedamping
-.
coefficient
maybe written
as
..
Cld =~cza
21+lLCl+c2+IC3
cosAk4-—
.)
(3 1 +–L.
I
—..
(A27)
where h isthetaperra~oj where I is_defined
by
.,. -.,
–
—.
1
1“=”4- fy*2dy*
(A28)
‘--
0
andis plottedin figure6(b),andwheretheexpression
-.
4
J’
1
OC
& y*2d~
21+3kf
mustbe-substituted
fortheterm —
31+L
—.—
or wingswhichdo notihsve
(A27)contains
a
lineartaper.Thevalueof C~d givenby..equation
fortheeffect
correction
fortaper(the Cl term)anda-correction
-.
of sweepon thespanwise
liftDistribution
(the C3 term).
—
‘- .~
..
33
NACATN 2751
Antisymmetric
LiftDistributions
Inasmuch
as equation
(A23)hasbeenderived
specifically
forlinear
antisymmetric
angle-of-attack
distributions,
it maynotyieldaccurate
results
whenapplied
to otherantisymmetric
angle-of-attack
distribuof theequations
for
tions,as is donein reference
3. Thederivation
symmetric
liftdistribution,
in thisappendix
suggests
a parallel
derivationforantisymnetric
distributions
whichhasas itsstarting
point
nottherelation
givenin reference
17,butratheritsanalogforantiTheresulting
equations
forsntisymmetric
symmetric
liftdistributions.
liftdistributions
arenotas simpleas equation
(A23)- theyimplya
separation
of theliftdistribution
intoa damping-in-roll
anda
residual
liftdistribution
analogous
to theadditional
andbasiclift
distributions
in thesy?muetric
case- buttheyareapplicable
to a
widervariety
of antisymmetric
angle-of-attack
distributions
thanis
themethodof reference
3.
.—
-.
theoryapplied
to thelinearantisymmetric
According
to lifting-line
case(a = y++),
(A29)
where ai2 is theinduced
angleof attackcorresponding
to thedistribution 72. Combiaing
thisequation
withequation
(A9)foranyarbitrary
liftdistribution
@.eldstherelation
7Y+ = 72a+
(7q2
.)
- 7Pi
(A30)
coefficient
of anyantiayrmnetric
liftdistribution
7
The rolling-moment
maybe writtenas
(A31)
so that,fromequation
(A30),
—
34
NACATN 2751
.-
By reasoning
similar
to thatemployed
in connection
withequation
(All)
thesecondintegral
maybe shownto be equalto zeroforanyantisymmetric7 distribution,
so that
—
+
—
(A32)
.
distribution
Yd forunitrolling-moment
coefA damping-in-roll
in a manneranslogo~to theadditional
liftdisficient
maybe defined
tribution
as
—.
..
.
?’2
)’d=
—
Czd
---
—-
since Cld istherolling-moment
coefficient
of the 72 distribution.
An effective
angleof attack%, whichis-theangleof attackat the
tipof a linearantisymmetric
distribution
withthesamerolling
moment .as thatof thegi-nnantisy?mnetric
distribution
7, maythenbe defined
as
(!2
Ue=—
Cid
*
.
‘-
or
(A33)
.“.- .-
.
—
35
NACATN 2’751
Theanalogof thebasicliftdistribution
(whichhaszerolift)in
thecaseof an antisymmetric
distribution
is theresidual
liftdistribution 7r,whichhaszerorolling
moment.Equationa
(A13)and (A32)
thensuggest
theformforthisdistribution:
()
a
7r = ‘3Cld—Y*
(A34)
ae 7d
Therolling
momentof 7r is thenzero,as required.
through
an analysis
of
Thefactork3 mayagainbe evaluated
elliptic
wingsby lifting-line
theory.For theantisymmetric
case,only
evenvaluesof n arecontained
in theseriesof equation
(A14).Furthermore,
since
~
..
..
.
,“
-cl=”:”
828?
~v—
‘A
A+4q
‘-’
b2”
.’
thenfor Yr thetermwith n = 2 mustbe zero,andtheseriesmust
beginwiththeterm n = 4. Butforan elliptic
wing
_——
..—
(A36)
Yd=:
sin2EI
and
ae = 2b2
—.
(A37)
36
NACATN 2751
.——
so thati-equat-ion
(A34)becomes
-L.,
.—
()
0.
A\a
7r=k3$cza~+4Ty* _.2b2~ 11sin20
.
or,as a result–of
equation
(A1.5),
—
(A38)
Thevaluesof Yr givenby equations
(A38)and(Alh)cannot--be
exactly
equalforallantisynnuetri.c
angle-of-attack
conditions,.,
butifthefirst
termof theseries(whichia usually
muchlargerthanallothers)
ismade
thesameforboth,theotherswillbe approximately
equal.Hence k3
musthavethevalue
k3 . A+47
A+8q
(A39)
-.
‘“
.
—
..
which,by thesamereasoning
as thatfor kl,maybe corrected
for
effects
by setting
lifting-surface,
sweep,andcompressibility
k3 = ‘m+’~cos’
~’
A~lLt_64fcjsAy+
“(A’O)
8qcosA
Thisfactoris plottedin figure2 against
theplan-form
parameterF.
“-:---
37
NACATN 2751
..-
—
APPENDIX
B
CALCULATION
OF AERODYNAMIC
INFLUENCE
COEFFICIENTS
Aerodynamic
influence
coefficients
areusefulin methods
of static
aeroelastic
analysis,
particularly
thoseemploying
matrices,
suchas
coefficients
maybe considered
thatof reference
18. Theseinfluence
to represent
thelift(indimensionless
form)at onepointof thespan
dueto a unitangleof attackat another
point.Suchinfluence
coefficients
canbe obtained
by meansof someof thetheoretical
methods
by calculating
theliftdistributions
corresponding
to angle-of-attack
distributions
whicharezeroeverywhere
on thespanexceptfor onepoint,
at whichtheyareinfinite
in sucha waythattheareaundertheanglesuchliftdistributions
are
of-attack
distribution
is zero. However,
difficult
to calculate,
andtheymaybe obtained
at pointsalongthe
spanwhichareinconvenient
foran aeroelastic
analysis.
.
.
.
The influence
coefficients
obtained
by theprocedure
of reference
s
arebasedon themethodof calculating
spanwise
liftdistribution
presentedin reference
2. Individually
theseempirical
”coefficients
do
nothaveanysignificance,
as do thecoefficients
described
in theprecedingparagraph,
butwhenmultiplied
by thevaluesof theangleof
attackat several
pointson thespanandsummed,
theydo yieldthelift
at a pointon thespanin thesamemanneras theotherinfluence
coefficients.Apartfromthefactthattheycanbe calculated
moresimply
andforanypointson thespanforwhicha setof integrating
factors
canbe calculated,
theempirical
influence
coefficients
havethesame
advantages
anddisadvantages
in comparison
withthetheoretical
coefficients
as do empirical
methodsin comparison
withtheoretical
methods.
In thisappendix
a procedure
is outlined
forcalculating
influence
coefficients
in a mannerwhichis similar
to thatemployed
in reference~, butwhichis basedon themethodof calculating
liftdistributionsgivenin thepresent
paperratherthanthatof reference
2. In
essence
theprocedure
presented
in thisappendix
consists
in formulating
thismethodof calculating
liftdistributions
inmatrixnotation
by using
integrating
matrices
of thetypepresented
in reference
18. Compared
withinfluence
coefficients
calculated
by theprocedure
of reference
5,
theonescalculated
by theprocedure
presented
in thisappendix
havethe
advantages
of greater
accuracy
andwiderapplicability,
because
the
methodof calculating
spanwise
liftdistributions
presented
herein,
on
whichthecoefficients
arebased,hastheseadvantages
overearlier
methods.The influence
coefficients
givenby theprocedure
of reference5 areapplicable
onlyto symnetric
loadings
andwingsof moderate
thosegivenby theprocedure
of thepresent
paper
aspectratio,whereas
..
..-
—
.-
—
—.
.——
—
..
38
, NACATN 2751
arenotsubject
to eitherof theserestrictions.
Theyshould,
therefore,be usefulformostpurposes
forwhichaerodynamic
influence
coefficients
arerequired.
—
.—
.
Thematrixnotation
usedhereinisthe.
sameas thatusedin
reference
18.
.
Symmetric
Case
to yield
Equations
(2)and(10)maybe combined
((
)
v
, = f!.
“%. 1 - kl &7a + k1u7a
)
.
or,inmatrixnotation,
.
.
{7}=%+- ‘Jhs{’}
+‘M+}}
(Bl)
L
of thevaluesof 7 atwhere {} 7 is a columnmatrixwhichconsists
a is a columnmatrixwhichconsists
several
stations
on thewing, {>
of thevaluesof a at thosestations,?i
{} is a columnmatrixof
matrixthenonzero
elementm
allequalto E, and ~7a~ is a diagonal
elements
ofiwhich
arethevaluesof 7a at thesamestations.
Theintegration
indicated
by equation
(11)maybe writt-en
in
.
matrixformas
(B2)
.
where H is defined
by
w/b
H=
Jo
78 ~
(B3)
.
NACATN 2751
39
andmaybe evaluated
withsufficient
accuracy
forthepresent
purpose
fromtherelation
H=~ ~b Ya
+4ya
() Y-&o () Y+=~+
(
()
‘aY+=:)
andwhere LI~ isanintegrating
matrix.Asdescribed
inreference
18,
a suitable
integrating
matrixmqybe obtained
by approximating
theintegrand(thefunction~Ya)by parabolic
segments.In viewof thefact
that ~a goesto zerowithinfinite
slopea special
typeof parabola
hasto be used,forinstance,
onewhichconsists
of a linearcombinafromthetip. With
tionof the1/2powerand3/2powerof thedistance
matrixfor
thisapproximation
to thecurveof a7ajtheintegrating
fromthewing
stations
at thewingrootandeverysixthof thedistance
maybe writtenas
rootto thewingtip,forinstance,
~$ =~.05556, 0.20833,0.15278,0.16667,0.14913,0.22500,0] (B4)
Root
TiP
Equation
(B2)canbe mitten as
wheretherowmatrix LI1~
L7al is obtained
by calculating
therow
matrix~
-.
LI~~al andaddingH to itsfirstelement.A square
whichconsists
of rowsall
matrix LIJ
l_7J canthenbe constructed
equalto plJki--l
equations
(Bl)and (B2)
. Withthissquarematrti,
canbe combined
andwrittenas
—
40
NACATN 2751
—
.
or
(B5)
. —-.
._,
.
andwherethematrix [Q.] defined
by
where LI~ is theunitmatrix,
~~al+ klL+ll
[Qs]‘ LY~ ~’ - kl)[II]
(B6)
is,in effect;
an aerod~amic-influence
-c=fficient
matrixforsymmetric
.7
liftdistributions.
In thesematrices
thevaluesof Y, _Yajand a arealltakenat
thesamesetof ~tations,
andtheintegratfig
matrixmustbe setup for
thesamestations.Stationa
at thewingrootandeverysixthof the
distance
fromthewingrootto thewingt+}(but%xcluding
thewingtap -“proper,
where Ya is zerounlesstiptanksor endplatesarepresent”} .—
havebeenfoundconvenient
formanypurposes.Forthesestations
I* = O, 0.1667,
0.3333,
0.5000,
0.6667,
0.8333
..
.
—-.
-—
.——.
-, -.
_-—
-. .
—
—
=. .
-.
where 7* is theratioof thelat~ral
dia~ance
of thegivenstation
fromthewingrootto thelength~.
Theintegrating
matrixgiven
2
by equation
(B4)maybe usedforthesestations.The lastelement,
0,
of thematrixof equation
(B4)canbe disregarded.
Antisymmetric
Case
. ~.
--..
—..
Theantisymmetric
liftdistribution
uaybe obtained
froma combiZnaionof equations
(13),(20),and(21)by useof matrixmethodsin the
samemanneras in thesymmetric
caae. Theresult%s
t
~a
1
1
{}
L
.J
(B?)
*
-.
.
41
or
.
.
where
(B9)
is an aerodmtic-influence-coefficient
matrixforantis=tric lift
distributions.
The~trix ~
111pdq iS a squarematrixwithrowsall
equalto w - b
~
@.j LYcilandwitha constantHI
addedto thefirst
elementOThisconstant
is defined
by
H’ .~b
2;0 f
.._
w~b
y*Yd
w
and is givenalprox~telyby therelation
H’ = ~
The~trix
relation
~
5
11
(
2(Yd)P ~ + (Yd)
=2b
Y*=;
)
canbe calculated
a-t the given stations
fromthe
___
.
.
42
NACA‘TN2751
AH’ENDIX
C
MOMENTDISTRIBUTIONS
.
Thespanwise
distribution
of thepitching
momentis oftenof less
interest
thantheliftdistribution;
foraircraft
loadscalculations
the
liftdistribution
usually
is of primaryin{eresty
and forstability
calculations
theuseof two-dimensional
centers
of pressure
oftenyields
sufficiently
accurate
over-all
aerodynamic
parameters.
Forsomepurposes,however,
suchas staticaeroelastic
calculations,
a knowledge
of
the spanwise
distribution
of thepitching
momentis impertanti-In this
appendix
someavailable
knowledge
concernl~g
thisdistribution
is
summarized
foruse in conjunction
withthem“ethod
of thispaper.
Thevariation
of thepitching
momentalongthespanmaybe obtained
fromthelift.
distribution
andthelocalcenters
of pressure.Thelocal
aerodynamic
center,
thatis,thecenterof pressure
dueto angleof
attackor twistof an uncambered
wing,andthecenterof pressure
due
to control
deflect-ion
areof most--general
interest-:-—Compared
withtheamountwfinformation
available
on spanwise
lift
distributions,
relatively
littleis knownaboutthelocalcenters
of
pressure
of theliftdistribution.
Thecommonly
usedWeissinger
method
(ref.13) forcalculating
liftdistributions
is characteristically
incapableof furnishing
localcenters
of presmre. Methodssuitable
for”
obtaining
suchinformation,
suchas Falkner’s
(ref.14),arevery
cumbersome,
andthelocalaerodynamic
centers
calculated
by these
methodsarenotaltogether
reliable.For instance,
by usingdifferent-vortexrepresentations
inFalkner’s
methoddifferentilocal
aerodynamic
centers
areobtained,
although
theliftdistributions
arenearly
the
.m..same. Thecenters
of pressure
dueto aileron
deflection
cannotbe
uncertainty
calculated
accur&tely
by Falkner’s
method.An additional
in connection
withthelocalcenters
of pressure
is thefactthatthese
centers
aremoresensitive
thantheliftdistribution
to deviations
from
thepotential-flow
conditions
(asa resultof boundary-layer
separation,
forinstance)
assumedintheanalytical
methods.
.“
——
——
—
As a resultof theseconsiderations
theinformation
concerning
....
localcenters
of pressure
presented
in subsequent
sections
shouldbe
-.-.
regarded
as qualitative
ratherthanquantitative
innature;it is
intended
as a roughguideuntilmorerefined
methodsof calculating
._, ,_
suchinformation
areavailable.
Theyalugsof thelocalaerodynamic
—
centers
andcenters
of pressure
dueto control
deflect-ion
arefor
incompressible
flow. In orderthatsubsonic
compressibility
effects
.
maybe takenintoaccount
approximately,
thevalues
should
be
estimated
.
times
.
forumequivalent
wingwithan aspectratioequalto fl~
43
NACATN 2751
theactualaspectratioandwithan effective
sweepangleA=, the
.
.
tangent
of whichis 1/~2
angle A.
of theactualsweep
timesthetangent
LocalAerodynamic
Center
Thelocalaerodynamic
centers
of an unswept
wingof veryhigh
aspectratioareat thequarter-chord
line. As theaspectratio
decreases
thelocalaerodynamic
centers
moveforward,
in particular
nearthetipof thewing. For constant-chord
wingsof vanishingly
low
aspectratio,virtual-mass
considerations
indicate
thatthelocalcentersof pressure
areneartheleading
edge,although
theapplicability
of low-aspect-ratio
theoryto sucha wingmaybe questioned.
Thelocalaerodynamic
centers
of several
unsweptconstant-chord
wingscalculated
by Falkner’s
methodandthoseof twoelliptic,
wings
calculated
fromtheresults
of reference
19 areshownin figure13(a).
At highandmediumaspectratiosthewingtaperhaslittleeffecton
thelocalaerodynamic
center,
as maybe seen,forinstance,
by comparing
theresultsfortheconstant-chord
andelliptic
wings.At verylow
as~ct ratios,
on theotherhand,virtual-mass
considerations
indicate
thattheeffectof taperis likelyto be morepronounced.
If theleading
edgeis sweptback,thelocalaerodynamic
centers
of a wingwithvanishinglysmallaspectratiomaybe estimated
fromthedatashownforthe lowaspect-ratio
deltawingin figure13(c)by considering
thepartof the
wingenclosed
by theleading
edgeandthelinewhichconnects
thetwo
endsof theleading
edgeto be a deltawing(seefig.5). The
aerodynamic-center
locations
shownin figurelk(a)wereobtained
in this
manner.
Thelocalaerodynamic
centers
of sweptwingsof veryhighaspect
ratioarealsoon thequarter-chord
line,exceptat therootandthe
tip. Nearthewingrootthelocalaerodynamic
center,
whichis at the
quarter-chord
pointforan unswept
wing,movesbackas thewingis swept
backandapproaches
themidchord
position
forveryhighsweepangles;
as thewingis sweptforward
thelocalaerodynamic
centermovesforward.
At thewingtip,on theotherhand,thelocalaerodynamic
centermoves
forward
towardtheleading
edgeas thewingis sweptbackandrearward
as thewingis sweptforward.
Thelocalaerodynamic
centersforsweptwingsof mediumaspect
ratiodependbothon theaspectratio(inmuchthesamewayas for
unswept
wings)andon theangleof sweep(ina mannersimilar
to that
described
in theprecedtig
paragraph).
Theresults
of calculations
by
meansof Falkner’s
methodforconstant-chord
wingsof varioussweep
anglesandas~ct ratios,
as wellas foronetapered
wingandtwodelta
.—
44
NACATN 2751
wings,areshownin figures
13(b)and13(c).Comparison
of thetwoplan
formsof aspectratio2 with600swe.epbackc
orro~ratesthepreviously
madestatement
thatta~r ha5an effecton––the
localaerodynamic
center
at lowas~ct ratios,
although
theeffectvis
fa-irly
small.
Forwingsof verylowaspectratiowithsweptback
leading
edges
andwithoutreentrant
trailing
edges,themethodof reference
8 maybe
used. In thecaseof deltawingsit yieldstheaerodynamic-center
variation
alongthespanshownin figure13(c).
Forwingswithanglesof sweepback
largerthanabout600the
results
of slender-wing
theorypresented
h reference
9 maybe usedas
a guide.Thelocalaera.dynamic-center
positions
obtained
in thismanner
areshownin figure5. Thesevariations
areonlyapyroximat-e,
particularlyin thecaseof thewingswithsmall’plan-form
parameter,
forwhich
onlythevaluesat therootandtipareknown.
Thelo.cal-aerodynamic-center
linesshownin figure13 havebeen
calculated
for.
constant
anglesof attackalongthespan. Fromthedata
of.reference
19,however,
a local-aerodynamic-center
linecanbe calculatedfora linearantisymmetric
angle-of-attack
distribution
of an
elliptic
wingof aspect-ratio
6. Thislineagreesperfectly
withthe
oneshownin figure13. Consequently,
itmaybe assumed
thatthenature
of theangle-of-attack
distribution
alongthes~n haslittleeffecton
theloc~l-aerodynamic-center
location
forunswept
wings,andprobably
forsweptwingsas well.
.
LocalCenterof Pressure
DuetWControlDeflection
Forunswept
wjngsof highaspectratioand,exceptat therootand
thetips,forsweptwingsof highaspect-ratio,
thetwo-dimensional
valueof thecenterof pressure
dueto control
deflection
maybe used.
Formediumaspectratiosnoresultscomparable
to thoseforthelocal
aerodynamic
centerareavailable.In orderthatsomeindication
of the
probable
location
of thecenterof pressure
relative
to thetwodimensional
valuemaybe obtained,
thedifference
between
thelift
actually
carried
at–anysection
of thewingandthatwhichwouldbe
carried
if thesection
wereintwo-dimensional
flowmaybe considered
to actat theaerodynamic
center.Thisassumption
leadsto theequation
.— .
.
,-..
-..
,
_-..
.
.
NACATN 2751
by themethodof thisPper,
where 7111 is thevalueof 7 calculated
a is thelocalaerodynamic
center,
and
.
7J__ = : Czaq
.
cos
A(b COSA)
c
—
= ;
CZE8 COS2A
c
where b is thecontrol
deflection
measured
in a planeperpendicular
to thehingelineand ~ is thetwo-dimensional
valueof thecontrol
effectiveness.
.
Forwingsof verylowaspectratiothemethodof reference
8 may
sometimes
be usedto calculate
centers
of pressure
dueto control
deflection,provided
boththewingandthecontrol
surface
havesweptback
leading
edges,thetrailing
edgeisnotreentrant,
andthehingelineis
of pressure
dueto a
perpendicular
to the freestream.Thecenters
symmetric
control
deflection
as givenby theapparent-mass
concept
on
whichreference
8 isbasedareshownin figurelk(b)fortwosuchwings.
Whentheentiretipof a low-as~ct-ratio
wingis rotated
abouta hinge
lineperpendicular
to thefreestresm(asshownin thethirdsketchof
fig.14(b))theresulting
spanwise
liftdistribution
canbe obtained
fromreference
15,butno chordwise
distributions
aregivenby thismethod.
In thecaseof thefirsttwoplanformsshownin figurelk(b),all
theliftcarried
aheadof thehingelinewhenthewholewingis at an
single
of attackIs carried
at thehingelinewhenonlythecontrol
surfaceis deflected.
Thecontrol
surface
itselfcarries
thesameliftas
itwouldif thewholewingweredeflected.
Forwingsof lowaspectratiotheapparent-mass
concept
alsoindicatesthatthe two-dimensional
valueof ~ cannotbe used;instead
a
valuecloseto 1 shouldbe usedforwingsof verylowaspectratiosuch
as thoseshownin figure14.
.
.
.—
‘-
46
NACATN 2751
—————-———
lW.WRENCES
1. Schrenk,
O.: A Simple
Approximation
~thod fcur
Obtaining
theSpan.
wiseLiftDistribution.
NACATM 948,1940.
2. Diederich,
Franklin
W.: A Simple
Approximate
MethodforObtaining
Spanwise
LiftDistributions
OverSweptWings. NACARM L7107,1948.
—-
—
.
..—
-.
3. Sivells,
JamesC.: An Improved
Approximate
MethodforCalculating
LiftDistributions
Dueto Twist.NACATN 2282,1951.
.-
4. Diederich,
Franklin
W.: A Plan-Form
Parameter-for
Correlating
Certain
Aerodynamic
Characteristics
of SweptWings.NACATN 2333,
1951.
.
5. Diederich,
Franklin
W.: Approximate
Aerodynamic
Influence
Coef-~
ficients
forWingsofArbitrary
PlanFormin Subsonic
Flow. NACA
TN 2092,1950.
.
.
6. Murray,
HarryE.: Comparison
withExperiment
of Several
Methods
of
Predicting
theLiftof Wingsin”Subs@iic
Compressible
F1ow. NACA
TN 1739,1948.
7. DeYoung,
John,andHarper,Charles
W.: Theoretical
Symmetric
Span
Loading
at Subsonic
SpeedsforWingsHavingArbitrary
PlanForm.
NACARep.921,1948.
-.
--
—
8. JonesjRobert-_T.
: Properties
of Low-As”~ect-Ratio
Pointed
Wingsat
SpeedsBelowandAbovetheSpeedof Sound.NACARep.835,1946.
.:
(Supersedes
NACATN 1032.
)
9. Lomax,Harvard,
andHeaslet,
[email protected].: “Linearized
Lifting-Surface
TheoryforSwept-Back
WingsWithSlender
PlanForms.NACATN 1992, ‘“
1949.
—
,—
. ..——.
10.Bird,JohnD.: SomeTheoretical
Low-Speed
SpanImading
Characteristicsof SweptWings inRollandSideslip.NACARep.969,1950.
(Supersedes
NACATN 1839,
)
-.
11.DeYoung,
John: Theoretical
Antisymmetric
SpanLoading
forWingsof
Arbitrary
PlanFormat Subsonic
Speeds.NACATN 2140,1950,
12.Toll,-Thoms
A.,andQueijo,
M. J.: Approximate
Relations
andCharts
forLow-Speed
Stability
Derivatives
of SweptWings. NACATN 1581,
lg48.
●
✝
✎
NACATN 2751
47
13. Weissinger,
J.: TheLiftDistribution
of Swept-I!.ack
Wings.NACA
TM 1120,1947.
14.Falkner,
V. M.: TheCalculation
of Aerodynamic
Loading
on Surfaces
of Any shape.R.& M. No.1910,British
A.R,C.,1943.
15. Diederich,
Franklin
W.,andZlotnick,
~rtin: Theoretical
Spanwise
LiftDistributions
of Low-Aspect-Ratio
Wingsat SpeedsBelowand
AbovetheSpeedof Sound.NACATN 1973,1949.
TheoryforSwallow
TailWin s of SmallAspect
16. Robinson,
A.: Aerofoil
of Aero.,Cranfield
Ratio
. Rep.No.41,College
?British),
Oct.1950.
of an Aerofoil
Dueto Variation
of
17. Gdaliahu,
M.: TheLiftIncrement
WcidenceAlongtheSp.n,anda SimpleMethodof Estimating
the
LiftDistribution.
R.& M. N~.2261,British
A.R.C.,1945.
18. Diederich,
Franklin
W.: Calculation
of theAerodynamic
Loading
of
SweptandUnswept
Flexible
Wingsof Arbitrary
Stiffness.
NACA
Rep.1000,1951. (Supersedes
llAcA
TN 1876.)
19. Swanson,
RobertS.,andCrandall,
Stewart
M.: An ElectromagneticAnalogyMethodof Solving
Lifting-Surface-Theory
Problems.NACA
ARR L5D23,1945.
—
—
TABLE l.- GEOMETIUC CHARMXERISTICS , CONSTANTS, A-NDLHYHXJRVE
SIDPES OF TRE SIX PIAN FOIU6 CONS13EREXlIN FIGURE 8 .,
— .
—
c~
la n
A
‘Orn
A, ). F
Leg
c1
C2
C3
‘o
— —
o ).5 6.00 ). 301
).441
I. 721
.405
.266
.792
O 8.~
6 45 L..
.405
4
3 30 L.5 3.46
5
3 45 )
6
3 60
—
1
6
2
6
3
45
Y+
1-J
Iethodof
fethod Of
Equation
3 refer- !quation9 reference13
ence 13
4.529
4.321 ! 0.430
0.425
3.517
3.444
.448
..55
.266
.792 3.517
3.2s~
.471
L
.0
.184
.6s2
.577 3.140
2.795
.4+9
.152
4.24
.221
.581
.6jh
2.817
2.813
.414
.407
.5 6.00
.301
.441
.94$
2.265
2.290
.445
.455
.5 8.48
I
v
.
,
.
,+
I
1’
,
,
,
Figure 1.- Section lift-curVe-EloPe correction.
*
.
.8
J%
.6
.4
.2
0
2
4
6
Plan-form
to
8
parameler,
Figure 2.- Factors
%J
‘1)
12
F
%>
ky
Em
k4.
14
/6
51
NACATN 2751
.=
I 1..1
”;I.I .; \
L.I..! 1:..!.1 :...1.l-...~I .1 ..l..-;
-—-..
.+.
... \
:
\
.;
~“-.:“--:
,;
,.
-,-- -- ,- .I
I
---—.--.
.-,,
.,
....-—
---.-..,1. :1”!
%lH=F==
‘-.
-.
I
—..—-
I
14 Y }+++1
--i-l
I
I
I
1 ...-....
I
t
I
I
1
,
-i
,.
I I I I
I
i
t
I
1. I
1 1-1-1-~
....1.1.. .1
/.
o ._+ + 4.,.. -.
!,.
.-,
-,-. .,
!
I
N
i..
._.
/.6
l\
-----
/.4
---~
‘--
..!
—- ..
-j””
. ,.,
I I I I
.-— — —..——
\\
I
.
\
i
.-
1
.~.~
i.
I
,.. , \
\
+
I
1“
I
.
/.2
I
‘t ‘-ti‘! i
I
>
—.
l-!
I
I
I
1
“’
.-t-, I
i=
. .
i A-+
I
1’
[
“’”l
~.
I
I
1
“j”” - ‘“i””
~-.
I
.
.,.
-~_:..
W.!”’’’L””L””
‘yk~
:
‘
i
~.819A”
,...
,--.-x
N
KJzEEL
‘x- 4zE1’av ‘#;;
,
/.0
v
“L 1 ; ‘ -’t--.
‘1““-z
!
--}-
vvl
l+-,, ,
I
.
(A..)
““
I
.
-A
.4
-3
k
\
/1
.
t’h
l’\ . :.
WliE3i
F-!-i-m 5fEEif
I
‘w
1
.
—
-—-:-
‘HE3!i
,4A
../
1
\\l
.’,
.
\
-+
;,
I
I
\
l\
“4tEEEL3azLL
E
.2 “““
-,4 !- ““”1--”
!“:::. i
,_,6
.
“l”””
“:”~
“
.,. —.-.
I
-“~.”
i/.
~~
‘
i
“\
j
.
:
I
‘.
‘“
\
;,.
—
J_
.-+
!
1.
~
-—-----—
.:
i
—
\
1
‘
-=E=-
-
,,’,
i\
,.
.
——
..-—
.
functionf.
Figure3.-Thelift-distribution
-—
UJ
C2
—-—. — C3
—.
\\
—
\
.2
F
/
L15
0
1111
1111
I
Ill
2468/0
Figure k - The Mft-distribution constants q>
.
%
and
C .
3
53
NACATN
2751
.
\
Two-dme’cmi?d
region
\
-.
T@ reglcm
—Ap=O
.
\
\
\
#
-
.
.
(a) Smallplan-form
parsmeter.
parameter.
(b) Largeplan-form
and section
aerodynamic
centers
liftdistributions
Figure5.-Approximate
on wingswithslender
planforms.
.-
54
NACATN 2751
.5
—
.
/
.4
.
J
.3
/ /
.g45
i5
30
45
E’ff@c+i
V@ ongle
ofSW@QpbUCk,Ae,deg
-30 -/5
O
60
.
(a) Spanwise
centerof pressureJ.
.
}2
.rtt
.6L
1
/
/
.4 /“ ‘
-/5
o
/5
30
45
00
(“b
) Momentof inertia 1.
Figure6.-Centroid
o~areaandmomentof inertia
of lift-distribution
functionf.
-.
.
.
—
55
NACATN 2751
.—
0
4
8
6
Plun-iorm pffrumerer, F
2
.
/0
+
~t
(a) Comparison
betweenthecorrection
factorK andthefactor2
-.
1.16
1./2
1.08
1.04
-
?b2”
/
K 1.00
/
.92
.88-40 .30 .20 -/0 o 10 20 30 40
Effechve ung/e of sweepbuck,.Ae,deg
(b)Effeetof sweepon thefactorK.
factorK.
Figure7.-The correction
50
60
mefhod
Approxlmak
/. 6r
———ldefhod
F
1-
1.2
–
—.
of reference /4
—-Method
of reference B
E
_
~s==-—
\.
\
:fi~
I
‘o
/6
r
“F
$.2: ;_
Plan form 3
F
---‘\
~
.8
CCL
4
:,@>
0
Plan form 4
.2
.4
.6
.4
/.0
.8
Dlmens/onlas
,,
Figure 8.- Additional lift distribution for six
l,;
‘.
1111
lateral
I
!
I
‘1!”
!!
,. ,..
I
plan forms.
A
I
,,
I
I
c
‘
I
1,
,,
,
,
E
,
,
Apprortmufe
———
method
.8 -
.8 -
.4 -
,4 -
Mefhod of reference
As
..
o
38
.;’,
I I
;’.8
.4
I
I
[0 ‘“80
I I I
I
.2
4
1 I
.6
Llmerwcwlesr
I
1
I
LO O
.8
16-
4-
.8 -
.8 -
3-
.6.4-
-3
(’v~b :
,2-
-.8 1 I
.2
1 I
.4
I
I I
.6
I
,8
I
I
.2
I
1
.4
I
I
.6
I
I
.8
[0
Ioi%rol w@70te, y *
[0
0
-/”
Unit psxabollc twist.
(a)
a
/
/
,/
-, 4 -
-.80’;’
/
[/
(’%%;
-. 4 -
/
i2
~
a-a
13
%2
-
—.=
-
c
/I I
fO ‘[60 .2
I
I
.4
I
I
,6
1
I
.8
I
I
I
I
I
I
I
I
I
I
I
100.2,4.6.810
DImerwIonloss Iafwal wdmuie, y*
(b) Unit deflection of half-semispm flap.
Figure
9..Symmetric lift distributions for plan form 2.
U
-4
I
1
.“
—
App-ox,m.demefho.4
———
Nefkf
drderance
2?2
R
12
[
‘[
t
-/
.2
I
1
1
.:
.6
,
,
.8
,
1
/0
3[
.,o~
.2.4
tlllllll,
.U.81D
Dlmm.swxa.cs
lolwal
02
Wdmatq
l’l
,4.6
m
u~
(a) Unit parabolic twist.
(b)Unit deflection of half-semispan flap.
Figure 10.- Symmetric lift distributions for
plan
form
1.
I
I
,
.
*
I
.:
I
1111
I
I
i’”
.1!
‘“ I
.
,
,
,
Approximate
rnefhod
‘=5’=
I
0
./
.2
.3
.4
.5
.6
.7
I
.8
I
I
.9
I
K’
(a) Linear twist (dmplng-in-roll ca~e).
Figure 11. - Antiaymmetric lift distribution for plan form 2.
I
I
I
U
w
60
NACATN 2731
2.
T
.
8
1.8
“
Fuirea’~
.4 -
0 I I I I I
I
I
I
I
J
I
.
I I 1 I
7.a-
.4
[
.—.==
—
0
-.4 -
-.8L J I
I
I
I
1
L30-
.-
—
..-*_
—
20ccl
-r-/.0
-
;-.
1
0
./
.2
.3
.4
.5
‘=== ----
----
.6
.7
.8
.9
——
[0
DlmerxrlonlessIuierd ordlfffff~ y*
(b)Unitdeflection
ofhalf-seinispan
flap.
Figure11.- Concluded.
.
61
N4CATN 2’751
.Sr
.—
.
.7 -
.
/
.6 -
.—----
_
.5 .4 ‘
.3-’
.
.2--
——-FYe+hod
of reference/5
.
0
./
.2
.3
.4
.5
.6
.7
Dtmenslonless Iuterol oro’rnute,y*
.8
.9
10
(a)Symmetriclineartwist.
Drmensionle.s.s luteru/ ordrnafe, y*
(ti)
Antisymmetric
quadratictwist.
Figure12.-Liftdistributions
for wingsof very low aspectratio.
—
J_
.20
&.
ii
t%
/0 -
\
—
—
~
—
~
/1=6
.20
\3-
—l-
1
Rectonguk.r wings
,0
(u)
Elliptic wings
Unswept wings.
7+6
.20
I
./0
&
(b)
.30
Q
y
l\“’A’=
’iii1
-!
4.5;
iV-F+
1 1
1
1,
1
\
\.
A =1
\
~
\
.40
~
\
“gzo
~
A=60°, A=2
(c)
.2
.4
.6
\
swept
/.0
.8
DimensiarWss
,20
lateral
Figure 13. - Local aerodynamic-cater
tapered
\
_
A=o
\
.30~
0.3
\
---%
\
\
0
1
.50
\
jj.lo
1
Swept canstanl-chcrd wings.
g.40
:$
0
Q
~,
c
: =6
\
_
\
2.3
\
/
De/to wings
=w=’–
rnngs.
0
ordinate,
.2
.?
~
L
b/2
positions for various plan form.
.8
1.0
CJ
,
,
(a) Aerodynamic-center locations
Ara
cc7rrg)ng
no /lft
Line
-----
of
.—
—
Ofce~fers
pressure
Hrflge
(b)Center-of-pressure lomtiona due to control deflection.
Figure 14.- Aerodynamic-center
locat
Ionsand localcentersof preBaure
‘due to control-&f lection on low-aapect-ratio winuB without-reentrant
traili~ edges according to low-aap~ct-ratiotheo~,
//fles