Lifting-surface-theory values of the damping in roll and of the parameter used in estimating aileron stick forces

NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS





WARI'TI ME RE PORT
ORIGINALLY ISSUED
August 1945 as
Advance Restricted Report L5F23

IJTING-SURFACE-THEORY VALUES OF THE DAMPING
IN ROLL AND OF THE PARAMETER USED IN
ESTIMATING AILERON STICK FORCES
By Robert S. Swanson and E. LaVerne Priddy

Langley Memorial Aeronautical Laboratory
Langley Field, Va.


WASHINGTON


NACA WARTIME REPORTS are reprints of papers originally issued to provide rapid distribution of
advance research results to an authorized group requiring them for the war effort. They were pre-
viously held under a security status but are now unclassified. Some of these reports were not tech-
nically edited. All have been reproduced without change in order to expedite general distribution.


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NACA ARR IHo. LEF23 RESTRICTED

TIATIO!IAL ADVISORY COMMITTEE FOR aERO.IA.UTICS


ADVANCE RESTRICTED REPORT


LIFTING-SURFACE-THEORY VALUES OF THE DAMPING

III ROLL AND OF THE PARAIT.ETER USED III

ESTIMATING AILEROni STICK FORCES

By Robert S. Swanson and E. LaVerne Priddy


SUMMU1ARY


An investigation was made by lifting-surface
theory of a thin elliptic wing of aspect ratio 6
in a steady roll by means of the electromagnetic-
analogy method. From the results, aspect-ratio
corrections for the damping in roll and aileron hinge
moments for a wing in steady roll were obtained that
are considerably more accurate than those given by
lifting-line theory. First-order effects of com-
pressibility were included in the computations.

The results obtained by lifting-surface theory
indicate that the damping in roll for a wing of aspect
ratio 6 is 13 percent less than that given by lifting-
line theory and 5 percent less than that gi-ren by
lifting-line theory with the edge-velocity correction
derived by Robert T. Jones applied. The results are
extended to wings of other aspect ratios.

In order to estimate aileron stick forces from
static wind-tunnel data, it is necessary to know the
relation between the rate of change of hinge moments
with rate of roll and rate of change of hinge moments
with angle of attach. The values of this ratio were
found to be very nearly equal, within the usual accuracy
of wind-tunnel measurements, to the values estimated
by using the Jones edge-velocity correction which for
a wing of aspect ratio 6 gives values 4.4 percent less
than those obtained by lifting-line theory. An
additional lifting-surface-theory correction was


RESTRICTED









NACA ARR No. L5F23


calculated but need not be applied except for fairly
large high-speed airplanes.

Simple practical methods of applying the results
of the investigation to wings of other plan forms are
given. No knowledge of lifting-surface theory is
required to apply the results. In order to facilitate
an understanding of the procedure, an illustrative
example is given.


INTRODUCTION


One of the many aerodynamic problems for which
a theoretical solution by means of lifting-line theory
might be expected to be inadequate is the case of a
wing in steady roll. Robert T. Jones has obtained in
an unpublished analysis similar to that of reference 1
a correction to the lifting-line-theory values of the
damping in roll that amounts to an 8-percent reduction
in the values for a wing of aspect ratio 6. Still more
accurate values may be obtained by use of lifting-surface
theory.

A method of estimating aileron stick forces in a
steady roll from static wind-tunnel data on three-
dimensional models is presented in reference 2. This
method is based upon the use of charts giving the
relation between the rate of change of hinge moment with
rate of roll Chp and the rate of change of hinge
moment with angle of attack Cha in the form of the

parameter ) = which is determined by means
parameter pCh rCh
of lifting-line theory. It was pointed out in reference 2
that the charts might contain fairly large errors which
result from neglecting the chordwise variation in
vorticity and from satisfying the airfoil boundary condi-
tions at only one point on the chord as is done in
lifting-line theory. A more exact determination of the
parameter p )Ch is desired. In reference 3 an addi-
tional aspect-ratio correction to Cha as determined
from lifting-surface theory is presented. In order
to evaluate the possible errors in the values of (ap)Ch








NACA ARR IT-. L5?23


as determined by lifting-line theory, it is necessary
to determine similar additional aspect-ratio corrections
to C p.

A description of the methods and equipment required
to solve lifting-surfa:ce-theory problems bj means of
an electromagnetic analogy is presented in reference 4.
An electromagnetic-analogy model simulating a thin
elliptic wing of aspect ratio 6 in a .teady roll was
constructed (fig. 1) and the magnetic-lield strength
simulating the induced downwash velocities was measured
by the rnethods of reference 4. Data were thus obtained
from which additional aspect-ratio corrections to Chp for
a wing' of aspect ratio 6 were determined.

Because of the small magnitude of the correction
to (aP)h introduced by the lifting-Furface calculations,
it was not considered worth whilee to conduct further
experiment on wings of other plan forms. An attempt
was therefore made to effect a reasonable generalization
of the results from the available data.

Inasmuch as the th6ery used in obtaining these
results is rather complex and an understanding of the
theory is not necessary in order to make use of the
results, the material presented herein is conveniently
given in two parts. Part I gives the results in a
form suitable for use without reference to the theory
and part IT gives the development of the theory.


SYrIEOLS


a angle of attack radiants, unless otherwise
stated)

cb section lift coefficient L q i


CL wing lift coefficient (
L\ q3 /

H/inge moment
Ch hinge-moment coefficienTL n o ent


CL rolling-moment coefficient Rolling moment








NACA ARR No. L5F23


ao slope of the section lift curve for incom-
pressible flow, per radian unless otherwise
stated

pb/2V wing-tip helix angle, radians

p circulation strength

CL damping coefficient: that is, rate of change
P of rolling-moment coefficient with rate

of roll 6' -
6(pb/2V)/

Chp rate of change of hinge moment with rate of
roll (
\ (pb/2V)

Cha rate of change of hinge moment with angle of
attack 1(

aC rate of change of wing lift coefficient
_with angle of attack (T-)

Vap)C absolute value of the ratio Ch
Ch Cha/

c wing chord

cs wing chord at. plane of-symmetry

cb balance chord of aileron

ca chord of aileron

Ca aileron root-mean-square chord
x chordwise distance from wing leading- edge

y spanwise- distance -from plane of -symmetry-
ba aileron span
b/2 wing semispan








"'ACA nRI; !


S

W'



8,

Oa

A

Ae








w
'F'

V

q

E

E'





-I
"n






KI, K2

Subscript T

LL


0o. LEF2S 5


area of wing

weight of airplane

stick force, pounds

stick deflection, degrees

aileron deflection, degrees, positive downward

aspect ratio

equivalent aspect ratio in conpressible

flow (A/ 2 )

taper ratio, ratio of fictitious tip chord
to root shord

free-stream Mach number

vertical component cf induced velocity

free-stream velocity

free-stream dynamic pressure (a1pi

edge-velocity correction factor for lift

edge-velocity correctiDn factor for rolling
moment

hinge-moment factor for theoretical load
caused by streailine-curvature correction
(reference 5)

experimentally determined reduction factor for
F to include effects of viscosity

trailing-edge angle, degrees

parameter defining spanwise location (cos-1 --\
\ b/2/
constants



lifting-line theory








'AC!.'-, ARR No. L5F23
J~ii 2


LS lifting-Gcrface theory

EV edge-velocity correction

SC streamline curvature

max maximum

o outboard

i inboard

e effective

c compressibility equivalent


I -AP P LI C ACTION OF METH OD T O

S T I C K F 0 R C E E S T I M AT I 0 N S

GENERAL METHOD


The values of the damping in roll C2p presented
in reference 2 were obtained by applying the Jones
edge-velocity correction to the lifting-line-theory
values. For a wing of aspect 6, the Jones edge-velocity
correction reduces the values of Cjp by about 8 percent.
From the data obtained on the electroma~nretic-analogy
model of the elliptic wing of aspect ratio 6, a more
accurate correction to CLp for this aspect ratio
could be calculated. The damping in roll was found
to be 13 percent less than that given by lifting-line
theory. The results were extended to obtain values
of CLp for wings of various aspect ratios and taper
ratios. These values are presented in figure 2. The

parameter \/ M2 is included in the ordinates and
abscissasto account for first-order cormpressibility
effects. The value of ao to be used in figure 2
is the value at M = 0.

The method of estimating aileron stick forces
requires the Use of the parameter a Ch









FTAc. AFR Do. LJoZ3


Because Ch can te fo'.nd from the static wind-tunnel
data, it is possible to determine Chp and thus the
effect of rolling upon the aileron stick forces
if (P h is known. Ir. order to av:,id measuring C
at all points to be computed, the effect of rolling is
usually accounted for by estisni.tin, an effective angle
of attack of the rolling wing such that the static
hinge moment at this angle is equivalent to the hinge
moments during a roll at the initial a;,gle of attacik-.
The effective angle of attack is equal to the initial
angle of attack correctted by an incre.nental angle (Aa)Ch
that accounts for rolling, where

a)Ch- (aP) C (1)


The valne of (Aa)Ch is added to the initial a for
the downgoing wing and zubtracted from .the initial a
for the upgoing wing. The value., of h corresponding
to these cor-ected values of a. ai-e then determined
and are converted to ctick force from the knowr:n dynamic
pressure, the aileron dimensions, and the mechanical
advc.ntage.

The value of pb/2V to be 'ised in equation (1)
for determining (&a)cU is (s explained in reference 2)
the estimated value fnr a ric'.i unyawed wing; that is,

pbt 7
2V C L,-,

The value of CL to be used in calculating IApb/2SV
should al.o be corrected for the effect of rolling.
The calculation of pb,/V is therefore ceterimined by
successive approx.imatiz ns. P For the first approxi-
mation, the static values of C, are visd witlh the
value of CL from figure 2. From the fI r t--appro:li-
matlon values of pb/2V, an incremental an-le of
attack (Aa)C is estimated. For. all practical ourrosec,

(ap)CI = (aP)Ch








NACA ARR No. L5F23


and from equation (1),

) = (P) pb
ePO Ch 2V

Second-approximation values of C0 can be determined
at the effective angles of attack a + Aa and a Aa.
The second-approximation value of pb/2V obtained from
this value of CL is usually sufficiently accurate
to make further approximations unnecessary.

In order to estimate the actual rate of roll,
values of pb/2V for the rigid unyawed wing must be
corrected for the effects of wing flexibility and
airplane yawing motion. An empirical reduction factor
of 0.8 has been suggested for use when data on wing
stiffness and stability derivatives are not available
to make more accurate corrections. Every attempt should-
be made to obtain such data because this empirical
reduction factor is not very-accurate actual values
varying from 0.6 to 0.9. The improvement in the
theoretical values of C p obtained by use of lifting-
surface theory herein is lost if such an empirical factor
is used. In fact, if more accurate corrections for
wing twist and yawing motion are not made, the empirical
reduction factor should be reduced to 0.75 when the more
correct values of C0 given in figure 2 are used.

The values of (a)Ch presented in reference 2
were obtained by graphically integrating some published
span-load curves determined from lifting-line theory.
Determination of this parameter by means of the lifting-
surface theory presented herein, however, gives somewhat
more accurate values and indicates a variation of the
parameter with aspect ratio, taper ratio, aileron span,
Fr
Mach number, Ch., and the parameter IcF-
(ca/c\2

In practice, a value of ( Ch equal to the
lifting-line-theory value of h(p) ChL (see appendix)
times the Jones edge-velocity correction
A + 4 AE + 2
parameter Ac + c AEc + 2 is probably sufficiently
Ac + 2 AcE'c + 4
accurate. The incremental angle of attack (Aa)Ch is then








rTcAM. AR. ; Ic. L5-5 192

(AQ)Ch = h

A + 4 cEc + 2 pb
\ P)/ChLL Ac c 2 cc + 4 2V

If further refinement in estir.atinr the stick force
is desired, a s.nall additional lifting-surface-theory
correction -Ch = A pChp L may be added to the

hinge moments determined. For wings of aspect ratios of
from about 4 to 8, values of this additional liftinz-
surface-theory correction are within the usual accuracy
of the measurements of hinge moments in wind tunnels;
that is,

Ch = pb 0.0 02


for a pb/27 cf 0.1 and therefore need not be applied
e:,cept for very accurate work at -hih rpe-d: on lar e
A^ + 4
airplanes. Values of la) Ac + are riven in
S/ ChLL Ac + 2'
figure 3. The effective aserect ratio Ac 1 Mv-
is used to correct for first-order compressibility effects
ArEc + 2
and valves of c are given as a function of Ac
AcE' c + 4
in figure 4. Values -of the correction
2
'hd p) (a/c) -12
LS
are given in f ig're 5 as a function of Ac, and values
of are given in figure 6. The value ,o ni is
(ca, c)
approximately 1 0.000~5". The values of ctb.Ca given
in figure 6 are for control surfaces with an external
overhang such as a blunt-noce or Frise ov.ert.,ng. For
shrouded overhangs such as the internal balance, the
value of cb,/ca should te multiplied -:- about 0.8 before
usin- figure 6.

If the wind-tunnel data are obtained in low-speed
wind tunnels, the estimated values of CZp and aPn)C
should be determined for the wind-tunnel Mach number









27ACA ARFr 14o. L5F23


(assume M = 0). Otherwise the tunnel data must be
corrected for compressibility effects and present
methods of correcting tunnel data for compressibility
are believed unsatisfactory.


ILLUSTRATIVE EXAMPLE


Stick forces are computed from the results of the
wind-tunnel tests of the 0.40-scale semispan model of
the wing of the same typical fighter airplane used
as an illustrative example in-reference 2. Because
the wind-tunnel data were obtained at low speed, no
corrections were applied for compressibility effects.
Because this example is for illustrative purposes
only, no computations were made to determine the effects
of yawing motion or wing twist on the rate of roll but
an empirical reduction factor was used to take account
of these effects.

A drawing of the plan form of the wing of the model
is presented in figure 7. The computations are made at
an indicated airspeed of 250 miles per hour, which
corresponds to a lift coefficient of 0.170 and to an
angle of attack of 1.30. The data required for the
computations are as follows:


Scale of model . .
Aileron span, ba, feet . .
Aileron root-mean-square chord ca, feet .
Trailing-edge angle, 0, degrees .
Slope of section lift curve, a, per degree
Balance-aileron-chord ratio, cb/ca ..
Aileron-chord ratio, ca/c, (constant) .
Location of inboard aileron tip, .. ..
b/2
Location of outboard aileron tip, Yo
b/2
Wing aspect ratio, A ...... ..
Wing taper ratio, A .
Maximum aileron deflection, 6amax, degrees
Maximum stick deflection, esmax, degrees .
Stick length, feet .. ..
Aileron-linkage-system ratio . .
Wing loading of airplane, W/S, pounds per
square foot . .


. 0.40
. 3.07
S0.371
. 13.5
S0.094
. 0.4
. 0.155
. 0.58

. 0.98
* 5.55
. 0.60
. 16
S. 21
. 2.00
. :1

. 27.2










NACA ARR Ho. L5F23


The required wind-tunnel tect results includJ
rol inr~-moimenit coefficients o.ndj hinge-moment coef'icientts
corrected f -r the effects of the jet oourdariecs. Typical
data plotted against aileron deflection are presented
in figure 8. The-e same coefficints cross-plotted
against anile of attacU: for orne-fi~rth, one-half, three-
fourths, and full aileron deflectfons are given in
figure 9. The value of C'Z/ao ~c determined from
figure 2 is 4.02 and th'e valIu of Clp is 0.373. The
A + 4 A + 2
value of a used in equation (2)
CL Ac + 2 A c- + 4
LL c c
tol determine (a) is fund from fiPures Z andh 4 to

be 0.565 and is used to compute both the rate of roll and
the -stick force.

In order ra facilitate the ,onputati:ns, simultaneous
plots of C? and (ia)Ch against pb/EV were made
(fig. 10) .

The stepor in the computation will be explained in
detail for the single case of equal up and down aileron
deflecuions of 40:

(1) From figure 9, the valuesof CL corresponding
to ba = 40 and 6a = -40 at a = 1.'' ae 0.005
and -0.0052, respectively, or a total :tatic C,
of 0.0110.

(2) A first approximation to (j,)Ch taken at the
value of pb/2V corresponding to Cj = 0.0110 in
figure 10 is found to be 0.9g.

(3) Sec:nd-approximation values of CL (fig. 9)
are determined at a = 0.350 for 6, = 40 and at a = 2.250
for 6a=-40, which ;ive a total C0 of 0.0112.

(4) The second approximation to (Aa)C is now
found front figure 10 to be 0.96, which is sufficiently
close to the value found in step (2) to make any additional
approx::iatILons unnecessary.










12 NACA ARR No. L5F23


(5) By use of the value of Cj from step (3), the
value of p = 0.0300 is obtained from figure 10.
2V

(6) From figure 9 the hinge-moment coefficient
corresponding to 6a = 40 and the corrected angle of
attack a = 0.34 is -0.0038 and for 6a = -40
and a = 2.260 is 0.0052. The total Ch is there-
fore 0.0090.

(7) The stick force in pounds is calculated from
the aileron-linkage-system data, the aileron dimensions,
the increment of hinge-moment coefficient, and the lift
coefficient as follows:

Stick force x Travel = Hinge moment x Deflection

where the hinge moment is equal to Chqbac 2 and the
motion is linear.

Substitution of the appropriate values in the
equation gives
2 x 21 16 a
FsF3 57.73 Chqb a

and the wing loading is

= qCL

= 27.2

Therefore,
27.2 3.07 (0.371\2 16 x 57.3
s CL h 0.4 0.4/ 2 x 21 x 57.3

or
Ch
F = 68.4 h
s CL

Thus, when Ch = 0.0090 and CL = 0.170,
0.0090
F = 68.4 x 0.0090
s 0. 170


= 3.62 pounds










[TACA AARE ,. LF...F


This -tick- fcrce is that due t c al.er-,.n deflection and
harE beer: corrected by (ap)0 as determirEd with the
EV
Jone- edge-'elocity correction apflisd to the lifting-
line-theoir value.

(,) The small add.tion-l lift ng---urtfacc correction
to the hine moment (fig. 5) is obtained fror:


,LS


(ac) a
= 0.0207
r i'


and nce = 13.50,

: = 1 0.00.'5(13.5)
= 0.91


From figure 6,


F0
(ca/c)


Therefore,


S(Chp) LS


= 0.0207 x 0.91 x 0.55


= 0.0103


and


C Ch )LS


= 0.010.. x 0.(3


= u. .U 5


(9) The AF, due to the addiition-.l lifting-
surface correction of step (8) may : be e:-presred as

60 l'-' /LS


= 0.124 pound









NACA ARR Io. L5F23


Then,

Total stick force = Fs + AFs

= 3.62 + 0.124

= 3.74

The stick-force computations for a range of aileron
deflection are presented in table I. The final stick-
force curves are presented in figure 11 as a function of
the value of pb/2V calculated. for the rigid unyawed
wing. For comparison, the stick forces (first-approximation
values of table I) calculated by neglecting the effect
of rolling are also presented. Stick-force characteristics
estimated for the flexible airplane with fixed rudder
are presented in figure 11. The values of pb/2V obtained
for the rigid unyawed wing were simply reduced by applying
an empirical factor of 0.75 as indicated by the approxi-
mate rule suggested in the preceding section. No calcu-
lations of actual wing twist or yaw and yawing motion
were made for this example.


II DEV EL O PM 1ENT OF METHOD


The method for determining values of CLp and Chp
is based on the theoretical flow around a wing in steady
roll with the introduction of certain empirical factors
to take account of viscosity, wing twist, and minor
effects. The theoretical solution is obtained by means
of an electromagnetic-analogy model of the lifting
surface, which simulates the wing and its wake by current-
carrying conductors in such a manner that the surrounding
magnetic field corresponds to the velocity field about
the wing. The electromagnetic-analogy method of obtaining
solutions of lifting-surface-theory problems is discussed
in detail in reference 4. The present calculations were
limited to the case of a thin elliptic wing of aspect
ratio 6 rolling at zero angle of attack.









UACA ARR 11o. LEP'2 15


ELECT ROM AGI ETIC-A!NALOGY MODEL

Vortex Pattern


In order to construct an electr.nmagnetic-analogy
model of the rolling :jing and wake, it is necessary
to determine fir.-t the vrtex pattern that is to
represent the rolling .ing. The desired vortex
pattern is the pattern calculated by means of the
two-dimensional t-1 eries thn-airfoil theory and
lifting-line theory. The :ad'ri tional a:pec-t-ratio
corrections are estimrnted by.r deermrrining the difference
between the act'i~l .sh~re of the wing an. the shape that
would be req.:.irecia t.:. sustain the lift distribution or
vortex pattern determined from ti-e two-dimensional
theories.

For the special cases of a thin elliptic wing at
a uniform angle of at-cick or in r. steady roll, the
lifting-line-tn-or- -'-.lues of the sp1:n load distribution
may be obtained bt, mriears. of C imle oal..:.lcatiCons i.refer-
ence 6). The span lo-a diw;,ri.o:.iaLjon, for bUth cases
are equal .. the span lead distributions deter.,;i.ned
from strip theory with a uniform reduction in all
ordinates of the span-load curves by an aerodynamic-
induction factor. This factor is -- for the wing
A +
at a uniform anic of attack- and for the winr in
,q + 4
steady roll. The equation for the load at any spanwise
station -J-- of a thin elliptic win- at zero angle of
b/2
a k -,J C
attack rolling steadily with unt wing-tip helix angle
pb/2V is therefore (see fig. 12)


cc0 *rrA \ / 2
c = 1 (3)
c (b,'2V) A + 4 b2,


where a = 2rr.








NACA ARR No. L5F23


The chordwise circulation function 21 from thin-
ccIV
airfoil theory for an inclined flat plate is


2r 1 [2 /X 2 + Cos 2x
ccV + D (4)
/

where x/c is measured from the leading edge. tSee
fig. 13 for values of 2 V

The vortex pattern is determined from lifting-
line theory as the product of the spanwise-loading
ccti
function and the chordwise circulation
cs(pb/2V)

function for all points on the wing and in the
ccLV
wake; thus,

2r ccI 21
cgV(pb/2V) c,(pb/2V) ccV

Contour lines of this product determine the equivalent
vortex pattern of the rolling wing. Ten of these lines
are shown in figure 14. The contour lines are given
in terms of the parameter

2P
c V(pb/2V)
S2r F
.V(pb/2V) ax

which reduces to -
Smax


Construction of the Model

Details of the construction of the model may be
seen from the photographs of figure 1. The tests were
made under very nearly the same conditions as were the
tests of the preliminary electromagnetic-analogy model
reported in reference 4. The span of the model was










:ACA ARR No. LEF23


twice that of the model of reference 4 (6.56 ft ir stec.
of 3.28 ft), but since the aspect ratio is twice as
large (6 instead of 3), the maximum chord is the same.

In order to simplify the consLtruction of the modJl,
only one semrispan of the vortex sheet was simulated.
Also, in order to avoid the large concentrations of
wires at the leading erfe and tips of the wing, this
semispan of the vortex sheet was constructed of two
sets of wires; each of the wires in the set representing
the region of high load grading simulated a larger
increment of A (- than the wires in the set
representing the'region of low load grading.


Downwash I.easurer.:ents

The magnetic-field strength was measured at 4 or 5
vertical heights, 1 spanwise locations, and 25 to 50
chordw.ise stations. A number of repeat tests were made
to check the accuracy of the measurements and satisfactory
checks were obtained.

The electric current was run through each set of
wires separately. With the current flowing through one
set of wires, readings were taken at pointL on the model
and at the reflection points and the sum of these readings
was multiplied by a constant determined from the increment
of vorticiuy A ( \ represented by that set of wires.
\ max /
Then, with the current flowing throu.g- the other set of
wires, readings rere taken at both real and reflection
points and the sum of these readings was multiplied by
the appropriate constant. The induced downwash was thus
estimated from the total of the four readings. The fact
that four separate readings had to be added together did
not result in any particular loss in accuracy, because
readings at the missing semispan were fairly small and
less influenced by local effects of the incremental
vortices. A more accurate vortex distribution was made
possible by using two separate sets of wires. The measured
data were faired, extrapolated to zero vertical height,

and converted to the downwash function b as dis-
max
cussed in reference 4. The final curves of wb are
rmax









IACA ARR No. L5F23


presented for the quarter chord, half chord, and three-
quarter chord in figure 15. Also presented in figure 15
wb
are values of 2 calculated by lifting-line theory
-max
and values calculated by lifting-line theory as corrected
by the Jones edge-velocity correction.


DEVELOPMENT OF FORMULAS

General Discussion


Lifting-surface corrections.- The measurements of
the magnetic-field strength (induced downwash) of the
electromagnetic-analogy model of the rolling wing give
the shape of the surface required to support the distri-
bution of lift obtained by lifting-line theory. Correc-
tions to the spanwise and chordwise load distributions may
be determined from the difference between the assumed
shape of the surface and the shape indicated by the
downwash measurements. Formulas for determining these
corrections to the span load distributions and the rolling-
and hinge-moment characteristics have been developed in
connection with jet-boundary-correction problems (refer-
ence 5). These formulas are based on the assumption
that the difference between the two surfaces is equivalent
at each section to an increment of angle of attack plus
an increment of circular camber. From figure 15 it may
be seen that such assumptions are justified since the
chordwise distribution of downwash is approximately
linear. It should be noted that these formulas are based
on thin-airfoil theory and thus do not take into account
the effects of viscosity, wing thickness, or compressi-
bility.

Viscosity.- The complete additional aspect-ratio
correction consists of two parts. The main part results
from the streamline curvature and the other part results
from an additional increment of induced angle of attack
(the angle at the 0.5c point) not determined by lifting-
line theory. The second part of the correction is
normally small, 5 to 10 percent of the first part of
the correction. Some experimental data indicate that the
effect of viscosity and wing thickness is to reduce the
theoretical streamline-curvature correction by about
10 percent for airfoils with small trailing-edge angles.









:.ACA ARE: i'. LEF23


Essentially the same final answer is therefore obtained
whether the corrections are applied in two parts (as
should be done, strictly speal:ing) or whether they are
applied in one part by use of the full theoretical value
of the stream?.ine-curvature correction. The added
simplicity of using a single correction rather than
applying it in t'vo parts led to the use of the method of
application of reference 3.

The use of the single correction worked very well
for the ailerons of reference 3, :'which were ailerons with
small trailing-edge angles. A study is in progress at
the Lanley Laboratories of the ITACA to determine the
proper aspect-ratio corrections for ailerons and tail
surfaces with beveaed trailing edges. For beveled
trailing edccs, in v',hich viscous effects may be much
more pronounced than in ailerons with small trailing-
edge angles, the reduction in the theoretical streamline-
curvature correction may be corsierably more than
10 percent; also, when Cha is po-sitive, the effects
of the reduction in the rtraaraline-curvature correction
and the additional down:ash at the 0.50c point are
additive rather tnan compensating. Although at present
insufficient data are available to determine accurately
the magnitude of the reduction in the streamline-
curvature correction for beveled ailerons, it appears
that the simplification of applying aspect-ratio correc-
tions in a single -tep is not allowable for beveled
ailerons. The corrections will therefore be determined
in two separate parts in crder to keep them general:
one part, a streamnlne-cturvacure correction and the other,
an angle-of-attack correction. An examination of the
experimental data available indicates that more accurate
values of the hinge moment resulting from streamline
curvature are obtained 1by multiplying the theoretical
values by an empirical reduction factor n wh-ch is
approximately equal to 1 C'.CO00502 where is the
trailing-edge angle in degrees. This factor will
doubtless be modified when further experimental data
are available.

Compressibility.- The effects of compressibility
upon the additional aspect-ratio corrections ..;ere iot
considered in reference 3. First-order compressi-
bility effects can be acco noted for by application of
the Prandtl-Glauert rule to lifting-surface-theory
results. (See reference 7.) This method consists in









iTACA ARR No. L5F23


determining the compressible-flow characteristics of an
equivalent wing, the chord of which is increased by the
ratio where M is the ratio of the free-
1l M2
stream velocity to the velocity of sound. Because
approximate methods of extrapolating the estimated
hinge-moment and damping-moment parameters to wings
of any aspect ratio will be determined, it is necessary
to estimate only the hinge-moment and damping parameters
corresponding to an equivalent wing with its aspect

ratio decreased by the ratio /I M2. The estimated
parameters for the equivalent wing are then increased

by the ratio -....
/1 M2

The formulas presented subsequently in the section
"Approximate Method of Extending Results to Wings of
Other Aspect Ratios" are developed for M = 0, but the
figures are prepared by substituting Ac = A/ M2
for A and multiplying the parameters as plotted
by Y M2. The edge-velocity correction factors Ec,
Eec, E'c, and E'ec are the factors corresponding to Ac.
The figures thus include corrections for first-order
compressibility effects.


Thin Elliptic Wing of Aspect Ratio 6

Damping in roll Cp.- In order to calculate the
correction to the lifting-line-theory values of the
damping derivative CLp it is necessary to calculate
the rolling moment that would result from an angle-
of-attack distribution along the wing span equal to
the difference between the measured downwash (determined
by the electromagnetic-analogy method) at the three-
quarter-chord line and the downwash values given by
lifting-line theory. (See fig. 15.)

Jones has obtained a simple correction to the
lifting-line-theory values of the lift (reference 1)
and the damping in roll (unpublished data) for flat









"ACA ARR ito. LEF23 21


elliptic wings. This correction, termed the "JonAs
edge-velocity correction," is applied by: multiplying
the lifting-line-theory values of the lift byr the
Ap +
ratio A+ 22 and the lifting-line-Lheory values
AcEc + 2
A + ,
of the damping in roll by Acc + with values of Ec
"cE' c+ 4
and E'c as given in figure 16.. As may be seen from
figure 15, the dov;nwash given by the Jones edge-velocity
correction is almost exactly that measured at the
0.50c points for flat elliptic wins. This fact is
useful in estimating the lifting-surface corrections
because the edge-velocity correction, w'jhich is given
by a simple formula, can be ued t.- correct for the
additional angle of attack indicated by the linear
difference in downwash at the 0.50c line.

The variation n dovrnwash between the O.25c line
and 0.75c line, apparently linear along the chord,
indicates an approximately circular streamline curvatLure
or camber of the surface. The increment of lift resulting
at each section from circular camber is equal to that
caused by an additional anrle of attack given by the
slope of the section at 0.75c relative to the chord line
or the tangent at 0.JOc that is, (- (.
'O. 7c 'O.50c
Because this difference in downwash does not vary linearly
along the span, a spanwise integration is necessary to
determine the rtreamline-curvature increment in rolling
monent; that is,

/ c B iQc '1 wb
(C)SC bV(AcE'c + 4) max/0.



-:~nS (-^\ L C d/^ (5)
naxOc.5c e b/2 \b/2)

An evaluation of rmax in terms of pb/2V is necessary
to determine the correction to the damping-moment
coefficient CLp. The lifting-line-theory relation








TfACA ARR No. L5F23


between rmax and pb/2V is, from equation (3),
n 2Vb(pb/2V)
"max A + 4

,7ith the edge-velocity correction applied
2Vb(pb/2V)
.max AcE'c + 4

The value of the streamline-curvature correction
to Cp is therefore
A 4T r( 1. wb \
-SC cc + 4) 2 [ Kmax/O.75c



wb d- (7)
rn0 /2 -/2 2)
KPmaxO.50c b/-2 x/

A graphical integration of equation (7) gives a value
of 0.022 for (AC .
\ p SC
By the integration of equation (5), the value
of (CL) for incompressible flow is found to
\ P/LL
be H A 0.471 for A = 6.

Application of the edge-velocity correction, for A = 6,
gives

P EV 4(AE' + 4)
= 0.433
and, finally, subtracting the streamline-curvature
correction gives a value of Cjp, for A = 6, as
follows:

C7p = (9 E) P C EV


= 0.411








TATCA ARR No. L5F23


' -- ri .- f 7 fl o' wing of 1 sc.c.-t r: tlo 6
" th.icrefore 1z percent les- than the vavlu- jiveI by
lifti -line thec.. and nercent e' :. tha. that
given by,- lift ing-line theory e .it'h the .Toners- ede-
velocit:,r ; -c rect.or. applied.

-invr e-nmoment per'-eter Ch .- The -trear'lin--
currvatiire corre-tion to 0h for c n: c;r~t-: fr i.'ntao-
chord -.;lerrrsn is. ifrm reference 5 an'r. wvith the value
of Pn c;lvern eia ntion (C),


.. vi .
.,...wrI/ i H -f T ( )
c( (x'c) (\cs
f ACE'c + 4 d .

where the inctecrA tonss Ear e mn.de acrcs t:-.e .Lleron spCn.
Pec.au'. e the .down .ash at the C.' 0 c jcint' is ,-'.en
satis ac'tori ly b appl.y.ing the edge-v:-locit.;,- .c-ctior
to the lift ing-lne-theory values :f thF d.: 'wnwa'.'h, the
par-t oD- the corr-ection: to C' which dependics upm: the
downwash at the 0.53c pc-int .ay be i.eter.minc bby rEisanc
of the eFde-'.eloc ty correctior. The effect -f aFre...d.rinamic
induct ion .'tas ner'.ec ted in ie" el3oiin' ecqu ;.. -.r ) tec i;
aerodynamic inductio,, has a ver-r small effect .'c.on the
binre-rmo.ent, c-r-ec ti.;n- au ca))sd oy; .tr,'&arline cu.rvatuLre.

V-.]ueS ,of the- factor -- for, ,arious tiler:.n-
(I2/c )'-
chord ratios ard ban st ,s as detcrr.ine'd fr;.-,- thin-
airfoll theory-- are gIven in r'l.re 6. ..s ;uent i ned
p~evi.1sl r, is a factor- th.st r.ap3 ro-iris ely aer:cou. nt.
fvr the cornmir ed effects of wvir:. thick ne s an'd visc:3si t
in "lte. in ch.e calciladted values of .. The

experiimcrntil &Seta vaivlatrle at .'rE'-ent inidica3te
that rn 1 C0.'C0-'5. Re lts th:- nt -ationr of
equation (8) for the ellipt.i- winl of .ect ratio 6
arc- -iven in f' qur e 17 w.- t r p.a ancter

,] + -- V, 1u s
A C ( ) + \/ l V.iL.L s







24 "A.- ARR No. L5F23
(ca//c2 ___
of (ACh) -- (A + 1) l M2 determined as
in reference 3 are given in figure 18.
The value of (Chp) is


(ChphLS


h ( ( LL A+ 4 + AChp)5
a~I ~ ChLL


Since


Ac + 2
AcEc + 2 ,Ch LL
c LL


: h~) LS


- (Ch) SC


A + 4 AE + 2
= (P)ChLL Ac' + 4 Ac + 2 (Ch!LS


+ (LCh) SC


(h) LS


- / Ch Ac + 4 ArcE + 2
SC /SC pchLL AcE'0 + 4 Ac + 2
SC LL


(P (EV a) L Chp)
-Oy 'Lb*L


The formula for the parameter ChLL is derived
for elliptic wings in the appendix, and numerical values
are given in the form (a \ c in figure 3,
\ /ChLL Ac + 2
together with values for tapered wings derived from
the data of reference 2.
It may be noted that use of the parameter (p)ChLS
determine the total correction for rolling would be
impractical because ChpL is not proportional
\Op is- no rpotoa
'J-ib


then


(Ch )LS








:"ACA i7r. :o. LEF23 25


to (bh \. Although the numerical values of (.)ChL

vary considerably with h the actual effect on
/ js
the stick forces is small because r C changes most

with h when the values 1of Ch l are small.
This effect is illustrated in figure 19, in 4hich
numerical values of (apCh for a thin elliptic wing

of aspect ratio 6 are given, together with the values
obtained by liftin.i-line theory., the values obtained by
applying the Jones edce-velocity correction, and the
values obtained. b- using the aileron midpoint rule
(reference 8). The values obtained by the use of the
Jones edge-velocity correction .re shi.w.n to be 4.4 percent
less than those obtained by the use of lifting-line theory.

The right-hand side of equation (9) is divided into
the following two parts:

Part I (ap)ch (Ch
Part I = hEV LS


Part II = A SChp)L


Part I of the correction for rolling can be applied
to the static hin7e-rorrite-t data as a change in the
effective angle of attack as in reference 2. (Also
see equation (2) .) Part II of equat in (9), however,
is applied directly as a change in the hinge-moment
coefficients,

'Ch A(11) LS ib

Inasmuch as part II of equation (9) is numnerically
fairly small Arh = 0.002 for = 0.1 for a
wing of aspect ratio 6), it need not be applied at all
except for fairly large airplanes at high speed.








TACA ARR No. L5F23


Approximate Method of Extending Results

to Wings of Other Aspect Ratios

Damping in roll Cp.- In order to make the results

of practical value, it is necessary to formulate at least
approximate rules for extending the results for a thin
elliptic wing of aspect ratio 6 to wings of other aspect
ratios. There are lifting-surface-theory solutions
(references 4 and 9) for thin elliptic wings of A = 3
and A = 6 at a uniform angle of attack. The additional
aspect-ratio correction to CL was computed for these
cases and was found to be approximately one-third greater
for each aspect ratio than the additional aspect-ratio
correction estimated from the Jones edge-velocity
correction.

The additional aspect-ratio correction to C1p
for the electromagnetic-analogy model of A = 6 was
also found to be about one-third greater than the
corresponding edge-velocity correction to C p. A
reasonable method of extrapolating the values of Cp
to other aspect ratios, therefore, is to use the
variation of the edge-velocity correction with aspect
ratio as a basis from which to work and to increase the
magnitude by the amount required to give the proper
value of CL- for A = 6. Effective values of E
and E' (Ee and E'e) were thus obtained that would
give the correct values of CL_ for A = 3 and A = 6
and of C;p for A = 6. The formulas used for esti-
mating Eec and E'ec for other aspect ratios were


Eec = 1.65Ec 1) + 1

E'ec = 1.65 (E'c 1 + 1

Values of Eec and E'ec are given in figure 16.

Values of -i /l M2 determined by using E'ec
are presented n ure 2 as a function of A/ao
are presented in figure 2 as a function of Ap/ao








NACA ARR No. L5F23


where Ac = A /,1 M2 and a3 is the incompress'ble
slope of the section lift c'-uve per degree.

Hinge-moment parameter Ch .- In order to deter-
mine Chl for other aspect ratios, it is necessary to
estimate the formulas for extranplating the streamline-

curvature corrections (ACCh and ACh Values
s' c \ P/sc

of (ACh S) for A = 3 and A =6 ere available in

reference 3. Values of Cha) might be expected to be

approximately inversei:,- proportional to aspect ratio and
an extrapolation fcrm.,ula in the for--( ICh,
L j C ha A + K
is therefore considered satisfactory. The values of K1
and K2 are determined so that the values of LCh
for A = 3 and A = 6 are correct. Values of K and K2
vary "with aileron scan. The values of K2, however,
for all aileron spans less than 0.6 of the .emispan are
fairly close to 1.0; thus, by assLuniin. a constant value
of K2 = 1.0 for all aileron spans and calculating
values of I1-, a satisfactory extrapolation formula
may be obtained. It is impossible to determine such a
formula for Ch~ because results are available
only for A = 6; however, it sels reasonable to assume
the same form for the extrapolati :in formula and to use
the same value of K. as for (iCh The value

of K1 can, of course, be determined from the results
for A = 6.

Although no proof is offered that these extrapolation
formulas are accurate, they are applied only to part II
of equation (9) values of Ci- which is numeri-

cally quite small, and are therefore considered justified.








JACA ARR No. L5F23


CONCLUDING EEMARkS


From the results of tests made on an electromagnetic-
analogy model simulating a thin elliptic wing of aspect
ratio 6 in a steady roll, lifting-surface-theory values
of the aspect-ratio corrections for the damping in roll
and aileron hinge moments for a wing in steady roll were
obtained that are considerably more accurate than those
given by lifting-line theory. First-order effects of
compressibility were included in the computations.

It was found that the damping in roll obtained by
lifting-surface theory for a wing of aspect ratio 6
is 13 percent less than that given by lifting-line
theory and 5 percent less than that given by the
lifting-line theory with the Jones edge-velocity correc-
tion applied. The results are extended to wings of any
aspect ratio.

In order to estimate aileron stick forces from
static wind-tunnel data, it is necessary to know the
relation between the rate of change of hinge moments
with rate of roll and the rate of change of hinge
moments with angle of attack. It was found that this
ratio is very nearly equal, within the usual accuracy
of wind-tunnel measurements, to the values estimated by
using the Jones edge-velocity correction, which for an
aspect ratio of 6 gives values 4.4 percent less than
those obtained by means of lifting-line theory. The
additional lifting-surface-theory correction that was
calculated need only be applied in stick-force esti-
mations for fairly large, high-speed airplanes.

Although the method of applying the results in the
general case is based on a fairly complicated theory, it
may be applied rather simply and without any reference
to the theoretical section of the report.


Langley Memorial Aeronautical Laboratory
national Advisory Committee for Aeronautics
Langley Field, Va.








7TACA ARR No. L5F23


APPENDTX


EVaLUATIOI7 OF o p) FOR ELLIPTIC WINGS
ChLL

It was shown in reference 2 that for constant-
percentage-chord ailerons the hinge moment at any aileron
section is proportional to the section lift coefficient
multiplied by the square of the wing chord; for constant-
chord ailerons, the hinge moment at any aileron section
is proportional to the section lift coefficient divided
by the wing chord. The factor (ap is obtained
LL
by averaging the two factors cjc2 and cl/c across
the aileron span for a rolling wing and a wing at
constant angle of attack. For elliptic wings, with
a slope of the section lift curve of 2n, it was
shown in reference 6 that strip-theory values multiplied
Ac A
by aerodynamijc-induction factors or c
'c +2 A, +4
could be used. (iHote that A is substituted for A
to account for first-order effects of compressibility.)
Thus, for constant-percentage-chord ailerons on a rolling
elliptic wing,

S2 A- sin28C 22n
c -Ac + 4 s V


2nrc 2Ae b sin20 cos 9
Ac + 4 2V

and for the same wing at a constant angle of attack a

cLc2 Ac sin2r Cs22na
A- 0C + 2
Uc

In order to find C ,the integral c d

across the aileron span must be equal for both the








:TACA ARR No. L5F23


rolling wing and the wing at constant a. Thus,


c Lc2 dy
u.


21- cs2Ac pb sin2e cos dy
Ac + 4 2V /d


2=Acs2Ac a sin28 dy
C Ac + 2


dy= 2 d(cos 9)

b sin e d9
2


Let


a = (ap cLL


Then


(p) ChLL


Ac + 2 sin3e cos e do
Ac + 4 in3e do


Ac + 2
Ac + 4


S[sin4eol

S1 o
7isin2o cos 0 + 2


9o
cos 90
4 i


where e0 and 9i are parameters that correspond to
the outboard and inboard ends of the aileron, respectively.


Values of


A + 4
a L Ac + were calculated for the
PChLL Ac + 2


outboard end of the aileron at Y = 0.95 and plotted
b/2
in figure 3.









:IACA ARR 1:0. LEF2


A similar develo.i.mrnent iv,.'e:, fc::r the c.nstant-ich:,rd
ailer.n-i,r


cl 2nrAc pb
f- + 4 =- -
Sc c ch + 4^ 2V
a c


cos 6
--J


II;
c s (A + 2 V 2 hLL -s In 0



/ \ + + 4 S os q d9
-' ChLL -c I'L




in 6


These value are aL so presented in fic.-re 3.









ITACA ARR No. L5F23


REFERENCES


1. Jones, Robert T.: Theoretical Correction for the
Lift of Elliptic Wings. Jour. Aero. Sci., vol 9,
no. 1, Nov. 1941, pp. 8-10.

2. Swanson, Robert S., and Toll, Thomas A.: Estimation
of Stick Forces from Wind-Tunnel Aileron Data.
NACA ARR No. 3J29, 1943.

3. Swanson, Robert S., and Gillis, Clarence L.:
Limitations of Lifting-Line Theory for Estimation
of Aileron Hinge-Moment Characteristics. NACA CB
No. 3L02, 1943.

4. Swanson, Robert S., and Crandall, Stewart M.: An
Electromagnetic-Analogy Method of Solving Lifting-
Surface-Theory Problems. NACA ARR No. L5D23, 1945.

5. Swanson, Robert S., and Toll, Thomas A.: Jet-Boundary
Corrections for Reflection-Plane modelss in
Rectangular Wind Tunnels. NACA ARR No. 3E22, 1943.

6. Munk, Max M.: Fundamentals of Fluid Dynamics for
Aircraft Designers. The Ronald Press Co., 1929.

7. Goldstein, S., and Young, A. D.: The Linear
Perturbation Theory of Compressible Flow with
Applications to Wind-Tunnel Interference.
6865, Ae. 2252, F.M. 601, British A.R.C., July 6,
1943.

8. Harris, Thomas A.: Reduction of Hinge Moments of
Airplane Control Surfaces by Tabs. NACA Rep.
No. 528, 1935.

9. Cohen, Doris: A Method for Determining the Camber
and Twist of a Surface to Support a Given
Distribution of Lift. NACA TN No. 855, 1942.










NACA ARR No. L5F23


0
o

0
0


0
1-4
1-4
N





ID

10




'-1
K'


0
01

*
0

0




0

01








8
*
0




0

O0

0








1-4


40







'II


t O 03 .-4
I- 0o .-4 t0 DIO to
0 0 N 4 '-4 C 0)
0 0 0



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0o -o
- 0 11 n0 4m ca I
0 0 N 1-i r- 0m I-4
.0 0 0 *
o to t*- 0
!
1 0 0 0





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0 .0 .
o N 1o 'o










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U

a0 0 0 N 0 .

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to t 0a t3 CO to -
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w 0 0- o H I
'r O r- r4 01o 002 D
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NACA ARR No. L5F23 Fig. la












-4





040









&4
e to















,-i
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4iZ* 4
o













bD -v
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NACA ARR No. L5F23 Fig. lb

















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NACA ARR No. L5F23


a, I I I












/






NATIONAL ADVISORY
COMMITTEE FOR AERONAUTICS


60

ao


60 /00
, l/-9d f
0


/2C /'fo /&0


Figure 2.- VariaLion of damping coefficient wilth aspect ratio
and taper ratio in Lerms of slope of section lift curve for
incompressible flow (per degree). tLift ne-line-tieory
values of reference 2 wi t an effective edge-velocitj
correction applied.)


Fig. 2








NACA ARR No. L5F23


Fig. 3


dv'Jc
3-I-i-

-4
-4


NATIONAL ADVISORY
COMMITTEE FOR AERONAUTICS
O .2 .4 .8 /o
Re/af/ve /oc4ton of /ndoorad ,//ron f/ ,


Ac + 4
Figure 3.- Values of (ap)hL Ae + for various aileron

tip locations. The corrections of figure 4 must be
used with these values.








NACA ARR No. L5F23 Fig. 4












0-,

Z
\ o




O(d
-- --l -






I c-
-- -- i



0.

---- ---- ----------- '4 %


--- ----- cg.










sL







-4
_______ ______ ______ ______ ______ ______ _______, ______ ______ _______ ------C .


,_ V
-t3 V








Fig. 5 NACA ARR No. L5F23


















01 0
-p r 1 4



o








S 0










>g \ o
I O

















/ 0
bo















r14
9 7 a

O a
-- -
^ s
-^' s-<
/^o~ 9 h
___ ^____ ___ ___ ___ ___ ___;SP <
---- --- --- --- --- o
,'" I








u-
,; ^ -^ J .






NACA ARR No. L5F23


/.f


A/

/.



/0






.7
I '
.6







.',


c 3.
---- -.








AY/
,///
-____-- ---_


/ /


















NATIONAL ADVISORY
CO{mITTEE roR MAIE AUTICS
//! / / z __ i_
/// i / / /^
I_ LL l _,1"_ _/_
___'1 /z ,z /_- _
_^ z _/Z _
l-il -/ i-
Ill/ i 1^
- lr -i-" -





; / ONHITEE 0 MlO~tTC
/i I /zzzz


A,/eron -ch/ord ra foo,


ca/C


Faw.-, 61. -/o frtation o the IAre -malewl correc/on factor le ,^s
wi th a leron- chord rat/o ,and e ter-nd/a'-oyerhany
aerodynawnl/c o/ance. -chord raf/o. For ilrternal/ erod namIc
bo/aoce, use an effectf/e c e 0.8 c, (reference 5).


Fig. 6









Fig. 7 NACA ARR No. L5F23














//- 0 |


I< >


Z I-





C o










IV
1 o

.U
*1






I
--o
ri

/ s


I 1 I 0



I / ^

l i t <
I / ri
I f PL







NACA ARR No. L5F23


.O4

- 0
QJ



-.08



Qa
S-./6


-25 -20 -15 -10 -5 0 5 /0 /5 20 25
Aileron deflection, SC, deg
Figure 6.- Aileron hinge- rind rolling- moment characteristics
of the 0.40-scale mode/ of the airplane used for the
///us/roaive example. Characteri/sthc p/ol/ed ogo/nsf
o//eron def/ection.


Fig. 8






NACA ARR No. L5F23


Fig. 9





.1C

C-)
:.08

S.04
0




08







S.03


.0
o

-:08


















-02
-.03


-4 -2 0 2 4 6 8 /0 /2 /4 /6 1/
Angle of attack, cr, deg
Figure 7.-Aileron hinge- and roll//ng-momnenl characferisic
of the 0.40-sca/e mode/ of the airp/one used for the
i//ustra//ye example. Character/stics p/oiled against
ang/e of attack.


NATIONAL ADVISORY
COMMITTEE FO AEROIIAUTICS








NACA ARR No. L5F23




.08


.07--

.06

o .0-----




.03-
F
0.02----

S- EE
N,


82


k
n


Fig. 10


NATIONAL ADVISORY
COMMITTEE FOR AERONAUTICS


0 .,' .03 .04 .05 .0o .07 .0 .09 ./0 .1
h'e/l/x ange .ob/2V rad/an
F-qure /0 .-Pre//l, .., ary cur ves5 ,sed n .he corr.oua/'a ..c
of the a ..!- t,o Ae,,x ang'/es andA /ne //c c forces
:or //e d//a/s-ra/lve examp/e.







NACA ARR No. L5F23


A/eron def/lecon
(de9)
20 / -

/6 (a) Effect of ro//in,yaw yawiny, and wng twist
neq/ected /n both and p6lV. ,

/26 ----------- -----/-


8--------------------______


0 -----------------------------------------*---
O -- --
Ile


0


Z 1(b) Effect of ro/llly accounted for in both / and pb/2Ky,
effect of yaw,yaw/ny, and wing twist accounted for in
Fs bu t not in b/









/ (c) Effect of ro/ly account ted for in both {and p, 2V;
effect of yaw,yawny, and wgnj twist accounted for
n s and assumed to --

25percent -- NATIONAL ADVISORY
COMMITTEE FOlR AERONAUTICS

0 __I

0 .0/ .O2 .03 .04 .05 .06 .07 .08 .0? ./0 .//
Hel/x anyle, fL 2V, radian
Fare P/. -Stc/A -force char~acterlsics esatmated for the
airplane based for /1e I//astra11;e ex amo/e.


Fig. 11










NACA ARR No. L5F23


Figs. 12,13


o










A----i I


















o=

A 0 7
*-!- t0^3 ^ UlSC/ft^/


F9^j? to~ ^ pI uodL


N


a





*1?


a

U


. 1,


,L
t L.



L
. -



C i



C
j.
C e-
- L


,s 1


" 1






- .-
F?.


'4












'I j


; 3


2


I






Fig. 14 NACA ARR No. L5F23-



IC




















tot








4,
w//ox ---------- --


^ \\^^^ --





^ \V ^^_ __^
\>. \ \. --
\ \ ^ ^-____ ^







NACA ARR No. L5F23


Liftiny-/iu theo-rA y wnrP'A Jones/
edge-ve/ocftu correction --
S(so/d //ne) -
.,5v c /-
Downwash \ / /
measured at .75c





//

'' lilfH -/,Ape











NATIONAL ADVISORY
COMMITTEE FOR AERONAUTICS


I I I I/ I I
oa*-v.v/se 1/s3c


/


Figure 15.- Induced dowr.wash function wb/2.8ax for
an elliptical lifting surface of aspect ratio 6 in
steady roll.


wb
2/3,Id


Fig. 15







Fig. 16 NACA ARR No. L5F23


i-- \ ---- i---- i------~--rrn O


g




k
ts












-- -- p 4 +-
o










'3t





,, ', Q
K VC










o




----l ix
i c, / l i



-- --y -~.




-------------/----^- 144
-- --q- G
_' a
__ ^
-^-^-^-^-^
-- -- ^ ^ -- r y -- --
-- ^^--- ^ ^ --^-------- ^ ^


3-7 ,'3c '7











NACA ARR No. L5F23


Figs. 17,18


a .3 .3








C







- .3 ..I

.3 .3


- 3.


SUJ

- .3




.* .3 -

0.3
C. C.
-) 3 U
I. -, a
C- .3 .3


CL.
0C

) -,
.3 .E
3. C- 1
C. 3l





.3 -
a .3



- --
-. ii, s


J 3S

^w^^-s^'P9 rq^{^ 7P)(a>?)


1J ~r

(o3v)







NACA ARR No. L5F23


r',# /
Indoad aleronh/2.~


Figure 19.- Values of parameter (ap)Ch from aileron-
midpoint rule, lifting-line theory, lifting-line theory
with edge-velocity correction applied, and lifting-
surface theory for an elliptic wing of aspect .ratio 6.
M = 0.


Fig. 19







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S262 0814 954 5





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