'Hk L'ii
NATIONAL ADVISORy COMMITTEE FOR AERONAUTICS
WARTIME REPORT
ORIGINALLY ISSUED
August 19^5 as
Advance Eestricted Eeport L5F23
LTFTUKJ-SaRFACE -THEORY VALUES OF THE BAMPIBCr
Df EOLL AHD OF THE PARAMETER USED Hi
ESTIMATIUG AHJEROU STICK FORCES
By Roljert S. Swaneon and E. LaTeme Prlddy
Langley Memorial Aeronautical Laboratory
Langley Field, Va.
^ MAC A
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. I
L - 53
DOCUMENTS DEPARTMENT
Digitized by tlie Internet Arcliive
in 2011 with funding from
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http://www.archive.org/details/liftingsurfacethOOIang
7/2. (f(7(^i-
^kCA ARR No. L5F23 RESTRICTED
NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS
ADVANCE RESTRICTED REPORT
LIFTING-SURFACE-THEORY VALUES OP THE DAMPING
IN ROLL AND OF THE PARAMETER USED IN
ESTIMATING AILERON STICK FORCES
By Robert S. Swanson and E. LaVerne Prlddy
SUMMARY
An Investigation was made by lifting- surf ace
theory of a thin elliptic v/ing of aspect ratio 6
In a steady roll by means of the electroraagnetic-
analogy method. Prom the results, aspect-ratio
corrections for the damping in roll and aileron hinge
moments for a wing in steady roll ?;ere obtained that
are considerably more accurate than those given by
lifting-line theory. First-order effects of com-
pressibility Vifere included in the computations.
The results obtained by lifting- surface theory
indicate that the dam.ping in roll for a v/ing of aspect
ratio 6 is 13 percent less than that given by lifting-
line theory and 5 percent less than that given by
lifting-line theory v;ith 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 v/ind-tunnel data, it is necessary to knov/ the
relation between the rate of change of hinge moments
with rate of roll and rate of change of hinge moments
with angle of attack. The values of this ratio were
found to be very nearly equal, v.'ithin the usual accuracy
of wind-tunnel m.easurem.ents, to the values estim.ated
by using the Jones edge-velocity correction, vifhich for
a wing of aspect ratio 6 gives values 4.4 percent less
than those obuained by lifting-line theory. An
additional lifting-surface-theory correction was
RESTRICTED
NAG A ARR No. L5P23
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- surf ace theory is
required to apply the results. In order to facilitate
an imderstanding of the procedure, an illustrative
example is given.
INTRODUCTION
One of the many aerodynamiic problems for which
a theoretical solution by m.eans 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 m.ay 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 momient v/ith
rate of roll Gj-, and the rate of change of hinge
of
Ch
moment v/ith angle of attack G]^ in the form of the
parameter lay.) = -^ , which is determined by m.eans
of lifting-line theory. It was pointed out in reference '.
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 (a^^. p is desired. In reference 3 an addi-
tional aspect-ratio correction to Ci, as determined
from lifting-surface theory is presented. In order
to evaluate the possible errors in the values of (ctpV
IIACA ARR No. L5P23 3
as dsterir.ined. by lifting-line theory, it is necessary
to determine similar additional aspect-ratio corrections
to Gi-,^.
A description of the methods and equipment required
to solve lifting-surface-theory problems by 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 steady roll v;as
constructed (fig. 1) and the magnetic-field strength
simulating the induced downwash velocities was measured
by the methods of reference 4. Data were thus obtained
from which additional aspect-ratio corrections to Chi^ ^ov
a wing ■ of aspect ratio 6 were determined.
Because of the small magnitude of the correction
to (^r,\n introduced by the lifting-surface calculations,
it was not considered v;orth while to conduct further
experiments on wings of other plan forms. An attempt
was therefore made to effect a reasonable generalization
of the results from, the available data.
Inasmxuch as the theory 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 m.aterial 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 II gives the development of the theory.
SYMBOLS
angle of attack (radians, unless otherwise
stated)
section lift coefficient f Mli
qc
c
I
C^ wing lift coefficient (-i^i^i
L ^ \ qS /
C^ hinge-mom.ent coefficient i^^inge momentA
V qca ha ^
r, TT. 4. wr.j^' • +- /Rolling m.oment\
C, rollmg-mom.ent coefficient f — ::^ -^r )
AC A ARR No. L5P23
i\
a slope of the section lift curve for incom-
pressible flow, per radian iinless otherwise
stated
ph/2V viflng-tip helix angle, radians
r circulation strength
C|, darnping coefficient: that is, rate of charge
P of roll ing-iTioment coefficient with rate
/ ^.C^ \
of roll '- )
\6(pb/2V)_/
C}^ rate of change of hinge moment with rate of
'a
(">0c.
ro
11 I
\d(pb,-^27) ;
Cyi rate of change of hinge moment with angle of
attack 1 -^^^ j
\oa J
Cj^ rate of change of wing lift coefficient
\6a / _
with angle of attack
absolute value of the ratio ( -— i- 1
c wing chord
c_ wing chord at. plane of " syrrmei:ry
c-]-, balance chord of aileron
Cg. chord of aileron
Ca_ aileron root-mean- square chord
X chordwls-e distance from .v;ing leading, edge
y spanwrsc distance from plane of -symmetry
ha aileron span
b/2 wing semi span
NACA ARR No. L5F23 5
S area of \ving
W ;veight of airplane
Pg stick force, pounds
9g stick deflection, degrees
5^ aileron deflection, degrees, positive downward
A aspect ratio
A_ equivalent aspect ratio in compressible
/
flow
(a/i - ir )
A. taper ratio, ratio of fictitious tip chord
to root chord
M fz'-ee- stream Mach nur.ber
v/ vertical component of induced velocity
V free-stream velocity
q free- stream dynamic pressure ; -^pY^ '
E edge-velocity correction factor for lift
E' edge-velocity correction factor for rolling
morae nt
F hinge-moment factor for theoretical load
caused by streamline-curvature correction
(reference 5)
T| experimentally determined reduction factor for
P to include effects of viscosity
trailing-edge angle, degrees
1 n r> n +". i rtT. / o r. ?? ~ ■'-
b/2^
8 paramieter defining spanwise location (cos"-'- — ^
K-]_, Kg constants
Subscripts:
LL lifting-line theory
6 ivIACA ARR Ko. L5F23
LS lifting- s'tirf ace theory
EV edge-velocity correction
SC streamline curvature
max maximum
o oufooard
i inboard
e effective
c compressibility equivalent
I-APPLICATION OP METHOD TO
STICK-PCRC3 ESTIMATIONS
GENERAL METHOD
The values of the damping in roll C^ 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 Gj ^ by about 8 percent
"P ''
^. _-_p(
J.
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 C^-Q for wings of various aspect ratios and taper
ratios. These values are presented in figure 2. The
/ ~
parameter \/l - H^ is included in the ordlnates and
abscissas to account for first-order compressibility
effects. The value of a^ to be used in figure 2
is the value at M = 0.
The method of estimating aileron stick forces
requires the use of the naram.eter (a-n\ - \ , P
\ ^/Ch ICh
a
7
NAG A ARR ITo. L5F25
3ecau.-e Ch can be found from the static wind-tunnel
data, it is^'possible to deteritdne Ch^ and thus the
effect of rolling upon the aileron , sticl: forces^
-:f In \ 1^ known. Tn order to avoid measuring 0-J3
at all points to be computed, the effect of rolling Is
uBuSlly'^accounted for by estimating an effective angle
of attack of the rolling wing such that txie static ^
hinge moment at this angle is equivalent to ^}^l f-_^Jf
moments during a roll at the initial angle 01 ^ttac^.
The effective angle of attack is equal to the ^^^tiai
angle of attack corrected by an incremental angle (Aa)^^^
that accovmts for rolling, where
£^ (1)
(^«^Ci, = («p)ct,
2V
The value of (Aa)c^^ is added to the initial a for
the downgoing wing and subtracted ^^f "-^..^^^^^'^L^^^^J^^aLg
for the upgoing wing. The values of o^ corresponding
to t^ese corrected values of a are then determined
and are coS?;?ted to stick force from the known dynamic
p?essSre, the aileron dimensions, and the mechanical
advantage .
The value of Fb/2y to be used in equation (1)
for determining (Aa)c. -^ i-^ explained in reference .)
the estimated value for a rigid unyawed wing; that is,
pb ^ _SL
2V Ct,^^
The value of C7 to be used in calculating pb/2V
■should also be corrected for the effect of rolling.
?he calculation of pb/2V is therefore determined by
.ScceJ5?v; apnroximatlons. For the first approxi-
matLnpthe static values of C, are usea with tne _
value of C^ from figure 2. From the f irst-approzi-
mat-: on valuel of pb/2V, an incremental ^""^l^t^f r^uvv^ses
Ittack (Aa)c, i^ estimated. For all practical purposes,
(''p)ci = ("p)ch
8 NACA ARR No. L5P23
and from equation (1),
Second-approximation va?.ues of O-j can be determined
at the effective angles of attack a + Aa and a - Aa .
The second-approximation value of pb/EV obtained from
this value of C7 is usually sufficiently accurate
to make further approximations urjiecessary.
In order to estimate the actual rate of roll,
values of pb/2V for the rigid unyavv-ed rnang 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^ 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 vi^hen the more
correct values of C7 given in figure 2 are used.
(. p <•- o
The values of (a-r\ presented in reference 2
were obtained by graphically integrating som.e published
span-load curves determined from lifting-line theory.
Determination of this parameter hj means of the lifting-
surface theory presented herein, however, gives somewhat
more accurate values and indicates a variation of the
param.eter v/ith aspect ratio, taper ratio, aileron span,
F Ti
Mach number, Ch„, and the parameter — ;— 7^
'^ ('■ca/c\2
In practice, a value of (a„\ equal to the
' ^h
lif ting-line-theorv value of (a^jp' (see appendix)
tiro.es the -Jones edge-velocity correction
A + 4 A S„ + 2
parameter — — 2 is probably sufficiently
accurate. The incremental angle of attack (Aa),'^, is then
^h
NACA ARIt Ifo. L5F25
(Aa)c
pk/:
(^p)cb,,.
h " 2V
A^ + 4 A^E^ + 2
LI.
- / \ ^'^c + 4 A ^Eg + 2 pb
- \"P;Ch A^+ 2 IcE'c + 4 2V ^^
If further refinement in estimating the stick force
is desired^ a small additional lifting- surf ace- theory
correction AC^ = A (Cii_\ •^y ma^T- be added to the
hinge moments determined. For wings of aspect ratios of
from about 4 to 8, values of this additional lifting-
surf ace- theory correction are within the usual accuracy
of the measurem.ents of hinge m.oments in vvind tunnels;
that is,
ACh = A(Ch \ || - 0.002
for a pb/27 of 0.1 and therefore need not be applied
e:^cept for very accurate work at high speeds on large
/ \ ^c + 4
airplanes. A/alues of (cCpl p 7 — ^^'^ given in
figure 3. The effective aspect ratio Ac = AV 1 - M^
is used to correct for first-order compressibility effects
and values of are given as a function of Ap
in figure 4. Values of the correction
^K).,^'^-%^/-''^
LS
are given in figure 5 as a function of A^ and values
w
of — — are given in figure 6. The value of n is
(ca/c.)2
approximately 1 - 0.0005JZ) . The values of Cb/ca given
in figure 6 are for control surfaces v/ith an external
overhang such as a blunt-nose or Frise overh.ang. For
shrouded overhangs such as the internal balance, the
value of cb/ca should be multiplied by about 0.8 before
using figure 6.
If the wind-tunnel data are obtained in low-speed
wind tunnels, the estlm-ated values of Gj and (ttp')p
should be determined for the w/ind-tunnel Mach nuiriber
10 2^ AC A ARR
(a&sujne M = 0) . Othervfise the tunnel data must be
corrected for corfipressibllity effects and present
methods of correcting tunnel data for compressibility
are believed unsatisfactory.
ILLUSTRATIVE EXAJ,iPLE
Stick forces are computed from the results of the
v/ind-tunnel tests of the 0.40-scale semdspan model of
the viflng 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 Vifere applied for compressibility effects.
Because this exam.ple is for illustrative purposes
only, no com.putations 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 accoiont
of these effects.
A dra'jying of the plan form of the wing cf the miodel
Is presented In figure 7. The computations are m_ade 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 l.S*^. The data required for the
computations are as follows;
Scale of model 0.40
Aileron span, ba, feet 3.07
Aileron root-mean-square chord ^a> feet . . . 0.371
Tralling-edge angle, 0, degrees ... 13.5
Slope of section lift curve, aQ, per degree . 0.094
Balance-aileron-chord ratio, c-b/cg, 0.4
Aileron-chord ratio, Ca/c, (constant) , . . . . 0.155
Location of inboard aileron tip, -^-4- 0.58
b72
Location of outboard aileron tip, -2^ ...... 0.98
b/2
Wing aspect ratio, A 5.55
V/ing taper ratio, a. 0.60
Maximum aileron deflection, Sa^vjax' degrees .... ±16
Maximum stick deflectlo.n, 9 Smax' degrees ±21
Stick length, feet '. 2.00
Aileron-linkage- system ratio It 1
VJing loading of airplane, Vs/s, pounds per
square foot 27.2
NACA ARR No. L5P23 11
The required wind-tunnel test results Include
rol.ling-momej.it coefficients and hinge-moment coefficients
corrected for the effects of the Jet boundaries. Typical
data plotted against aileron deflection are presented
In figure 8. The'^e same coefficients cross-plotted
against angle of attack for one-fourth, one-half, three-
fourths, and full aileron deflections are given in
figure 9. The value of C^ /aQ as determined from
figure 2 is 4.02 and the value of Cj is 0.573. The
A + 4 A„E,, + 2
value of /a-^\ -r-- ^ ■ „ ■ .^r ~ used in equation (2)
^ ^)Cy, ^c +^ A^^'c + 4
to determine (Aa)^ Is found from fip-ures 3 and 4 to
^h
be 0.565 and is used to compute both the rate of roll and
the -stick force .
In order to facilitate the computations, simultaneous
plots of Cy and {^a)r against pt)/2V were made
""h
(fig. 10). .
The steps in the computation will be e.xplained In
detail for. the single case of equal up and dov/n aileron
deflections of 4°:
(1) From figure 9, the valuesof C^, corresponding
to 6a = 4*^ and 5a = -4° at a = 1.5° are 0.0058
and -0.0052, respectively, or a total static C7
of 0.0110.
(2) A first appro xlm.at ion to ('^^Iq taken at the
value of pb/2V corresponding to Cj = 0.0110 In
figure 10 is found to be 0.95°.
(3) Second-approximation values of C-j (fig. 9)
are determined at a = 0.35^^ for 5a = 4°' and at a = 2.25°
for 5a ---4°, which give a total Cj of 0.0112.
(4) The second approxim.ation to (La) n is now
^h
tovxi-d from figure 10 to be 0.96°, which i.s sufficiently
close to the- value found in step (2) to make any additional
approxim.ations ijnnecessary.
12 NACA ARE Ko . L5P23
(5) By use of the value of C^, from step (3^ , the
value of £^ = 0.0300 is obtained from figure 10.
(6) From figure 9 the hinge-moment coefficient
corresrionding to 5g^ = 4*^ and the corrected angle of
attack a = 0.34^ is -0.0038 and for 5^ = -4°
and a = 2.26° is 0.0052. The total G^ is there-
fore 0.0090.
(7) The stick force in pounds is calculated from
the ailerpn-linkage-system data, the aileron dimensions,
the increment of hinge-moment coefficient, and the lift
coefficient as follovi/s;
Stick force x Travel = Hinge moment x Deflection
where the hinge mom.ent is equal to ^v^qbg^c - and the
motion is linear.
Substitution of the appropriate values in the
equation gives
^s -57T3- - 5T3 C^qb^Ca^
and the wing loading is
1 = qC,
S ^ L
= 27.2
Therefore,
p - S'^'S Q 3.07 /0.371^^ 16 x 57.3
, _3^07 ( 0.57lY
'h 0.4 y 0.4 /
or
s - Ct i^ 0.4 \ 0.4 / 2 X 21 X 57.3
F„ = 68.4 _S
s Cl
Thus, when Cy^ - 0.0090 and C;^ = 0.170,
^s - 68.4 X Q^^^Q
= 3.62 pounds
NACA ARR No. LbFPZ 13
This .•^tick force is that due to aileron deflection and
has been corrected by ('ap\ as determined with the
Jones edge-velocity correction applied to the lifting-
line-theory value.
(o) The small additional lifting- surface correction
to the hinge moment (fig. 5) is obtained from
= 0.0207
ind since gf = 13.50,
•n = 1 - 0.0005(13.5)2
= 0.91
Prom figure 6,
^ =0.55
(Ca/c)
Therefore,
A/Ch ) = 0.0207 X 0.91 x 0.55
= 0.0103
and
"(=h)Ls =
0.0103 X 0.03
= 0.0003
(9) The AP^ due to the additional lifting-
surface correction of step (8) may be e::pressed as
= 0.124 pound
14 NACA ARR No. L5P23
Then,
Total stick force = F^ + AP3
= 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/'^V calculated, for the rigid unyav/ed
wing. For comparison, the stick forces (f irst-approxlraation
values of table I) caloulatedi 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 unyav/ed 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 v/ing twist or yaw and yawing motion-
were made for this example.
II - D E V S L P M E N T OF
The method for determining values of C7 and Ch„
p -t^p
is based on the theoretical flow around a wing in steady
roll with the introduction of certain empirical factors
to talce account of viscosity, wing twist, and minor
effects. The theoretical solution is obtained by means
of an electrom.agnetic-analogy m.odel 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 v/ing. The electromagnetic-analogy m.ethod of obtaining
solutions of lifting- surf ace- 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 sero angle of attack.
1\^ACA ARR Wo. L5F23
ELECTROMAGNETIC -ANALOGY MODEL
Vortex Pattern
In order to construct an electromagnetic-analogy
model of the rolling wing and wake, it is necessary
to determine first the vortex pattern that is to
represent the rolling wing. The desired vortex
pattern is the pattern calculated "by means of the
two-dimensional theories - thin-airfoil theory and
lifting- line theory. The additional aspect-ratio
corrections are estimated hy determining the difference
between the actual shape of the wing and the shape that
would "be required to sustain the lift distribution or
vortex pattern determined from the two-dimensional
theories .
For the special cases of a thin elliptic wing at
a uniform angle of attack or in a steady roll, the
lifting-line-theory values of the span load distribution
may be obtained by means of simple calculations (refer-
ence 6) . The span load distributions for both cases
are equal to the span load distributions determined
from strip theory Vi/lth a uniform redaction in all
ordinates of the span-load curves by an aerodynamic-
induction factor. This factor is -,; p^ for the wing
A -r d
at a uniform^ angle of attack and -, — '■ — 7 for the wing in
A + 4
steady roll. The equation for the load at any spanwise
station — 'V- '^^ ^ thin elliptic v/ing at zero angle of
b/2
attack rolling steadily with unit wing-tip helix angle
pb/2V is therefore (see fig. 12)
cc
b/2V) - A + 4 (^/2)^ \W2j ^ '
where a^ = 2Tr.
16
NACA aRR IIo. L5F23
The chordwise circulation function _£l_ from thin-
CCjV
airfoil theory for an inclined flat plate is
2V
cc^V
1
TT
/f - (f )
' + cos"Vl -f^
(4)
where x/c is measured from the leading edge
fig. 15 for values of
2V \
ccjVy
/
I:
ee
The vortex pattern is determined from lifting-
line theory as the product of the spanwise- loading
CCi
function : — '^ and the chordwise circulation
Cg(pb/2y)
function — ^
CC^v
wake ; thus ,
r^ for all points on the wing and in the
q p
cc-
CgV(ph/2V) Cg(pb/2V) cc^V
Contour lines of this product determine the equivalent
vortex pattern of the rolling wing. Ten of these lines
are shov/n in figure 14. The contour lines are given
in term.s of the Toarameter
2r
c<,V(pb/SV)
2V
CoV(pb/2V)
-i max
which reduces to
r
"■ max
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 v/as
!TACA ARR No. L5P23 17
tv/ice that of the model of reference 4 (6.56 ft Ir. stead
of 2.28 ft), but Flnce the aspect ratio is tv/ice as
large (6 Instead of 3), the maxiraurn chord is the rame.
In order to simplify the construction of the model,
only one seriispan of the vortex sheet v/as simulated.
Also, in order to avoid the large concentrations of
wires at the leading edf;:e and tips of the wing, this
sem.ispan of the vortex sheet v/as constructed of two
sets of Vi^ires; 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.
Downwa sh M e a sur erne nt s
The m.agnetic-f ield strength was neasured at 4 or 5
vertical heights, 15 spanwise locations, and 25 to 50
chordv/ise stations. A number of repeat tests vv'ere made
to check the accuracy of the measurements and satisfactory
chec'rs were obtained.
The electric current was run through each set of
v/ires separately. With the current flowing through one
set of v;ires, readings were taken at points' on the model
and at the reflection points and the sum of these readings
was multiolied by a constant determ.ined from the increment
of vorticity h (•— — \ represented by that set of wires.
Vrmax /
Then, with the current flowang through- the other set of
wires, readings v'ere 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 semi span m^ere f airier 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' -^-^ as dis-
^max
cussed in reference 4. The final curves of -^ are
"^■^max
18 HACA ARR No. L5F23
presented for the quarter chords half chord, and three-
quarter chord in figure 15. Also presented in figure 15
are values of -^ calculated by lifting-line theory
^ '- max
and values calculated oy lifting-line theory as corrected
hy 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 b:/ the
downwash measurements. Formiulas for determ.ining these
corrections to the span load distributions and the rolling-
and hinge-moment characteristics have been developed in
connection vflth 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 theor7/" and thus do not take into accoujit
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
norm.ally 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 ARR No. LtP23 19
Essentially the same final answer is therefore obtained
v/hether the corrections are applied in two parts (as
should be done, strictly speaking) or v^hether they are
applied in one part by use of the full theoretical value
of the streamline-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 Langley La.boratorles of the NACA to determine the
proper aspect-ratio corrections for ailerons and tail
surfaces with beveled trailing, edges . For. beveled
trailing edges, in which viscous effects may be much
m.ore pronounced than in ailerons with si.iall trailing-
edge angles, the reduction in the theoretical streamline-
curvature correction may be considerably more than
10 percent; also, when Cho; is positive, the effects
of the reduction in the streamline-curvature correction
and the additional downwash at the 0.50c point are
additive rather than 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 step is not allowable for beveled
ailerons. The corrections v/ill therefore be determined
in two separate parts in order to keep them general:
one part, a streamline-curvature 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 by multiplying the theoretical
values by an empirical reduction factor n, wh^ch is
approximately equal to 1 - 0.000502 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 were not
considered in reference 5. First-order compressi-
bility effects can be acco\aited for by application of
the Prandtl-G-lauert rule to lifting- surf ace- theory
results. (See reference 7.) This method consists In
20 NAG A ARR No. L5F23
determining the compressible -flow characteristics of an
equivalent wing, the chord of v;hich Is Increased by the
ratio where M Is the ratio of the free-
/
1 - M^
stream velocity to the velocity of sound. Because
approximate methods of extrapolating the estimated
hinge-moment and damping-mom.ent 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 /l - M^ . The estimated
param^eters for the equivalent v;ing are then Increased
by the ratio
\/l - M'^
The formulas presented subsequently in the section
"Approxim.ate Method of Extending Results to Wings of
Other Aspect Ratios" are developed for M = 0, but the
figures are prepared by substitutin-g Aq = A \/l - M
for A and multiplying the param.eters as plotted
by yl - M^ . The edge-velocity correction factors E^,
Eecj E'c, and E'ec ^^® '^^^'^^ factors corresponding to A^
The figures thus include corrections for first-order
compressibility effects.
Thin Elliptic Wing of Aspect Ratio 6
Damping in roll C? .-In order to calculate the
correction to the lifting-line-theory values of the
damping derivative C7„ it is necessary to calculate
the rolling m.oment that would result from an angle-
of -at tack distribution along the wing span equal to
the difference betv/een 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 slm.ple correction to the
lifting-line-theory values of the lift (reference 1)
and the damping in roll (unpublished data) for flat
NAG A ARR Fo. L5F23
21
elliptic wings. This correcticn, termed the "Jonjs
edge- velocity correction/' is applied by multiplying
the lifting-line-theory values of the lift hj the
ratio
An + 2
AcEc
+ 2
of the dam.ping
and the lifting-line-theory values
A,
in roll by
+
^c^'c
+ 4
with values of E,
and E'^ as given in figure 16. As may be seen from.
figure 15, the dov/mvash given by the Jones edge-velocity
correction is almost exactly that measured at the
0.50c points for flat elliptic v/ings. This fact is
useful in estim.ating the lifting- surface corrections
because the edge-velocity correction, which is given
by a simple formula, can be used to correct for the
additional angle of attack indicated by the linear
difference in downwash at the 0.50c line.
The variation in doi^nv/ash between the 0.25c line
and 0.75c line, apparently linear along the chord,
indicates an approxim.ately circular stream.line curvature
or camber of the surface. The increment of lift resulting
at each section from circular camber is equal to that
caused by an additional angle of attack given by the
slope of the section at
or the tangent at 0.50c
Because this difference
0.75c relative
- that is.
to the chord line
(v)o.75c " ^"^/0.50c'
in dov;nwash does not vary linearly
along the span, a spanwise integration is necessary to
determine the stream.line-curvature increment in rolling
moment; that is,
r-i)
sc
ST'
max-^c
bV(AcE'c + 4) Jo
ri
/ wo
l2r
max
-/0.75c
An evaluation of T
m.ax
0.50c
m berms ox
Cs b/2
pb/2V
(5)
IS necessary
to determine the correction to the damping-moment
coefficient C:
'P
The lifting-line-theory relation
22
NACA ARR IIo. L5F23
between F
^^^ and p'!d/2V is, from equation (3),
_ 2\T3(pb/2V)
"max ~ A + 4
V/lth the edge-velocity correction applied
^ _ 2Vt(pb/2V)
^ rnax - "A^E'c + 4
(6)
The value of the streamline-curvature correction
to C;^ is therefore
P
(f^^p)sc
16 Ae .'1
^(_wb_^'^
\ QP
niS-X,
0.75c
^ wb ^■
,' WD \
\2^ max/
0.50c
_£. _Z_ d/-l
b/2 \b/2
(7)
A graphical integration of equation (7) gives a value
of C.022 for /aCt
By the integration of equation (3) , the value
of (Ct I for incompressible flovi' is found to
^e -? -r-~—7 - 0.471 'for A = 6.
4 A + 4
Application of the edge-velocity correction, for A
= 6,
p;ives
N
''p
lA
EV
4(AE' + 4)
= 0.433
and, finally, subtracting the streamline-curvature
correction gives a value of ^l-r.) ^^^ A = 6, as
follows :
Cl. = Cj
^ \ p/ev
= 0.411
AC-
'P
NACA Aim Ho. L5P23
'j'he value of
'^l
for a Y/ing of aspect ratio 6
5,3 therefore 13 percent less than the value given by
lifting-line theory and 5 percent less than that
given by lifting- line theory Tivith the Jones edge-
velocity correction applied.
^ling.e-moriient parameter Ch .- The rtreamllne-
curvature correction to Cv, for- con3tant-"Dercentage
chord -\llerons i;
■P
Oi
max ■^"
.ven
;, from reference 5 and with the value
.n equation (6) ,
^h
P/SG
l'Fr\
r
i
6(x/c) .
C \"
\C;
. ( J \
\^
+ 4
(8)
'7^
where the integrations are made across the aileron span.
Because the downvirash at the 0,50c point is given
satisfactorily by applying the edge-velocity correction
to the lifting-line- theory values of the downvvash, the
part of the correction to Cv,p -which depends upon the
downwash at the 0.50c point raay be determined by means
of the edge-velocity correction. The effect of aerodynamic
Induction was neglected in developing equation (8) because
aerodynam.ic induction has a very sraall effect upon the
hinge -m.o3r.ent corrections caused by streamline curva.ture.
Values of the factor
various
(oa/c)2
chord ratios and balan^^e rati.os as determined from thln-
a.irfoll theory are given in figure 6. As mentioned
previously, rj is a factor that approximately accounts
for the combined effects of wing thickness and viscosity
m
in altering the calculated values of
experim.ental ^ta available at pre 5^ en t indicate
that T] r 1 - O.OOOSjZf'^. Results of the integration of
equation (8) for the elliptic wing of aspect ratio 6
are given in figure 17 as the parameter
.^2
f^Chp)
/SO
(V
+ ij\/i - m'
Values
24
NACA AKR No. L5F25
of (AC^ ^
'hrr j __ p^?r — (Ac + l) /l - 1^ determined as
in refsreiice 3 are given in figure 18.
is
The value of fChS]
f^p)
/LS
'Mll ^'P)ci
Ac + 4
^^LL '^e^V
— ^(^^^P).
Since
A,3 + 2
AcEc +
5(=ha),,= (=^-.),S-e=^-a)sc
then
(°bp)^3 = (-P)ch
A + 4- A. f^ + 2
j^^ A^E'^ + 4 A^, - 2 \ -a/rs
+
(^°'^p)sc- ('°'^a)3, Nc^^I
Af, + 4 A.^Ep + 2
or
The formula for the parameter
+
(=-"p)
("p)<
LS • (9)
is derived
VChLT.
for elliptic wings in the apjpendix, and numerical values
A_ + 4
^ in figure 3,
ChLL ^c + 2
are given in the form (a \
together with values for tapered wings derived from
the data of reference 2.
It -CTi&j he noted that use of the parameter (o.j}\
determine the total correction for rolling would he
impractical because (Cj-^ \ is not proportional
to
'^LS
To. L5F23 25
;>, \ . Although the numerical values of /j-ni
^ /IS . ,. V ^Z <
varv considerably with y^\y ) , the actual effect on
\ Vls
the stick forces is small because /ctp\ changes most
with Ch ) when the values of [0,^^ \ are small.
< VLS \ «/LS
This effect is illustrated in figure 19, in which
numerical values of ('cip\ for a thin elliptic wing
of aspect ratio 6 are given, together virith the values
obtained by lifting-line theory, the values obtained by
applying the Jones edge-velocity correction, and the
values obtained by using the aileron midpoint rule
(reference 8) . The values obtained by the use of the
Jones edge-velocity correction are shown 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 1 = (ap) (ch„)
ChEv V 7lS
Part II = Afch„)
\ VLS
Part I of the correction for rolling can be applied
to the static hinge -moment data as a change in the
effective angle of attack as in reference 2. (Also
see equation (2) ,) Part II of equation (9) , however,
is applied directly as a change in the hinge-moment
coefficients.
AC
h
Inasmuch as part II of equation (9) is numerically
fairly small { Aq, =0.002 for £| = o.l for a
wing of aspect ratio 6y , it need not be applied at all
except for fairly large airplanes at high speed.
26 I-TACA ARR No. L5F23
Approximate Method of Extending Results
to Vvings of Other Aspect Ratios
Damping in roll Ct,^.- 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 Cj; 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 C7
P
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^,--. A
P
reasonable method of extrapolating the values of Cj
to other aspect ratios, therefore, is to use the
variation of the edge-velocity correction with aspect
ratio as a basis from v/hlch to work and to increase the
magnitude by the amount required to give the proper
value of C^,^ for A = 6. Effective values of E
and E' (^Eg and E'e) were thus obtained that would
give the correct values of Ct, for A = 3 and A = 6
and of Ct, for A = 6. The formulas used for esti-
mating Ee^ and E'q_ for other aspect ratios were
See = 1.55(Ec - l) + 1
= 1.65 (s'c - l) + 1
Values of Ee^. and E'e^. ^-^^ given in figure 16.
are presented in figure 2 as a function of Ac/ao
Values of — ^ v'l - M'^ determined by using E'e^
aQ c
NACA AP.R No. L5F25 27
where A^ = A /l - M"^ and a.^ is the incompre ssfhle
slope of the section lift cjirve per degree.
Hinge-moment parameter Ch .- In order to deter-
2_P
mine Ch for other aspect ratios, it is necessary to
estimate the formulas for e.xtrapolating the streamllne-
7sc \ '''^ysc'
of (ACh^'l for A = 3 and A = 6 are available in
SC
curvature corrections (ACh \ and (ACh ' • Value;
reference 3. Values of (AC]-, \ might he expected to be
^ / 3C
approximately Inversely proportional to aspect ratio and
/ \ K-|
an extrapolation formula in the form ! ACv, i = -; =rr—
V N SC ^ + ^2
is therefore considered satisfactory. The values of K]_
and K.2 are determined so that the values of (ACh i
\ V SC
for A = 3 and A = 6 are correct. Values of K]_ and K2
vary with aileron span. The values of li2, however,
for all aileron spans less than 0,6 of the semi span are
fairly close to 1.0; thus, by assuming a constant value
of Kg = 1.0 for all aileron spans and calculating
values of llj^, a satisfactory extrapolation formula
may be obtained. It is impossible to determ.lne such a
formula for (ACh I because results are available
\ P/SC
only for A = 6; however, it seems reasonable to assume
the same form, for the extrapolation form.ula and to use
the same value of Kq as for (ACv, \ . The value
^ \ S'sc
of K-| can, of course, be determined from the results
for A = 5.
Although no proof is offered that these extrapolation
formulas are accurate, they are applied only to part II
of equation (9) (values of A(Cit \ ), v/hich is numeri-
\ V "P/ls/
cally quite sm.all, and are therefore considered justified.
28 IJACA ARR No. L5F2;
CONCLUDING REMARKS
Prom the results of tests made on an electromagnetic-
analogy model simulating a thin elliptic v^ing 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 mom.ents for a wing in steady roll were
obtained that are considerably m.ore accurate than those
given by lifting-line theory. First-order effects of
compressibility were included in the computations.
It v;as 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 a.nd 5 percent less than that given by the
lifting-line theory with the Jones edge-velocity correc-
tion applied. The resul.ts are extended to wings of any
aspect ratio.
In order to estimate aileron stick forces from
static wind-tunnel data, it is necessary/- to Irnow the
relation between the rate of change of hinge moments
with rate of roll and the rate of change of hinge
moments vi'lth angle of attack. It was found that this
ratio is very nearly equal, within the usual accuracy
of wind-tumiel m.easurements , 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
additiorial 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 cornxDlicated 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 Pie id, Va.
lACA ARR llo. L5P23 29
APPENDIX
EVALUATION OP /ar,\ FOR ELLIPTIC V/INGS
It was shown in reference 2 that for constant-
percentage-chord ailerons the hinge moment at any aileron
section is pi'oportional 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 v/ing chord. The factor /a-n\ is obtained
V"°-^LL ,
by averaging the tv;o factors CjC^ and Cy/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 2Tr, it was
shown in reference 6 that strip-theory values multiplied
A A
by aerodynamic-induction factors -, ^^^—^ or
A^ + 2 A^ + 4
could be used. (Note 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,
^£_, sin^e c 22TT M
o
^l'^" ~ Ac + 4 ""'' ^ "s '^" ~V
= — — ^ — 2 ££ sln"^9 cos 9
Ac -i- 4 2V
and for the same wing at a constant angle of attack a
A
cz,c2 = - — -2__ sinSe 03^2
Ac + 2
Tra
r
In order to find /ap\ , the integral /CtcS dy
across the aileron span must be equal for both the
30
".;TACA ARR No. L5P23
rolling v/ing and the wing at constant a. Thus,
jcic^ dy
"-^ i V^m^e .0= 9 *,
STrcs^Ac /■. 2q ^
— — £: — — a /sxn'^Q dy
dy = 2 d(cos 0)
§ sin e de
Let
^ = Ho^.„, i
^^LL
(«p)
Ac + 2 (7
/sin'^e cos 6 d9
Ch
LL
Ac + 4
Ac + 2
3r
d9
T^i^^eje
1
■ijsln^B cos 9+2 cos 9j
I'o
where 9^ and 9^ are parameters that correspond to
the outboard and Inboard ends of the aileron, respectively.
/ \ A^ + 4
Values of (a-n\ — ?t were calculated for the
outboard end of the aileron at —7— = 0.95 and plotted
b/2
in figure 3.
JACA ARR No. L5F23
A similar development gives, for the const ant -chord
aileron,
^^-
SrrAr
pb /cos G
c /A + 4 \ 2V
^ s {^^c J
sin
^y
JttA,
r
C3(^^C + 2)
ph / \ / ;
\ 2V V^'P/ChLi^ "^'^^
dy
Ln G
fa.
Vci
^^c
+
+
_4 _ <
2
ycos 9
ae
^-c
/de
r i'°
sin G
^
[^r
These values are also presented in figure 3.
32 I'lACA ARR No. L5P23
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. Swan:^on, Robert S., and. Toll, Thomas A.: Estimation
of Stick Forces from Wind-Tijxinel Aileron Data.
NACA ARR No. 2 J2 9, 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, Stevifart 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 Models in
Rectangular Wind Tunnels. NACA ARR No. 3E22, 1943.
6. Munlc_, Max M. : Fundamentals of Fluid Dynamics for
Aircraft Designers. The Ronald Press Co., 1929.
7. Goldstein, 3., and Young, A. D.; The Linear
Perturbation Theory of Com.pressible Flow with
Aiopllcations to Wind-Turmel Interference.
6865, Ae. 2262, P.M. 601, British A.R.G., July 6,
1943.
8. Harris, Thomas A.t Reduction of Hinge Moments of
Airplane Control Surfaces by Tabs. NACA Rep.
No. 528, 1935.
9. Cohen, Doris: A Method for Determdning the Camber
and Twist of a Surface to Support a Given
Distribution of Lift. NACA TN No. 855, 1942.
NACA ARR No. L5F23
33
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Fig. 3
NACA ARR No. L5F23
4-
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NACA ARR No. L5F23
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COMMITTEE F0« AERONAUTICS
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NACA ARR No. L5F23
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aileron deflection.
Fig. 9
NACA ARR No. L5F23
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NACA ARR No. L5F23
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Fig. 11
NACA ARR No. L5F23
Aileron det/ection
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NACA ARR No. L5F25
Figs. 12,15
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NACA ARR No. L5F23
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Fig. 15
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COMMITTEE FOR AERONAUTICS
1 i
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Figure 15.- Induced downwesh f\anctlon wb/z/^gx ^°^
an elliptical lifting surface of aspect ratio 6 in
steady roll.
Fig. 16
NACA ARR No. L5F23
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Figs. 17,18
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NACA ARR No. L5F23
/a
Zn6ocfr</ a/Zeron t/p^ hJz .
Figure 19.- Values of parameter Ka^^Q 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.
UNIVERSITY OF FLORIDA
DOCUMENTS DEPARTMENT
1 20 MARSTON SCIENCE UBRARY
P.O. BOX 1)7011
GAINESVILLE, FL 32611-7011 USA