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FROMDistribution authorized to U.S. Gov't.agencies and their
contractors;Administrative/Operational Use; Nov 1957.Other requests
shall be referred to WrightAir Development Center,
Wright-PattersonAFB, OH 45433.
AUTHORITY
Air Force Flight Dynamics Lab ltr 10 May1976
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CONFIDENTIAL
WADC TCLlt REPORT 56-285/2.qASTIA DOCIRMET NO.AD128(UmCLABInFWK
TrTlM
f) Fi.UTTER MODEL TESTS OF A SWEPT-BACK, ALL-MOVINGLU -
BEMZOtJTAL TAIL AT SUPERBSOWC SPEEDS
* Gif Vo- . Amber*4; ~John 11 Aztaccoui
*Wwrrrm a H. Jleatherili
UABSACRUSETTlJS IlTUTE OF TECHNOLOGY
NOVEMBER 195?
FEB 2 nWRIG!LT AIR DEVELOPMENT CENT V.
56 ,VCLS 10322~-2
JAN a~ N
CONFiDENTIAL
-
This document is the property of the United StatesGovernment. It
is furnished for the duration of the contract and
shall be returned when no longer required, or upo-arecall by
ASTIA to the following address:
Armed Services Technical Information Agency, Arlington Hall
Station,Arlingon 12, .J-g.ni.
NOTICE: THIS DOCUMENT CONTAINS INFORMATION AFFECTING THE
NATIONAL DEFENSE OF THE UNITED 3TATES WITHIN THE MEANING
OF THE ESPIONAGE LAWS, TITLE 18, U.S.C., SECTIONS 793 and
794.
THE TRANSMISSION OR THE REVELATION OF ITS CONTENTS IN
ANY MANNER TO AN UNAUTHORIZED PERSON IS PROHIBITED BY LAW.17 :ý
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CONFIDENTIAL
- ADC TECHNICAL REPORT 56-285 /ASTIA DOCUMENT NO. AD-142088
(UNCLASSIFIED TITLE)FLUTTER MODEL TESTS OF A SWEPT-BACK.
ALL-MCVING
HORIZONTAL TAIL AT SUPERSONIC SPEEDS
Gifford W. Asher
John R. Martuccelli
Warren H. Weatherill
Massachusetts Institute of Technology
NOVEMBER 1957
Aircraft Laboratory
Contract AF 33(616)-2751
Project No. 1370
Wright Air Development Command
Air Research and Development Command
United States Air Force
Wright-Patterson Air Force Base, Ohio
56 WCLS 10322
r-' - - - I-'
CONFIDENTIAL
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FOREWORD
This report, which presents the experimental and theoretical
results of aprogram conducted to Investigate the supersonic flutter
characteristics of aswept-back all-movable surface, was prepared by
the Aeroelastic and StructuresResearch Laboratory, Massachusetts
Institute of Technology, Cambridge 39,Massachusetts for the
Aircraft Laboratory, Wright Air Development Center,Wright-Patterson
Air Force Base, Ohio. The work was performed at MITunder the
direction of Professor R. L. Halfman, and the project was
supervisedby Mr. G. W. Asher. The research and development work was
accomplished underAir Force Contract No. AF 33(616)-2751, Project
No. 1370 (UNCLASSIFIED
TITLE) 'KAeroelasticity, Vibration and Noise," and Task No.
13479,(UNCLASSIFIED TITLE) "Investigation of Flutter
Characteristics of All-MovableTails," with Mr. Niles R. Hoffman of
the Dynamics Branch, Aircraft Laboratory,WADC as task engineer.
This research was started in January 1955 and completedin September
1956. Additional supersonic flutter testing of swept
all-movablestabilizers may be performed at a later date to obtain
further information.
The authors are indebted to Mr. 0. Wallin and Mr. C. Fall for
their help inbuilding the models and in keeping the experimental
equipment in good order, andto Mr. G. M. Falla for his help In
making the high speed photographs. Theauthors are also indebted to
Miessrs. A. Heller, Jr., J. R. Friery and H. Moserfor their help in
preparing the necessary calculations, tables, and figures forthis
report.
Portions of this document are classified CONFIDENTIAL since the
datarevealed can be employed to establish design criteria for the
prevention of flutter
of swept-back all-movable tails of aircraft in the supersonic
speed range.
WADC TR 56-285
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CONFIDENTIAL
ABSTRACT
This report describes the flutter testing at supersonic speeds
of a series ofswept-back all-moving stabilizers. An attempt was
made to define the flatterboundaries, for one location of the
pitching axis, over the Mach number range of1. 3 to 2. 1, by
testing at a number of different levels of stabilizer stiffness,
andat a number of different pitching frequencies.
The results indicate that large increases in the region of
instability canoccur due to the introduction of the pitching
degrees of freedom. The test results
follow the trends of theoretical calculations, but the
quantitative correlation be-tween the theoretical and the
experimental results is only fair.
PUBLICATION REVIEW
This report has been reviewed and is approved.
FOR THE COMMANDER:
-f*... PRANDALL D. KEATOlHColonel, USAF
Chief, Aircraft Laboratory
WADC TR 56-285 iii
CONFIDENTIAL
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TABLE OF CONTENTS
PageSection I INTRODUCTION ............. .............. 1
Section II DISCUSSION OF RESULTS ......... .......... 31
Discussion of Theoretical Results ...... ....... 32 Discussion of
Experimental Results ......... .. 7
Section III CONCLUSIONS ....... ............... . 17
Bibliography ...................................... 18
Appendix I THEORETICAL CALCULATIONS .......... 20I Introduction
....... ............... 202 Flutter Equations Based on Velocity
-Componeiit
Method .......... ................ 203 Solution of Equations of
Motion for Flutter . . . . 29
Appendix II EXPERIMENTAL DATA. ........ ........... 37
WADC TR 56-285 iv
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LIST OF ILLUSTRATIONS
Fig. No. Page
1 Flutter parameters Vf/WCa b0 .7 and Vf/ofb0 .75 versus
Mach
number from three-degree-of-freedom supersonic calculations
4
2 Flutter parameters Vf/aI b0 .75 and Vf/wfb .75 versus (w(b/W
Cl
and (w0 /whl) from three-degree-of-freedom incompressible
calculation .................. ................... 6
3 Flutter parameter Vf/wh Nb 0 7 5 b F0.5 V versus Mach number
fromexperimental tests and comparison with theory .... .......
9
4 Flutter parameter Vf/wa Nb0 .75 Vf versus Mach number from
experimental tests and comparison with theory ......... ..
10
5 Flutter parameter Vf/wfb 0 .75 N1f versus Mach number from
experimental tests and comparison with theory ......... ..
11
6 Flutter parameter (b0 . 7 5w2/af) V(jf/65)0 . 75 versus Mach
numberfrom experimental tests ........ .............. 15
7 Flutter parameter (b0 .7 5w2 /, al) r( :,'/50.."versus Mach
numberfrom experimental tests ........ .............. 15
8 Axis system for swept stabilizer ..... ............ 21
9 Flutter parameters Vf/1wal b0 .75 and Vf/wfb 0 .75 versus (w0
/Wh1 2
from two- and three-degree-of-freedom calculations ..... ..
32
10 V-g curves from four-degree-of-freedom calculations . ...
33
11 Sketches of V versus g versus Mach number curves from
four-degree-of-freedom calculations ...... ............ 34
12 Flutter parameters Vf/wi lb 0 . 75 and Vf/wfb0 .75 versus
Mach
number from four-degree-of-freedom calculations ........ ..
35
13 Variation of Vfi/:v I b,.75 versus Mach number with change
in
structural damping from four-degree-of-freedom calculations. .
36
WADC TR 56-285 v
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LIST OF ILLUSTRATIONS (Cont.)
Fig. No. Page
14 Swept stabilizer design drawings ..... ............ .. 38
15 Pictures of root mounting block ...... ............ 40
16 Pictures of flutter of SWS-1-98 model from high speed movie
44
17 Pictures of flutter of SWS-3d-87 model from high speed movie
45
18 Analysis of high speed movies of SWS-1-98 model ........ ..
46
19 Analysis of high speed movies co SWS-3d-87 model ... .....
46
20 Vibration frequency data for swept stabilizer models .....
... 47
21 Location of influence coefficient stations ... ..........
.... 48
List of Tables
Table No. Page
I Design parameters for swept stabilizer models ... ....... ..
39
2 Static data for swept stabilizer models .... .......... ..
42
3 Pitching frequency data ....... ............... 42
4 Experimental flutter data ........ .............. 43
5 Experimental vibration data .......... ............. .
6 Experimental influence coefficient data .. ........ 55
WADC TR 56-285 vi
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LIST OF SYMBOLS
NOTE
All quantities marked with * are measured in an unswept
reference system
except when appearing with subscript 9. They are then being
referred to a
reference system swept with the elastic axis (see Fig. 8 for
unswept x, y and
swept xD, ya reference systems).
a *Location of elastic axis in semichords aft of stabilizer
midchord
a Speed of sound (ft/sec), a - 49.1 V'T
AR Panel aspect ratio
b 'Semichord of stabilizer (ft)
c *Chord of stabilizer (ft)
C0 Flexibility influence coefficient of pitching mechanism
(rad/ft-lb)
d *Distance between pitch axis and elastic axis at the root,
positive aft (ft)
ea Elastic axis or shear center position (% chord)
E Modulus of elasticity in bending
El *Bending stiffness
f Frequency (cps)F Assumed mode shape for calculationg
Structural damping coefficient (ref. 10)
G Modulus of elasticity in torsion
GJ *Torsional stiffness
h Vertical displacement of stabilizer elastic axis (ft)
hl, h2 See Appendix I, Eq. (1)
la *Mass moment of inertia of stabilizer per unit span about the
elastic axis
(slug-ft2
I0 Mass moment of inertia of rigid stabilizer about pitch axis
(slug-ft2
k *Reduced frequency, bw/V; (k = kD)
I *Semi span of model (ft)
L, M Aerodynamic coefficients (see Appendix 1)
LE Leading edge
m *Mass of stabilizer per unit span (slug/ft)
M *Mach numberr *Section radius of gyration (r2 = I /mb 2 ) in
semichords
Sa *Static mass unbalance per unit span about elastic axis
(slug-ft/ft)
WADC TR 56-285 vii
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t Time (secs)
T Absolute temperature (OR)
TE Trailing edge
V *Velocity (ft/sec)
x, y, z *Coordinate distances (shown in Appendix I, Fig. 8)
x a *Distance section center of gravity of the stabilizer lies
aft of elastic
axis in semichords
a *Torsional deflection of the stabilizer, positive nose up
(radianm)
ci *See Appendix I, Eq. (1)
*Nondimensional spanwise coordinate, Y? = y/1
A Taper ratio, tip chord/root chord
Jq Angle of sweep of elastic axis positive for sweep-back,
(degrees)
a *Relative density, p = m/wp b' (constant along the span)
p Air density (slug/ft3)
0 Rigid body pitching about pitch axis, positive nose up
i See Appendix I, Eq. (1)
W Frequency (rad/sec)2Z Flutter parameter (w al/wf) , Eq.
(23)
Z a Deflection of the mean surface of the stabilizer (ft)
SUBSCRIPTS
f Conditions at start of flutterhI, h2 First and second
uncoupled bending modes of the stabilizer
hN First measured cantilever or "pitch locked" bending mode of
the
stabilizer (Nominal first bending frequency)
L Pertaining to pitch-locked-out conditionM Experimentally
determined parameter0 Parameter evaluated at the root of the
stabilizer (y = 0)
0. 75 Parameter evaluated at the 75% span station of the
stabilizerr Reference station for theoretical calculations (75%
span station of the
stabilizer)
T Parameter evaluated at tip of stabilizer (y =
oIl First uncoupled torsional mode of the stabilizer
cYN First mpaturped erntilever, or "pitch locked, " torsional
mode of thestabilizer (Nominal first torsional frequency)
.9 Parameter measured in reference system swept with the elastic
axis
ks Rigid pitch degree of freedom1,2 First and second measured
coupled modes
(1I,4)c Quarter chord
WADC TR 56-285 viii
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CONFIDENTIAL
SECTION I
INTRODUCTION
This report covers the experimental flutter teits and associated
theoreticalcalculations made on a swept, all-moving horizontal
stabilizer at supersonicspeeds. The configuration tested is
becoming a comma one for high speed air-craft and missiles.
At present, the methods of theoretical supersonic flutter
analysis usingtwo-dimensional 4erodynamic forces derived from
linearized theory do notappear adequate to predict the absolute
levels of the flutter boundaries.Reference 3 shows that even for
the simple cantilever straight wings analyzedin that report such
analyses give results that are conservative in one Machnumber range
and unconservative in another. It may be suspected that the
poorcorrelation between theoretical and test results shown in Ref.
3 arises fromthe use of two-dimensional aerodynamic forces on a
three-dimensional liftingsurface, and so the use of more powerful
methods of analysis, such as theaerodynamic influence coefficient
methods of Refs. 5 and 6, may improve thecorrelation. However,
correlations between theoretical and experimentalresults, where the
theoretical calculations have been based on
three-dimensionalaerodynamic forces, are not common in the
supersonic regime. Until suchcorrelations have been made, it is not
certain that the added labor of the in-fluence coefficient methods
will be worth while in terms of improved results.The designer will
probably rely on the simpler two-dimensional calculationto supply a
description of the trends to be expected when various parametersare
changed, and will probably depend for some time on what
experimental datais available or can be obtained to define the
absolute levels of the flutter
boundaries.
The present program is intended to define experimentally the
level of theflutter boundaries for an all-moving, swept horizontal
stabilizer. The canti-lever, or "pitch locked,- boundary is defined
by tests of the cantilever con-figuration of the model shown in
Fig. 14a, and through the use of data fromprevious flutter testi.
Various levels of wing and pitching restraint stiffnessarp then
combined in an attempt to define the effect of the pitching degree
of
Manuscript released by the authors September 1957 for
publication as aWADC Technical Report.
WADC TR 56-285
CONFIDENTIAL
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CONFIDENTIAL
freedom on the flutter boundaries, over the Mach number range
1.3 to 2.0. Theresults are discussed in Section 11 of this report,
and a complete compilationof the experimental data is found in
Appendix I1.
Along with the experimental program, a large number of
theoreticalcalculations have been made on the basis of
two-dimensional aerodynamiccoefficients, both supersonic and
incompressible. The major effort was ex-pended on
three-degree-of-freedom calculations employing assumed wing
bend-ing and wing torsion structural modes and a rigid pitching
mode. Four-degree-of-freedom calculations were also made which
included an assumed second bend-ing mode as well as the previously
mentioned modes. The results of the calcula-tions are discussed in
Section II, and the equations used for setting up the cal-culations
are described in Appendix I.
WADC TR 56-285 2
CONFIDENTIAL
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SECTION n
DISCUSSION OF RESULTS
I Discussion of Theoretical Results
A considerable number of calculations were made during the
course of the
program for the swept stabilizer models to determine, if
possible, what trends
might be expected in the flutter boundaries for models of
various stiffness levelsand varying pitch frequencies. The
calculations used the velocity component
method of Ref. 8 with two dimensional aerodynamic coefficients.
The model forcalculation of the aerodynamic integrals was assumed
to be untapered in orderto avoid variations of reduced frequency, k
= buf/v, along the span, but the"mass and stiffness parameters were
assumed to vary in the same manner as theexperimental model. A
description nf the calculations is found in Appendix I.
Most of the theoretical effort was expended on
three-degree-of-freedomcalculations employing supersonic
aerodynamic coefficients. The three degreesof freedom used for this
analysis were wing first bending (parabolic),wing firsttorsion
(linear),and rigid pitch about the rotation axis. The results of
thesecalculations are given in Fig. I for various values of the
pitching to torsion
frequency ratio (w /W )l2 and pitching to first bending
frequency ratio
(W 0 /W h ) In all of the calculations (w hl /Wa,) 0.25, where h
W
and w refer to the frequencies in the assumed uncoupled modes.
Note thatthe reference semichord for the calculations, br, is that
of the 75 % spanstation, b0 . T5 The reference axes for measuring
the semichord as well
as other similar quantities are aligned with the stream unless a
subscript .Q
is used. In that case the reference axes are swept with the
elastic axis (SeeFig. 8). Many of the parameters, such as g, are
constant along the span.
Perhaps the most interesting feature shown in Fig. I is the
sharp increase
In the region of instability that occurs below Mach numbers of
about 1. 8 for avalue of (w/• C)2 = 0. 20. For this value of (w )2,
the boundary actually
crosses the M = 1. 7 line three times giving the shape shown. As
noted in Ap-
pendix I, the four-degree-of-freedom calculations indicated that
this sharp dropor "bucket" in the boundary probably occurs because
of a change in the mode of
WADC TR 56-285 3
-
AR 1.67 m - to(bA/) 2 5.0
(1/4)c " 45, S- S. (b/b0 )3
2 W--I,=2
5.0 5 0. 5- "
~~~~~~~~~'# _--f2 I.I (/ 0) ý)O3 t) 4.8, 4.0r•)L0,25.(b/bo) -0
*0
4.0 eaofa 40% chord (.O.20
cg of 50% chord(d/bo) - 0.487
a .0 -pf " 30, .21.8.. .
> UNSTABLE2.0..ILL 2. >
1.3 1. 15 .6 .7 1.(19 . 0 2.10 1.6
1.02 w
2
•' MACH NUMBER
J~iT- ..- ; -030--
1.3 1.4 1 .5 1.6 1.7 1.8 1.9 2.0 2.1 2.2
MACH NUMB~ER
6.0 1 W# 1.25,.o •.• -.d' Z.,•'s 30 , 4.8 '
5.0 Wll
"aj2 2
S, I !0.75
> -. 1 1. . 16-71. . 0 .102.
Fig. 1. Flutter parometers Vf ;o= bO-• and Vfti,.fb 0 .. •
versus Moch n~m~ber from •,ree-degree-of-freedom
supersonic: calculations.
\,ADC I1R 56-2854
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flutter. Figure 13 of Appendix I shows that the "bucket" may be
very much af-fected by the level of structural damping, g, to the
extent that for g about 0.6i theboundary for (w, /WO) 0.20 may
follow smoothly the trend established by the
1 2cantilever, or "locked" case fw o/W a ) = ) .
For high pitch frequencies the velocity - Mach number trend is
Nimilar tothe locked pitch case with only a moderate lowering of
the flutter velocity. Forlow pitch frequencies the velocity - Mach
number trend appears to be nearly aconstant equivalent airspeed
(Nr'aV = constant) through a wide range ofMach number.
For M - 1. 5, corresponding to a cross flow Mach number
perpendicular tothe 40% chord line, Mj2, of 10/9, a sharp increase
in the region of instabilityoccurs even for the cantilever case, (w
e/ 22 )= o. This increase in theregion of instability may arise
from the use of the linearized supersonic aero-dynamic theory at
such a low cross flow Mach number. It must be rememberedthat the
Mach number used for the determination of the aerodynamic
coefficientsof the calculation is the cross flow Mach number, not
the free stream Machnumber.
The cantilever curves, (w./w 1)2 = ac, of Fig. 1 were determined
from twj-
degree-of-freedom calculations in which first bending and first
torsion mode.were used without the pitching degree of freedom.
Figure 2 shows curves of the flutter parameters Vf/W l b0 .7
andVf/ b0 . 75 versus the frequency ratios (w 0 /X CI ) and (wf/wh
) calculated by
using inrompressible aerodynamic coefficients and three degrees
of freedom:wing first bending, wing first torsion, and riid pitch.
A sharp increase in theregion of instability for values of (w /w
&) less than about 0. 5 can be seen.
More significantly, the decrease in stability appears to be
related to thenear equality of pitch and bending frequencies. This
effect has been observedby other investigators and depends on pitch
axis location.
Calculations were also made using an assumed second bending mode
alongwith the first bending, first torsion, and pitch modes and the
results are dis-Lu.zed ii Appendix I. The addition of the second
bending mode does not affectthe shape of the flutter boundaries
signmficantlv in the Mach number range .studied.Changing the ratio
of second bending to first torsion frequency, (th2/ wylfrom
slightly greater than I. 0 to slightly less than I. 0 alio has
little effect onthe flutter boundaries. It appears, then, that
sufficient accuracy was obtained
NADC TR 56-285 5
-
16.0
.0-1 - - 12.0
3 i4jUSTABLE
0--0.20. 0.40 0.60 0.80 1.00 1.20 1.40 1.60 1.80 2.00 2.20:
0 0.80 1.60 2.40 3.20 4.00 4.80 5.60 6.40 7.20 8.00 8.80
3, AR 1.167 -30, -21.8
1(I/4)c g 4 - - 0
>h,/Ic., -0.25 m - m0(bibd)2
0. 21 5 . S% 6,
1, - "(b1ibo) 4"ea at 40% chordi .... ... cq at 50% chord(d b.)
- 0.487 (bib0) 2 - 7,
0.20 0.40 0.60 0.80 1.00 1.20 1.40 1.60 1.80 2.00 2.20
0.80 1.60 2,40 3.20 4.00 4.80 5.60 6.40 7.20 8.00 8.80
F., 2 Flutter parameters Vý -. , b0.,, and Vf, •(b0 -5 versus
(&ij/ma and h from three-degree-of-freedom incompressibie
calculation.
WAI)C TH 56-285 6
-
CONFIDENTIAL
in the calculations with three degrees of freedom for the wings
studied. As noted
in Appendix I, the V-g solutions for the four-degree-of-freedom
calculations didfurnish valuable insights into the modes of flutter
and th.c effect of structural
damping.
2 Discussion of Experimental Results
During the test program nine cases of flutter occurred for the
sixteenconfigurations tested. Two models fluttered in a cantilever,
or "pitch locked,"condition and the remaining models at various
levels of wing stiffness andpitching frequency.
Reference I describes the M. I. T. -WADC supersonic variable
Mach numberBlow-Down Wind Tunnel facility in which the tests were
conducted. Reference 2describes the techniques of testing that were
used to obtain the data. No majorchanges were necessary in either
the wind tunnel facility or the testing techniquesto obtain the
experimental data presented in this report.
The planform of the stabilizer models tested is shown in Fig. 14
ofAppendix I1. They incorporated a pitching degree of freedom with
a pitch axisperpendicular to the root chord, 64. 3 %of the root
chord aft of the leading edge.The stiffness of the pitching
restraint could be varied at will. The model con-struction was
similar to that described in Ref. 2 with a single spar providing
therequired stiffness. Balsa fairings glued to the spar gave the
required 6% thickdouble wedge airfoil shape and suitably spaced
lead weights provided the requiredmass parameters. A more complete
description of the models Is given in
Appendix H1.
Before flutter testing, each model was given vibration and
static tests. Theresults of these tests, as well as the tabulated
results of the flutter tests, arecontained in Appendix Ir. With the
pitching mechanism "locked cut, " thecantilever condition, the
lowest natural modes of vibration were determined foreach model. In
general three modes were easily excited, the first bending,first
torsion, and second bending modes. The first bending mode and
firsttorsion mode determined In this ruature" were used to plot the
flutter data ofFigs. 3 and 4 are the w and wN of the figures. The
rigidities in bending and
torsion, EIr and GJr, at the root were also determined for most
of the mdxJels inthe cantilever condition. This data is not too
satisfactory since it is difficult toassess accurately the effects
of root fitting deformation. As can be seen fromTable 2 there seems
to be considerable scatter in the El and GJ data sincer rmodels
with ebbentially the same cantilever frequencies appear to have
widelydifferent values of El r and GJr' With the pitching mechanism
in operation
NADC TR 56-285
CONFIDENTIAL
-
CONFIDENTIAL
vibration data was also taken for various pitch restraint
stiffnesses. In general,only the first three modes of vibration
could be excited easily as can be seen
from the data of Table 5. This data furnished the coupled
vibration frequenciesW1 and w2 for the plots in Figs. 6 and 7.
Influence coefficient data was alsotaken with the pitching
mechanism in operation. The :ncoupled pitch frequency,W, was
determined from the measured rigidity of the pitch mechanism
andfrom the measured total mass moment of inertia of the wing and
root fitting, I1.A few of the frequencies so determined were
checked by fitting a rigid disc ofknown moment of inertia to the
flexure and measuring the resulting vibration fre-quency. The check
on frequencies was satisfactory. Pitching frequency data canbe
found in Table 3 of Appendix II, while the frequency data and all
of the flutterdata is summarized in Table 4.
Figures 3. 4, and 5 ccmpare the experimei ,1 flutter data and
the theoreticalpredictions when plotted versus Mach number. ,t is
presumed that W the
first measured cantilever torsion frequency, corresponds fairly
closely to theuncoupled first torsion frequency, w a, used as a
parameter in the calculations
and similarly that whN corresponds closely to wh ' Since the
different models• fluttered at somewhat different relative
densities, gi, and since the value of
used in the theory is lower than for most experimental points,
the factorI,/Vk-i has been included in the ordinates to reduce the
effects of these variations,
The tests of the SWS-2 model, which fluttered In a locked
configuration,along with the data of Ref. 3 were used to establish
the cantilever, or bending-torsion flutter boundary; ( ,v /WhN )2
or (w, /W CN) 2 = c. (The SWS-Id model
also fluttered in a cantilever condition but, since the
vibration data of Table 5show! that this model had a low torsion
frequency quite different from therest of the stabilizer models, it
was used only as a guide in drawing the "locked"boundary.)
The SWS-I series of models, had a slightly higher stiffness
level than SWS-2and thus had a margin of safety of about 7% in
bending-torsion flutter. Themargin of safety Is defined as the
ratio of w ,N necessary to prevent flutter in
the cantilever condition to the w (YN of the actual model. The
SWS-1 series models
were flutter-free in the cantilever condition but when the pitch
frequency was low-ered to about 98 cps, flutter occurred at M -- 1.
35, as can be seen from theSWS- 1-98 model test point. Two other
SWS-l series models were flutter testedat lower values of pitch
frequency, the SWS-Ic-48 and the SWS-le-74 models.The vibration
data shows that these models were similar to the SWS-1-98
W'ADC TR ,56-285 8
CONFIDENTIAL
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CONFIDENTIAL
A - S! ._-_ -P.- t•
- - o £.
-A - --- - -
iN_ __ . ._
To I , 9
0
f; st
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alN,
inCT 5-8
COFDETA
-
CONFIDENTIAL
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co
""-- ® o1
-., • : -, --__ _ - .-
U-1
o o I ., •
I a -
----. - ' - - l
k,? IAA' I, A,•,.
C,
A -2 --
0O -IE TI
LZi
-
LL
G~e -D
vVA~crR 5628S 1CONIDETIA
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(N
C,9*
In z
M -0
00C N
0 c i c;
CD
-6-6TR5628
CONIDETIA
-
CONFIDENTIAL
model in a cantilever condition. The SWS-Ic-48 model fluttered
on injection
when practically in the tunnel whereas the SWS-le-74 model
fluttered in the
middle of a test run.
The three flutter points for the SWS-l series models c(..ver
quite well the
Mach number range available in the wind tunnel so that further
tests of this
series of models at ittermediate values of the pitching
frequency were not
attempted. Instead a third series of models, SWS-3, were
designed with a
margin of safety in bending-torsion flutter of about 45%based on
the curvesof Figs. 3, 4 and 5. In order to achieve the higher
frequency and stiffnesslevel required by the increased margin of
safety without increasing the model
thickness ratio, it was necessary to modify the design
parameters of the SWS-3models. The frequency ratio (whN / wa N) was
lowered from an average of 0. 29
to 0. 26 rather than change the mass parameters. The test data
and calculations
of Ref. 3 show that there is little variation in the level of
the cantilever flutterboundaries for straight and swept wings for
variations in (w. U1/ "N) over this
range.
Three of the SWS-3 series models, SWS-3b-53, SWS-3a-63, and
SWS-3c-74fluttered on or very close to injection. The SWS-3c-74
fluttered when fully in
the tunnel but before the Mach number had started to change and,
therefore, is
not shown as an inject!on flutter. The SWS-3a-63 and the
SWS-3b-53 were almostin the tunnel when flutter occurred and are
shown as injection flutter. In sketch-
ing the experimental boundaries,the data for SWS-3d-87 and
SWS-3c-74 wererelied on more heavily than the data for SWS-3a-63
and SWS-3b-53.
The SWS-3 series data show that there can be a very large
increase
in the region of instability if the ratio (Uw7/uwh )2 is near
unity. In fact, it2 N
appears that for a given value of (w 0W/• ) the stiffer SWS-3
models will flutter
at higher Mach number than their SWS-l counter parts. Thus, the
experimentallxbundaries for a given (wO/hN) 2 appear to bend back
and form deep "buckets*'
in the curves just as they do for the calculated results.
Ingeneral, however,
the calculations predict larger regions of instability than the
experimental
i esults indicate.
It is interesting to note that the SWS-3 series flutter
apparently occur ina different flutter mode than the SWS-l series.
Figures 18 and 19
.-how the analysis of the high speed movies for the SWS-l-98 and
the SWS-3d-87n•)•d;., taken from the excerpts from the high speed
movies shown in Figs. 16
anrd 17. The SWS-1-98 mode~l should have a different mode of
flutter than the
ý'A'X" TiR- 56-285 12
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SWS-3d-87 if the results of Appendix I are correct in that the
"buckets" of the
boundaries are formed by a new flutter mode. Examination of
Figs. 18 and 19shows that while the relation between the tip
vertical translation amplitude topitch amplitude is of the same
order of magnitude for the two models, therelationship between the
tip angle of attack anipiitiide and the pitch amplitude ismuch
different. The SWS-1-98 model shows a much larger ratio of tip
angle ofattack amplitude to pitch amplitude than does the
SWS-3d-87. This fact indicatesthat the flutter mode for the
SWS-1-98 is composed of important pitch-bending-torsion motions
while the flutter mode for the SWS-3d-87 is mainly
pitch-bending.
It would appear, then, that for small margins of stability in
bending-torsionflutter the addition of a high frequency pitch
degree of freedom causes a decreaseinwhat is essentially a
bending-torsion fCutter speed largely because of thedecrease In the
coupled torsion Frequency. However, if (w / 1} is low enoughto be
near unity a bending pitch mode develops which may increase the
regionof instabili ty to as high as M = 2.
Before discussing some of the other curves drawn from the test
data, someattention should be given to the SWS-3-53 model. This
model, although practi-cally identical with the SWS-3b-53 model
insofar as vibration frequencies areconcerned, was tested in the
same range of Mach number and density as theSWS-3b-53 model but
failed to flutter. :1owever, the structural damping of thefirst two
important coupled vibration modes is about twice as great for
theSWS-3-53 model (average g of 0.04) as it is for the SWS-3b-53
model (average gof 0.02). The SWS-3 series flutter points form the
sharp increases in theregions of instability or "buckets" of Figs.
3, 4 and 5; thus, the mode of fluttermay be one that is very
sensitive to g variations, Since it was predicted theo.retically
(Fig. 13) that the mode which forms the "bucket" i' very sensitive
tochanges in g, it then seems possible that the higher structural
damping of theSWS-3-53 model may have prevented flutter for this
model down to a Machnumber of 1.8 where it was destroyed ty a
failure of the inboard leading edgecaused by a root seal failure.
This possibility that the "buckets" in theexperimental curves are
sensitive to g variations may point the way towardselimination of
large regions of instability by use of damping. It should be
notedthat for most of the stabilizer models tested the value of g
for the first twoimportant coupled modes is about 0. 02.
The data for the SWS-3e-120 model is also particularly
interesting becausethis model faiied to flutter over the Mach
number range 1. 25 to 2.00. Thisfailure to flutter means that the
curve for (u0/WhN/)2 2 0. 16 must be drawn as
shown in Figs. 3.4 and 5 and shows that at these higher values
of (W/WhN/ ) the
"bucket" is not evident. Comparison of the data for the
SWS-3e-120 and the
WADC TR 56-285 13
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other SWS-3 models helps to set upper and lower limits of (wlb /
WaN ) for flutter
in the Mach number range 1. 27 to 2. 10.
In Fig. 6 and Fig. 7 the first two coupled frequencies with the
pitchingmechanism In operation, wI and w2 , were used to form the
flutter parameters(b0 . 75 w/af) '(/65) 0 . 75 and (b0 . 75
w2/af)/(jA/65) 0 " 75' where af is the speedof sound at flutter.
The use of the relative density correction in this form isbased on
the previous experimental results of Ref. 3 and not on any
firmtheoretical basis. Figures 6 and 7 may be useful as design
charts; a straightline parallel to the abscissa being a constant
altitude line, and a straight line fromthe suppressed origin being
a line of constant dynamic pressure.
In Fig. 6, the first'coupled vibration frequency, w,, is used to
somal i zethe data. This vibration mode, as can be seen from Table
5, is essentiallya combination of the rigid pitch and the first
bending modes of the model. TheSWS-le-74 and the SWS-3d-87 both
have the same value of the parameter (w 0 /W1 )and hence must fall
along the same boundary. Thus, the curves must be drawnas shown in
Fig. 6 with a narrow stable region between the (I/w 1 ) = cc and
the(W /W 1 ) 1.60 curve.
Figure 7 shows curves similar to those of Fig. 6 except that the
secondcoupled vibration frequency, w 2 , is used as a parameter.
This vibration mode,as can be seen from the data of Table 5 is
largely a combination of the rigidpitch and first torsion modes of
the model except for the lowest pitch restraintstiffnesses where it
may involve appreciable bending.
For the various experimental plots, curves have been drawn on
the basis ofa bare minimum of data. The fairing of such curves is
subject to some question,and Figs. 3, 4, 5, 6, and 7 therefore
represent only rough sketches of where theflutter boundaries lie.
The general outlines of the curves are probably correct,and enough
experimental data has been obtained to show that there are
largeincreases in the regions of instability with sufficiently low
values of the pitchingfrequency. Furthermore, these increases
appear to follow the general trendsestablished by the theoretical
results.
In one respect the theoretical results do not match the
experimental resultseven qualitatively. This is at the lower Mach
number of the calculation M = 1.52or Mg = 10/9. For this case the
calculated results show that even the "locked"case has a sharp
increase in the region of instability and predicts that the
SWS-land the SWS-3 series models will flutter in the cantilever or
"locked" configura-tion. The failure of the theoretical
calculations to predict flutter correctly in thisregime is probably
due to the failure of the linearized aerodynamic theory topredict
aerodynamic forces correctly in the high-transonic - low
supersonic
".%ADC TR 56-285 14
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CONFIDENTIAL
0.12AR 1.67 t 0..
ea at 40% chord0.5 cg at 5M' chord ______________
0 ~~g - j__ __ __SWS -3c - 74
ISWS -5d-8T I 481
(130
sws -li 48I2
t UNTAL INJECTION FLUTTER,
STABLE
MACH NUMBERFig. 6. Flutter parameter ( b0.75 (-2/0f) V75)j.7
versus Mach number from experimental tests.
0.60 -_______________
AR~ 1.67 a 02
ea af 4M~ chord
0.50 A . cg at 50% chord __r, 0 ?5 g - 0. 02
0.40 ___
S". -3 -120 - W-,-7 5S36'
K.0 ~No FLUTTER -1040 8) (028t)(028)
swsto Silt -I, -- S* 74 S*S - 0(0.54) (0.43) 1
STABLE INJECTION FLUTTER
UNSTABLE
MACH NUMBER
Fig 7. F lutler onror'-eter "b,- of) %, ("f 65),..,~ versus Vach
num-ber fror. experimental tests.
WADC Ili 56-285 15
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regime. Similarly, it seems probable that the failure of the
theoretical
calculations to make good quantitative predictions throughout
the Mach numberrange for the various pitching frequencies and model
stiffnesses is due to the
failure of the aerodynamic terms in describing accurately the
actual forces on
the wing.
WADC TR 56-28M 16
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SECTION III
CONCLUSIONS
Some conclusions may be drawn from the theoretical and the
experimental
results of the present program. They may be summarized as
follows:
1. Three basic assumed modes appear to be sufficient to
define
qualitatively the flutter boundaries when the
velocity-component
method of Ref. 8 is used, These modes are wing first
bending,
wing first torsion, and rigid pitch. Addition of wing second
bend-
ing does not change the results of the calculation
significantly.
2. For low margins of safety in bending-torsion flutter, the
inclusion of a high frequency pitch mode results in minor
reductions in flutter speed.
3. For both low (7% and high (45% margins of safety in
bending-
torsion flutter, the inclusion of a critical pitch mode (w
causes large regions of instability in an essentially pitch-
bending flutter mode which may extend as high as M = 2.
4. The theoretical calculations do not give a good
quantitative
correlation with the experimental results. The theoretical
calculations predict larger regions of instability than are
observed experimentally. They also predict that the rapid
itj-
creases in the regions of instability will occur at higher
values
of (wO!/h ) than were observed experimentally.
5. The theoretical calculations indicate that the mode of
flutter
which causes the large increases in the region of intability
may be very sensitive to changes in structural damping coef-
ficient. Some of the test data obtained from the SWS-3
series
models confirm this conclusion.
XVADC TR 56-285 17
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BIBLIOGRAPHY
1. Halfman, P. L., McCarthy, J. F., Jr., Prigge, J. S.,Jr.,
Wood, G. A., Jr.,A Variable Mach Number Supersonic Test Section for
Flutter Research,
WADC Technical Report 54-114, December, 1954.
2. McCarthy, J. F., Jr., Asher, G. W., Prigge, J. S., Jr.,
Levey, G. M.,
Three-Dimensional Supersonic Flutter Model Tests Near Mach
Number 1. 5, Part 1, Model Design and Testing Techniques,
WADC
Technical Report 54-113, Part 1, December, 1955.
3. McCarthy, J.F., Jr., Zartarlan, G., Martuccelli, J. R.,
Asher, G.W.,
Three-Dimensional Supersonic Flutter Model Tests Near Mach
Num-
ber 1. 5, Part II, Experimental and Theoretical Data for Bare
Winesand Wings with Tip Tanks, WADC Technical Report 54-113, Part
II, to
be published (Confidential).
4. Jones, G. W., Jr., DuBose, H. C., Investigation of Wing
Flutter at
Transonic Speeds for Six Systematically Varied Wing Plan
Forms,NACA RM L53G10a, August 13, 1953 (Conflidential).
5. Zartarlan, G., Hsu, P. T., Theoretical Studies on the
Prediction of
Unsteady Supersonic Airloads on Elastic Wings, Part I,
Investigationson the Use of Oscillatory Supersonic Aerodynamic
Influence Coefficients,
WADC Technical Report 56-97, Part I, December 1955
(Confidential,
Title Unclassified).
6. Zartarian, G., Theoretical Studies on the Prediction of
Unsteady
Supersonic Airloads on Elastic Wings, Part U, Rules for
Application ofOscillating Supersonic Aerodynamic Influence
Coefficients, WADC
Technical Report 56-97, Part 11, February 1956 (Confidential,
Title
Unclassified).
7. Spielberg, I., Fettis, FJ E., Toney, H. S., Methods for
Calculating
the Flutter and 'vibration Characteristics or Swept Wings. M. R.
No.
MCREXAS-4595-8-4, Air Materiel Command, USAF, August 3,
1948.
WADC TR 56-285 18
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-
8. Barmby, J. G., Cunningham, H. J., Garrick. I. E., Study of
theEffects of Sweep on the Flutter of Cantilever Wings, NACA
Technical
Report 1014, 1951.
9. Scanlon, R. T., Rosenbaum, R., Aircraft Vibration and
Flutter, FirstEdition, The MacMillan Co., New York, 1951.
10. Smilg, B., Wasserman, L. S., Application of
Three-DimensionalFlutter Theory to Aircraft Structures, U. S. Army
Air Corps, Techni-
cal Report 4798, July, 1942.
WAVC [H b6-285 19
-
APPENDIX I
THEORETICAL CA LCULLATIONS
I Introduction
In setting up the flutter equations for the all-movable swept
stabilizer, the
authors examined the relative merits of the strip-theory method
(Ref. 7) and
the veloclty-component method (Ref. 8). For the strip-theory
method, the
aerodynamic forces are applied to sections parallel to the
free-stream whilefor the velocity-component method, they are
applied to sections normal to theelastic axis. The former method is
more rational when the wing ribs areparallel to the free stream,
and gives a better representation of the aero-dynamic conditions at
the root and wing tip. The latte," method, however,appears to be
more suitable for the swept stabilizer model which derives all
itsstiffness characteristics from a single spar. The simple spar
type of construc-tion, the relatively high length to chord ratio as
well as the results of vibra-tion tests suggest that the concept of
the root being effectively clamped perpen-dicular to the elastic
axis, which is a basic assumption of the velocity-componentmethod,
is well justified. Therefore, it was decided that the
velocity-component
mcthod would be used in deriving the equations of motion.
In the derivation and solution of the equations of motion by the
velocity com-ponent method all quantities, mass parameters and
aerodynamic forces, are re-ferred to a reference system (xg, yD )
swept with the elastic axis (Fig. 8).In particular the Mach number
used in obtaining the aerodynamic coefficientsmust be the crossflow
Mach number MD . In the presentation of the results,however, all
the theoretical flutter parameters have been referred to anunswept
reference system (x, y) for convenience when comparing with
experi-
mental results.
2. Flutter Equations Based on Veloc'ity-Component Method
The flutter equations are derived following the method of
Section 16. 2 oflief. 9. The assumption that the wing displacement
is a superposition of fourni),les gives as the deflection of any
point (Fig. 8)
A'ADC TR 56-285 20
-
Za(X , YQ ,t) --Fhl (y )hl(t) + Fh2 (yS)h 2 (t) + x Fa(y )5Q
(t)
4 (yD sinQ + xD cos -d) 0 (t) (1)
where (see Eqs. 40-42)
Fh , Fh are the first and second assumed cantilever bending
modes2
Fa is the first assumed uncoupled torsion mode
El H2 } are reference tip amplitudes for the first and second
bending modes,a ,_ first uncoupled torsion mode, and rigid body
pitch mode, respectively.
ACTLIAL ROOT
2111
./ 0.. MOYO NOTATOO f" AXIS
It_
X .. E L.ASTIC AJIS
"NI Na
Fig. B. Axis sy-tern for swept stabilizer.
From Eq. (1) it is seen that the rigid body pitch is equivalent
to a bending ofthe elastic axis plus a rotation about the elastic
axis, so that only the aero-dynamic forces due to the translation
and rotation of sections normal to theelastic axis are needed.
Application of the Lagrange equations of motion to thesystem as
given by Fig. 8 along with the assumption of simple harmonic
motionand the Introduction of the dimensionless variable
(2)
leads to the following dimensionless set of flutter
equations:
A(h ' C 5 + Dl =0 (3)ba0 b 0
A'ADC TR 56-285 21
-
E( h F(h 2 Gdg. Hi0O (4)
I /h ~2 K 5bg0 0 o
bo 0 D
where
(b 3b d'V 1 b ~ LhF2h d~jq + fl (b9ý ( ~o 1 Fd.A ( b9 h h 02
di7
+[Z Wg zi di(~- JQ ýWa 0
(7)
1(b 2Ib2)3 bo )dFh
fj'("Q ) r d n +(9 f - _ LhhFh di7a0 1 hh~hl h2 7 gQ
/ g0 t d17 1
00
+ f(" 2 l) 9 XFh Fh d~ (8
0 (9)9
\ADC TIR 5t6-285 22
-
D d~ f~.Q _h. Fh diQ~h+co
0 0 4.F 1Q+Cs0 b Q0
f bl )b ) P.QFh d 77Q(00)
0 9 0 9
b(Q b1 ~~.Q ()dhiLhh drIDELhhFhFh d,7a + 0l ba AQh
f , -Q A.Fh Fh dIr lf
F r Lb hd + f/bQ )3( bO-) " h Lhh , Fh2 dn,
0~ b-- 90 2' 0 b 1* d772
22 ~QF t 2
.j~F(Od?7 + Zf (I (bg 2 d,1;2
0a 0ba D 2 ((13)
H7 = flb )L ,Fd7 + Io~ _ )3a ~ c Fh d i7.Q
0 0 0 a0
0 l ~LI~d7 (14)
0 bo0 bD0 a
~WADC TR 56-285 23
-
3 1(g4 b dFl b g M ah F Fh dirg f £ _£2(. 0 ) h l , Fad v7£2
iwf( )g 0 1a 0Q d71 l(b£2 3 A9 X aF FhdT£} (15)
I (b 3 1(b.Q 4 b Do)dFh2
lbhFaFhd,72 + f _- ) ( a'F i.0 b-9 b
9 d1
+ ~ 9 QXaa F aF hd 72 (18)
K Mib ) 2 Fd779 + f~ (±)~ Moa Fa d'7g
+ [I -ZI 1(bg9 ) r, ~ a a0 b(17)
L f1b Ma# Fad?7~ ll -Q _ )"Q Fa d 7g
+ Cos9 f (bR M aa£2 (180 D
WADC TR 56-28ý, 24
-
1b 4 bQ d FhIm-~o~ C0.9 .- ) Mah Fh dii. + Cos Qf( Q. ) ( - )~
di
b 2 bp b 3/•)0Q dFh
(. ~ )LhhFhd,79 - f d~)(~ )\ 7 ,) dr1 Lhh'd?7.Q1p 0 bQrj
0 0 go
I d bD 2 1b tQxbf A~ .... Fh dnD + cosQf J (Y.) iix h Ij
(20Q
0 0 D
(bg 3 ) b~Fdl. 4 bQ dF, hM dl,
N()2 = QXa l a F hdJ7 n .coa Df t"Q) 2 "
0 0 0go
11 (±_)(bQ F iw +Cos f (bg gx Fh dI.Q(20)
WAD Tb 56-28 2520b 1
-
C 1 M bD• 0b0 0 DO D0
2 2 2 2
-2 Z (b+o) ) ),rua dd(2
and23(23)
2 2 (constant along span) (24)
b ~ b
and (constant along span)2
"LY (4 f Lh (23)
h , = (cota nstanQt a (27)
k
LhG = Lt-L~h (1 + a) (28)2
= 2 tan.Q ( s a s( p _a)1 (29)
Lhh " -• -T ,- (26)
Moh = Mh_ Lh(! + a) (30)
2
_____ -it " [ h(1 - )(31)
Mah, -i L[h(- + a)I (3)k 2
WADC TR 56-285 26
-
Mao Ma - (Mh + L0,)(I + a) + Lh( + a)2 (32)
2 2
Macy = -itang 3 1 - Lh(I _ a2) (33)k 8 2k 4
Lho = bsn Q d Lhh Lho CosQ + Lhh , sin Q (34)
y.sin9 -d Mh + Mao cos.- Mah, sing (35)
where L., Lar, M. and Mh are as defined by Ref. 10.
The above equations were written using the actual mass
distribution for thestabilizer being studied. These relationships
are
b 2mgQ = m. (#- (36)
3 (7
bQa
r2 ba' (38)D r2 m90 bP2
•0
where
b__ = ( I 'g2Q (39)However, to simplify the aerodynamic
calculations, the tapered planform was re-placed by a rectangular
planform of constant chord so that the aerodynamic coef-ficients
would remain constant along the span at a given value of reduced
frequency.A check calculation has shown that if reference
semichord, br' is taken at the 75(j:,span station of the actual
mode perpendicular to the elastic axis, the differencebetween the
values of the aerodynamic integrals as given by the rectangular
*It should be noted that this equation is given incorrectly in
Ref. 9.
WADC TR 56-285 27
-
planform and those values found by an actual numerical
integration along thespan of the tapered wing are very small.
rhe first bending mode was taken as
W2 (40)Fhl f
the first torsion asF = '7Q (41)
and second bending as
Fh - -12.209 + 25. 488-17 12.279t7 (42)
The second bending mode was obtained by assuming a power series
in 77g whichsatisfied
(1) the boundary conditions for a cantilever mount,(2) the
condition of orthogonality with the first bending mode, and
(3) the condition of zero deflection at the 75 percent span
location.
Condition (3) was obtained from observation of the node line for
the secondbending mode of the actual mode during vibration
tests.
The parameters used in the analyses were
ig = 30 a r -0.20
1r2 0. 250 Q= 43014'
x 0.20 b.0 0.66798
- 0.21233 a sini A 3. 22603b90
WADC TR 56-285 28
-
which resulted in the following values for the coefficients of
the flutter equations.2
A = 0. 078,125 Lhh + 0. 025, 919 L hh' + 2.071,439 L (r_' ) Z
(43)a 1
B = 0.0 2 0 , 3 3 2 Lhh + 0.051,958 Lhh, (44)
C = 0.061,035 L~he + 0.010,799 Lha, + 0, 342, 857 (45)
D = -0. 228,066 Lhh + 0. 059,291 Lha + 0. 055,743 Lhh, + 6.
582,193 (46)
E = 0. 020, 331, 530 Lhh - 0. 000,120, 555 L hh, (47)
F =0. 091,120 Lhh + 0.025,919 L~hh. + 3.417,742 1l - (% 2w iyj 2
Z] (48)
G = -0.000,283,89 Lha - 0.004,972,8 LhI, - 0.212,126 (49)
H = 0.038,584 Lhh -0. 027, 302 Lha -0. 025,668 Lhh, -1. 161,620
(50)
I =0.061,035 M ah + 0.021,599 M ah, + 0. 342,857 (51)
J = -0.000,283,89 Mah + 0.037,371 Mah, -0. 212, 126 (52)
K=0.050,863 Maa + 0.010, 124 Maa' + 0. 441,964 [1 - Z (53)
L -0.180, 995 Mah + 0.055, 585 Maa + 0.052, 259 M ah, + 1. 691,
275 (54)
M 0.0 5 9 ,2 9 1 M ah + 0.0 2 3 ,604 Mah' + 0. 22 8 ,06 6 Lhh
+0.0 7 6 ,860 Lhh, +6.582,193(55)
N = -0.027,302 MUah + 0.023,604 Mah, + 0.038,584 Lhh + 0.
158,271 Lhh, - 1. 161, 620(56)
0 = 0. 055,585 Mcy + 0.014.753 Ma * 0.180,995 Lha + 0. 030,618
Lhh' + 1.691,275(57)
P = -O.1 6 8,0 9 8Mah +0.080,996 Maa + 0.076,149 Mah, - 0. 68 7
,6 4 8 Lhh
+ 0. 168,098 Lha . 0. 158,038 L", + 23.195.949 1 o..._ _ )2Z]
(58)Orl
3 Solution of Equations of Motion for Flutter
The flutter equations for the swept stabilizer were solved for
two (bending-
torsion), three (bending-torsion-pitch) and four (first
bending-second bending-
torsion-pitch) degree-of -freedom systems. The flutter
determinants for each
system are respectively
WADC TR 56-285 29
-
A C
I K
A C D
I K L =0
M 0 P
(59)
A B C D
E F G H
J K L =0
M N 0 P
where A, B, ---- P are given by Eqs. (43) through (58). In each
case theaerodynamic terms were evaluated by selecting specific
combinations of M and
k.
The general pattern of solution of the three flutter
determinants was thesame. A given determinant was first expanded
into a complex polynomial. Sincethe right-hand side of the equation
was zero, two separate equations werewritten by setting both the
real and imaginary parts of the polynomial equal to
zero. These two simultaneous equations were solved for any two
desiredeigenvalues.
In the two-degree-of-freedom case, the complex polynomial
resulting fromthe expansion of the determinant was solved for the
eigenvalues (wh /W ) and
2 1(V1 /Wf)2. The cross flow Mach numbers used were M2 = 0,
10/9, 5/4 and
10/7. This case corresponds to the pitch-locked condition.
The three-degree-of-freedom system was solved for the two
eigenvalues(W0 / 2f) and (w V/Wf)2. (whlwal )2 was set equal to 0.
0625, a value which
corresponded closely to the average value for the actual
stabilizer models.Again M2 z 0, 10/9, 5/4, and 10/7 were used.
Finally. for the four-degree-of-freedom systems, Z = (w a /f)2
(1 + ig) was
used as the eigenvalue. Here, it was necessary to specify values
of (Wh /W a22 2 1
h2"' ) and (w 0 !n ) in advance. The solutions of the
fourth-order deter-minant,; were carried out on a 650 IBM computer
using a program developed byNorth American Aviation in Columbus,
Ohio, and the results ploted on a
WADC TR 56-285 30
-
V-g diagram. Value of the constants for which solutions were
found were(wh /wal) 2 = 0. 0625, (wh2 / 1)2 = 0. 9025, 1. 155625
for (w /, al)2 = 0. 30
and (wh/I 1 1)2 . 0.0625, (Wh2l/al) 2 1. 155625 for (w0/W a. 1)2
= 0.20.
The results from the two- and three-degree-of-freedom cases are
plotted inFig. 9 and then crossplotted in Fig. 1. The pitch-locked
values shown as as-ymptotes as (w0/Whl) 2 --w c in Fig. 9 and as
the (w / Whl) 2 __w co boundary in
Fig. 1 are the results of the two-degree-of-freedom
calculations.
The most interesting feature of these analyses are the very deep
"buckets"that occur at low values of (w/Wh l)2 in Fig. 9 and
correspondingly at low
values of (w / W)2 in Fig. 1. In some cases the curves actually
double back
on themselves giving two regions of stability at a given value
of (WO/ whI) or
Mach number. The presence of these "buckets" is apparently due
to a change influtter mode shape and can be explained by looking at
sample four-degree-of-freedom calculations in some detail.
Each solution for the four-degree-of-freedom problem at a given
set ofvalues for ((oh /1wall), (1wh2 /al )2, and (w 0 /w,,)2 and
Mach number, yields
four separate curves on branches on a V-g plot and several
values of k for theflutter condition of g = 0 (see Fig. 10). Since
each branch represents aparticular mode of flutter, it appears from
Fig. 10 at Ma = 10/9 (M = 1. 525)that the stabilizer is capable of
flutter in the Ist mode and the 2nd mode. AtM . 5/4 (M = 1. 716)
the stabilizer has one unstable region along the V axisin the 2nd
mode and two unstable regions in the 3rd mode. A set of
three-dimensional sketches of V versus g versus M is shown in Fig.
11. To avoidconfusion, each sketch contains only one type of
flutter mode. Because of thedifficulty in following the "various
possible flutter modes from a V-g diagram toa Vf/W a b0. 75 versus
M plot, the results of the four-degree-of-freedom
analysis of Fig. 12 were ultimately drawn after looking at
three-dimensionalplots of V versus g versus M with interest
concentrated on the traces of thed4ifferent modes in the g = 0
plane.
The lowest set of curves on Vf/w al b0 . 75 versus M in Fig. 12
form the
critical flutter bounidary. This boundary, as can be seen by
looking at Fig. 12is formed by three different flutter modes each
becoming the critical boundaryof instability over a particular Mach
number range, The flutter boundary fromthe four-degree-of-freedom
analysis for (w 0lWal = 0. 30 appears to compare
WADC TR 56-285 31
-
0
2 ~ ;
0( 1/4, - 5u 2.
-~21
F19 9 Fltt r affieters bf~ aund V1 ,Wb versus hrntw. n
fedcui¶ Calculations. hl2fo Io n three-dogree.o.jWADC TR 56-285
32
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o )
* • •__ aC; C
o, g -
WAC TR 56-285 3
4-'1
cm I
3AD R 5628 33
-
70
(00
MOE I MOE
Nqb' ,
MODE 3 NODE 2
V
//
'/
II It
MOO! S tOOS 4
Fig. 11. Sketches of V versus g versus Mach number curves from
four-degree-of-freedom calculations.
WADC TR 56-285 34
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12.0 " : " - .
AR * 1.67-
11.0 1(/14 )c - 450 MODE 3'A. 0.5
10.0 (wh/ - 0.25
9.0 2.. 0.25 - - - . -8.o ea, at. 40,% chard8.0 aot4~eh~d - - -
MODE 3
cg at 50% chord7.0 ----. 72
(dA, . 0. 4873 6. - 2. MODE I
v ga 1o 9h- go - 0>• s ro,, fboijrp,)2ý , ...
4.0 So 0S .(b,)/b) )3.
"d.0 1. 1.. (bf)/oD)4 ... .
2.0 . ... (1.0 (G/w)2 0.30 j :
0 "- " • 30 C
06.0
I MODE 3
IMODE 4 MODE I
4.0
MODE 2
"3.0 h ,
-rh: 2,,a, 1 0.9025 | MODE 31
0 1.655625
1.0 -- (,h2,/woal)" - 1 1 5
1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0
MACH NUMBE P
Fig. 12. Flutter paromn iers Vt/i" j b0.-75 and Vf/ifb 0 .. ;,
versus Mach number from four.degree-oF.freedom
calculations.
WADC TR 56-285 35
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very fzvorably with the three-degree-of-freedom flutter boundary
[(W h2/W1)2 =0
which Is also shown in Fig. 12. Thus, the second-bending degree
of freedomVf
apparently has little influence on the flutter parameter of aW
aI b0. 7 5
swept stabilizer below M = 2. 0 except for a general lowering of
the curve. In
using the four-degree-of-freedom analysis to interpret the
points found in the
three-degree-of-freedom analysis, it is seen that the S-shaped
curves of Fig. 9
do indeed appear reasonable and are a direct result of a change
in critical
flutter modes in going from low Mach number to high Mach
number.
Another interesting characteristic of the V-g solutions of the
four-degree-
of-freedom analysis is the variation in the flutter boundary
with small changes
in the structural damping coefficient, "g. " By referring to
Fig. 13, it is seen
that increasing the structural damping from g = 0 to g = 0. 06
moves the "bucket"
on the flutter boundary due to the 2nd mode from about M = 1. 75
to about
M 1 1. 65. The general level of the flutter boundary as
determined by the Ist
and 2nd modes will not be changed.
7.0- AR - 1.67
1) - 0 (I/4)c 450
6.0 A - 0.5S-, *.oo h I/.. . 0.25
5.I I • .a 0.255.0 ,.o.os -o at 40% chord
c I c at 50% chord-I - I\ jI~ -" d/bo) -0.487
4. # Lf o 30, F - 21.8
> .0 000 0 0 )
S .6 . 1b.. .bs0)3
00
1.5 1.6 1.7 1.8 1.9 2.0 2.1MACH NUMBER
Fi,. 13. Voriotion of Vf'" , -b.. versus Ma'ch number with
chongein structural
domping from four-degree-of -. o,'om calculations.
WADC TR 56-285 36
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APPENDIX 11
EXPERIMENTAL DATA
This appendix gives the detailed tabulation of both design and
experimentaldata. Since both the model design and the testing
techniques are essentially thesame as those described in Ref. 2,
little discussion (f them is included in thisappendix.
The planform of the stabilizer models is shown in Fig. 14a and a
crosssection of the root of the SWS-I series models is shcn in Fig.
14b. Rootcross sections for the 9WS-2 and the SWS-3 series models
are nut shown sincethey differ only in minor details from the SWS-I
model. As can be seen inFig. 1 4b the spar, which contributes
essentially all of the model bendingstiffness and most of the
torsional stiffness, is constructed of a pine corearound which is
wrapped an aluminum skin. Steel caps are then cemented to thespas.
Both the steel and aluminum are tapered linearly along the span
giving,when combined with the taper of the height and width of
spar, the requircd fourth-power distribution to the bending and
torsional rigidities, EIQ and GJ Q
ElI Q E19 o (60)
_,/ \b Q 4
GJi GJ.Q o( Q-_-_) (61)
where o
bQ
Balsa wood cemented to the spar was used to give the aerodynamic
shape re-quired, and suitably spaced lead weights were used to give
the mass parametersrequireJ. Table 1 gives a summary of the design
parameters for all cf thestabilizer modtels.
WADC TR 56-285 37
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1O00 • •--- •
" " • • M0 CHORDCHNORD
/ N AXIS OF ROTATION
..... 5o-----'-A Fig. 14a. Model plonform
OO55-= STEEL VAR CAP
0 0.020 TOP 9s0-1101
24-ST
SPAR CROSS -SECTION DETAIL AT ROOT
- -~50 "
LEAD WEIGHTS •
6000
007 0,36 DG
•6o 4,33 -- 10j 6 ; 5 0 0 --- 4 0 3 - - 0
3 3
Fig. 14b,. Model crossL-section detail at root
Fig. 14. Swept stab•£hzef design dirawings ('all dimensions in
inches).
WADC TR 06-285 38
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Table 1. Design parameters for ,. .ept stabilizer models.
The parameters presented in this table are common to all the
models built in this progra.Geometric Parameters
Panel aspect ratio, AR .......................................
.. 1-2Taper raio, A
..................................................... 1"2Sweep
angle of 1/4 chord . ..........................................
45.0rMean aerodynamic chord (in.), MAC
................................ 7.7778Section mar. thickness (%
chord) . ............................... .. .0.Line of max.
thickness (% chord) ..................................... .0. s
Design ParametersS ection cemer of graviry location (% chord).
(cg) .......................... 50.0.Radius of gyratior (traction
of semichord), r. I............................ 0.50Calculated
locus of shear cenrters(% chord), (eo) ..........................
40.0%
Properties of Balsa Wood (average values)Modulus of elasticity
in bending (lb/in2 ), E .. ........................ (400Modulus of
elafticity in torsion (lb/in 2 ), G .......................... .
20,000Density (lb/in'). PsA .............
................................... .... 0.00)900
Properties of Pine CoreModulus of elasticiry in b.Jdirn
(lb/in
2), E . ......................... 1.329 , 10'
Modulus of elasticity in t,.;qion (lb/in2
), G ........................... 0. o107 tO'Density (lb/in ), p
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . .. 0.014
The rnot fitting and the mounting block with the pitching
mechanism areshown in Fig. 15. The spar was glued and screwed to
the root fitting, shownremoved from the mounting in Fig. 15a.
Pitching frequency was controlledby changing the thickness of the
flexure shown on the end of the root fitting inFig. 15a. Figure 15b
shows the rear of the mounting block with the flexurein place. The
angle of attack of the model could be changed by rotating thewhole
clamp shown in Fig. 1Sb. Drag and lift loads were carried
adequatelyby three ball bearings in the mounting block. The gap
between the root and themounting block was sealed with aluminum
foil for all tests.
With the pitching mechanism "locked Out, " static tests were
made on mostof the models in an attempt to determine the cantilever
properties of the model.The properties determined were measured
elastic axis, (ea)M. as discussed inRef. 2. and the root values of
EIg and GJg. The results oif these measurementsare given in Table
2. There is considerable scatter in the El• I GJv and (calM
data.
The measured mass per unit len',ngi, at the root (mo)M is also
given in Table 2.This quantity was indirectly measured using the
assumed mass distribution
2
m (y) (me)M (0 - Y.-) (63)21
By just measuring the total mass and then computing (me)M
from
(me)M total mass (64)
t' (2I - - 2 dy0 21
WADC TR 56-285 39
-
Fig. ISa.
Fig. 15b.
Fig. 15. Picluresof root mounting block.
WADC TR 56-285 40
-
In Eq. (64) the total mass does not include the mass of the root
fitting, so thatthe value for (me)M includeb only the mass of the
balsa, lead, glue and the
spar.
Data for the pitching frequency is found in Table 3. The mass
moment ofinertia of the whole model, including the root fitting,
was obtained by swingingthe model with a bifilar pendulum. The
pitching mechanism flexibility Mifluencecoefficients, C., was
measured with a transit and mirror arrangement. Thepitching
frequency was then calculated as:
f - 1 • 1 (65)2v: I€ C0
The results of the flutter tests are given in Table 4. For the
sake ofconvenience, most of the important experimental natural
still-air-vibrationfrequencies are included as well as the tunnel
conditions at flutter. If flutteroccurred, the conditions at the
start of flutter are given. If no flutter occurredduring the test
run, the conditions at the start and end of the test are
given.Figures 16 and 17 are excerpts from the high speed movies
taken during theflutter of the SWS-l-98 and the SWS-3d-87 models,
respectively. These portionsof the movies have been analyzed and
the results are presented in terms of thepitching motion at the
root and the motion of the tip sections in Figs. 18 and 19.These
flutter modes are typical of those encountered for the stabilizer
models.
Complete vibration data, including sketches of node lines,
frequency, andstructural damping of the lower modes of vibration
are found in Table 5. Allof the models were vibration tested in
both the "locked, " or cantilever, condi-tion and with the pitching
mechanism in. Figure 20 is a plot of the normalizedcoupled
frequencies with the pitching mechanism in. The lowest
cantileverbending frequency, fhN' was used as the normalizing
frequency. It is interestingto note that the frequencies of the
first coupled modes, ft for must of thestabilizer models fall along
a common curve with not too much scatter. Thesame is true for the
frequencies of the second coupled mode, f2 '
Table 6 gives the influence coefficient data for the models with
the pitchingmechanism in and Fig. 21 shows the location of the
stations at which influencecoefficients were taken.
WVADC TR 56-285 41
-
Table 2. Static data for swept stobilizer models.
Model (mo). (Me)M 1 (GJo()• (ElI-u(Slut/ht) (% chord) I
WS- 1 0.0214 3.690 x 104 8.493 x 104
SYS-2 0.0214 2.696 . 104- 6.860 M 104-SVS-Ib 0.0220 2.877 . 104
6.661 x 104SlrS-ic 0.0225 46.5% 3.459 x 104 7.422 x 104
SI'S-Id 0.0225 49.0% 2.380 x 104 5.84 X 104
SWS-Ie 0.0246 39.0% 4.220 x 104 7.836 x 104
SWS-.3 0.0240 35.36% 7.572 . 104 10.448 . 10'SWI" 3a 0.0231
43.0% 3.680 x 104 8.59 . 104
SWS- 3b 0.0224 50% 5.23 x 104 7.139 x 104
SWS-3C 0.0233 1.942 x 101- 6.754 x 104.
S1TS-3d 0.0247 50% 5.234 . 104 7.489 x 104SIS-.e 0.0274 3.89 x
10- 8.89 x 104.
Date fot spar only.
Toble 3. Pitching frmquoncy data.
Model odý).~ (Cdea(Slup-qt2) (rsdilb~tt) (cps)
SWS.. I-L 0.00244 0.0000225WS-1-138 0.00244 0.000543 138
5Wt1- 1- 105 0.00244 0.000915 105STS- 1-98 0.00244 0.001072
98.3
WS-r 2-L 0.000022
SVS. lb.L" 0.00223 0.C00022S15. i.-48 0.00238 0.004460 47.6
S•I'S ld.L 0.00002SW; le- 74 0.10226 0.002132 ?3.7SW' 3- 5
0.00206 0.004460 52.5STS-3.-63 0.00224 0.002854 63. 1S.S- 3b-53
0.00212 0.004249 53.3S.S-,c. 74 0.00205 0.00224 74.4
STSW.d.87 0.00225 0,001498 8A.7s
1-,e.I120 0.00275 10.000628 120.0
WADC TR 56-285 42
-
CON FIDENTIAL
1 B6 1 d d 6 -61 1 s
d id c H I 0 0
a - - - C; aI
I!Ig Thn~~
0A 0k 0-8 43 0 0
ICOFFIDENTIAL
-
CONFIDENTIAL
Fig. 16. Pictures of flutter of SWS.1.99 model fromi high speed
movie
WADC TR 56-285 44
CONFIDENTIAL
-
I
CONFIDENTIAL
I
Fig. 17. Pictures of flutter of SWS-3d.87 model from• high speed
movie
WADC TR 56-285 4
CONFIDENTIAL
-
LE
L.2 hy __ T
/O
04 /
-08
-1F ig. 18. Analysis of high speed movies of SWS-1.98 model.
12 T~~ L
-ROOT Pi T>4.06 ~ FILM SPIEED- .
2760 PiCTuRES/ArC2,
PIC T URES
428 32
-04
Fig 19 Anclysi% of high spe~ed no-es of SWS-Ad-27 model,
WADC TPL 56-285 46
-
8.0-
SWS. .I -0138
7.Co - -- SWS-1-10 157SWS-I-95 -
SWS-Ie-74
SWS-3c-74
SWS- 3a 63
SWS-3b-53
zSWS - IC-48 ,
- 4°0--l -
zMODE 3 3i910
3)
MODE 2
-- _ --- - -
M.oo El • ____ _MODE IS(f)
0 0.4 0.8 1.2 1.6 2.0
f o h,
Fig 20 Vbroton frequency doto for swept stobilizer models
WADC TR 56-285 47
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AL
SArmed Services Technical Information AgencyARLINGTON HALL
STATION!
ARLINGTON 12 VIRGINIA
' FOR
MICRO-CARD 20F2CONTROL ONLY
NOTICE: WHEN GOVERNMENT OR OTHER DRAWINGS, SPECIFICATIONS OR
OTHER DATAARXTE-MD FOR ANY PURPOSE OTHER THAN IN CONNECTION WITH A
DEFINITELY RELATEDGOVERNMENT PROCUREMENT OPERATION, THE U. S.
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USE OR SELL ANY PATENTED INVENTION THAT MAY IN ANY WAY BE
RELATED THERETO.
C ONFIDENTIAL 'Si'-IA
-
SPAN STATIONiS
CiOO*IS( STATIONS
Fig. 21. Location 0# influence Coefficient stotions.
WADC TB 56-285 48
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z z
00 0
0
, ow
I I
£- -°2
0 2 0
0 0
I7
m -A 0
It 7
WADC TR 56-285 49
-
24
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ANN
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ofT V
0 o i
aa
W'ADC TR 56-285 50
-
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D
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X 0
W,,DCx "R 56-285 51
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wAJ w
UU0.0
4n
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WAIC T* 56-285 52
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"WADC TR 56-285 54
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p ic
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CONFIENTIA