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AD142088

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TOApproved for public release, distributionunlimited

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

THIS PAGE IS UNCLASSIFIED

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AD__Z088

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SCIENTIFIC AND TECHNICAL INFORMATION

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CLASSIFICATION CHANGEDTO UNCLASSIFIEDFROM CONFIDENTIAL

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Bul. no. 66-20 15 Oct. 1966

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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

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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

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

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

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

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

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

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

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

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

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

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- "

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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

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•' 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

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

CONFIDENTIAL

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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

CONFIDENTIAL

CONFIDENTIAL

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

CONFIDENTIAL

CONFIDENTIAL

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

CONFIDENTIAL

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

CONFIDENTIAL

CONFIDENTIAL

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

CONFIDENTIAL

CONFIDENTIAL

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

CONFIDENTIAL

CONFIDENTIAL

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

CONFIDENTIAL

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

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

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

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

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

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

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

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. GOVERNMENT THEREBY INCURSNO RESPONSIBILITY, NOR ANY OBLIGATION WHATSOEVERT AND THE FACT THAT THEGOVERNMENT MAY HAVE FORMULATED, FURNISHED, OR IN ANY WAY SUPPLIED THEG IVD DRAWINGS, SPECIFICATIONS, OR OTHER DATA IS NOT TO BE REGARDED BYIMPLICATION OR OTHERWISE AS IN ANY MANNER LICENSING THE HOLDER OR ANY OTHERPERSON OR CORPORATION, OR CONVEYING ANY RIGHTS OR PERMISSION TO MANUFACTURE,

USE OR SELL ANY PATENTED INVENTION THAT MAY IN ANY WAY BE RELATED THERETO.

C ONFIDENTIAL 'Si'-IA

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CiOO*IS( STATIONS

Fig. 21. Location 0# influence Coefficient stotions.

WADC TB 56-285 48

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WADC TR 56-285 49

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W'ADC TR 56-285 50

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