FOR AERONAUTICS/67531/metadc...determiningthe naturalvibrationmodes given in the present report. ... NACA TN 4010 3 9 semispanand the ordinateis the acceleration. The cross-hatthingon

.’kJ---. ...-

FOR AERONAUTICS

TECHNICAL NOTE 4010

EXPERIMENTALLY Determined NATURJiL VIBRATION

CANTILEVER-WING FLUTTER MODELS BY USING AN

ACCELERATION METHOD

By Perry W. Hanson and W. J. Tuovila

Langley Aeronautical LaboratoryLangley Field, Va.

Washington

April 1957

lRNATIONAL ADVISORY COMMMT!EE

TECHNICAL Nom 4-010

EXI?EKDWWULY DETERMINED NATURAL KIBMTION MODES OF SOME

C!ANTILEVER-WINGFLUTTER MODELS BY USING AN

ACCELERATION METHOD

~ Perry W. Hanson and W. J.

Three-dimensional viewsvibration mode shapes of ten

SII’MMARY

TuOvi.la

are presented of the first three naturalcantilever-wiu models. A table of nor-

malized deflection at six spanwise and fiv~ chordwise stations isincluded for each mde. These mode shapes were measured by a ratherunique experimental technique using grains of sand as accelerometers.The ‘technique,which is particularly-suitedof sma~ wing models, is described and someto be encoun~red in-applying the technique

INTRODUCTION

for measuring nmde shapesof the difficulties likelyare discussed.

Ilnorder to perform a mdal-me flutter analysis of a ~, it isnecessary to know-the first few nat&l vibration ties. Severai experim-ental methods can be used to determine them. A photographic techniqueis described in reference 1. Mirrors @aced on the wing can be used tomeasure slopes from which deflections can be computed; accelerom%ers ormicrometers which incorporate various sensing devices can be used to meas-ure amplitudes directly. Bkde shapes can also be calculated from measuredinfluence coefficients. (See, for example, refs. 2 and 3.) These ~thodshave disadvantages, however, when used on small wing models.

This paper describes a technique that is particularly suited for themeasurement of mode shapes of small wing models. This technique is anextension of the technique described in reference k with the “chatterbar” replaced by sand. ~ addition to a description of the new tech-nique, the first three nmde shapes of ten cantilever-wing models arepresented.

2 NAC!ATN kOIO

SYMBOIS

A aspect ratio

b wing stichord at

f~ natursl vibration

root measured parallel to airstresm

frequency for the nth mode, cps

2 length of semispan of model measyred normal to stresm direction

t thickness of wing

x chordwise station, measured psmllel to root chord fromleading edge

Y spanwise station,the root

z airfoil thickness

A sweepback sngle of

A taper ratio

measured perpendicular

()t 100‘rtinate’ m

quarter-chord.~ine

EXPERIMENTAL TECHNIQUE FOR DETHMXING NATURAL

General Concept

A new technique, which maybe termed “the 1 g

to root chord from

.

Vibration MODES

methdj” is used indetermining the natural vibration modes given in the present report.The technique is based on the principle that the”acceleration of a psr-

—

tidle vibrating in shnple harmonic motion is the product of its emplitudeand circular frequency squsred. At my given frequency, therefore, par-ticles having equel accelerationsw$ll also have equal amplitudes. Anacceleration correspondingto that of gra@ty occurs when a psrticleplaced on a vibrating wing just begins to rise from the surface.

In the 1 g method, the wing surface is sprinkled with ssnd and vibra-ted at one of its natural frequencies. A line, or lines, along whichthe acceleration is equal to gravity snd the amplitude is, therefore,constant is obtained between regions of bouncing and stationary sand.By changing the magnitude of the exciting force the wing amplitudes arechanged and the location of the 1 g line or lines is changed. This is J

illustrated in figure 1 which shows a hypothetical wing vibrating in—

first bending at three discrete amplitudes. The abscissa is the wing u

NACA TN 4010 3

9 semispan and the ordinate is the acceleration. The cross-hatthing oneach curve represents regions where the sand is bouncing (accelerationsgreater than that of gravity). The boundaries between regions of bouncing

4 and stationary sand we shown at spantise locations a, b, and c. Foreach of the three wing amplitudes shown h fi~e 1, the strain is meas-ured at the wing root. It is assumed that the wing amplitude varies lin-esrly with root strain and that the relative distribution of amplitude(mode shape) is independent of absolute smplitude. TQe following analysisshows how the root strains are used to convert the 1 g lines into ~icontourlines” that define the wing mode shape. (Refer to figure 1.)

Let ~ be deflection at sendspan location x for wing smpli-

tude n and an be strain associated with smpli.tude n, and its corre-

sponding 1 g line. Then,

.

b

If similar relations are written for the other deflections, based onthe arbitrary wing amplitude 3, at .spanwisestations b and c, thequantity in the parenthesis is seen to be a constant. Since the deflec-tions defining the mode shape are relative, this constant mqf be takento be 1. The 1 g lines thus become contour lines with amplitudes equalto the reciprocal of the strain associated with the line. By varyingthe magnitude of the wing eccitin.gforce in six or seven steps, a suf-ficient number of contour lines sre located to define the mode shape.

Equipment

The wing models were cantilevered from a large backstop and wereexcited acousticallyby a ptilic address system loudspeaker with thehorn removed. The loudspeaker was tiivenby a wide-range oscillatorthrough a direct-coupled amplifier. The signal.from stiain gages onthe wing root was fed through an snrplifierto an electronic voltmeterplaceilbeside the wing. A modified aerial csmera mounted on the backstopabove the wing was used to photograph the ssnd pattern and the voltmeterreading for each contour line. The sand used was washed =d throughlydried to minimize stickiw to the w@.

Technique

After clamping the model to the backstop, it was csrefully cleanedto remove dirt smd fingerprints which might case the sand to stick tothe wing. The loudspeaker unit was placed beneath the wing and the output

4

frequency varied until resonance @cated by a maximum deflection of the

NACA TN 4010

the desired natural mode was indi- ●

voltmeter needle.

The loudspeaker was turned off and ssmd was sprinkled lightly overthe surface of the model. The volume of the lou&pesker was slowlyincreased until the sand just started moving scmewhere on the wing,usually at the tip or along one of the edges, depending on the mode.The csaera shutter was opened for a time exposure to record the movementof the sand and the corresponding root strain indicated on the dial ofithe electronic voltmeter. Also recorded in the photograph were the win&identification and nunbers indicating the attenuation setting of thestrain-gage smplifier. The volume of the loudspeaker was furtherincreased to move the contour line to a different position on the wingand another photograph was taken. The process was repeated in stepsuntil all the sand on the wing was moving except along the node lines.Usually, six or seven photographs were sufficient to give a good spacingof the contour lines from the wing tip or edges to the node lines.

Interpretation of Data

The raw data are a series of photographs, each showing regions wherethe sand is blurred (moving) and sharp (stationary) snd a meter readingand amplifier attenuation showing the corresponding rmt strain. A ssmplephotograph is shown in figure 2. Here the-~ is vibrating in the thirdmode and two regions of stationary sand, A and C, and two regions ofmoving sand, B and D may be seen. Regions ‘Aand C!sre vibrating withan acceleration less than that of gravity, and regions B and D with anacceleration greater than that of gravity. The lines separating theseregions sre vibrating with an acceleration equal to that of gravity sndtherefore are lines of equal amplitude. The meter reading multipliedby the amplifier attenuation, 30, gives the relative strain at the root

....

associated with these particular lines. The numbers and letters at thetop of the photograph me wing, mode, and frequency identifications.

Lines separating the regions of moving and stationsxy sand weredrawn on each photograph and a composite ~awing of the wing was madeshowing the locations of the various lines of equal.smplitude. Thereciprocal of the root strain for each of these lines was noted on thecomposite drawing. These nunibersindicate the relative deflection ofthe wing along that particular line. A ssmple composite drawing isshown in figure 5. The whole nunibersbesid& each line identify thephotograph from which the lines were obtained and the fractional numbersindicate the relative deflection.

Each wing was divided into six spanwise stations at 0.1, 0.3, 0.5,0.7, 0.9, and 1.0 semispan, and five chordwise stations at 0, 0.27, O.N,O.~, and 1.00 chord, as shown in figure 3, A plot was made of relatiw

—

——

—

—

.

.

.—

NACA TN 4010 5

● smplitude of deflection against semispan distance y/Z at each of thefive chordwise stations. Another plot was made of relative smplitudeof deflection against chord, x/2b, at each of the six semispan stations.

4 After fairing each of these plots by cross reference to the other, thevalues of the relative deflections at the intersections of the variousspanwise snd chordwise stations were normalized on the msximum deflectionand arranged in tabular form. Ekamples of such plots are shown in fig-ures 4 and 5. A three-dimensional composite view, as shown in figure 6,was chosen to illustrate the mode shapes because of the complexity of someof the nmdes.

Limitations

The acceleration technique described in this paper h.aEcertaininherent limitations.

The surface of the wing must be nearly horizmtsl so that sandparticles do not roll and thereby give the impression of bouncing.!I%iscondition is encountered at the leading edge of most airfoil.shapes;however, with a little experience, allowance for this effect canbe madein the interpretation of the results. A thick symmetrical dotile-wedgesection, however, might present this problem over the entire wing surface.

. Low frequency modes are difficult to measure by using this technique dueto the large srrrplitudesrequired to obtain accelerations equal to gratity.

Models that sre very light in weight and also vexy 10W2.Ydsraped(having sharply tuned resonances ) present difficulties because of therapid increase in amplitude near the resonsnt frequenqy. The sand thatis sprinkled on the wing increases the mass of the wing enough to causeits frequency to shift slightly and thereby chsnge its amplitude con-siderably. This condition makes control of the wing smplitude difficultas sand bounces off. Control of the amplitude can be maintained byadjusting the shaker frequency and output.

EXI?ERIMENTALVIBRATIONMODES OF SEVERALFIDTTER MODEIS

Description of Models

A complete description of the test nmdels is presented in figure 7.In order that thickness distribution of the nmdels may be readily avail-able, the ordinates of the section shapes, except for the flat platemodels, are presented in table 1.

6

The modelsas a backgroundwere mounted at

were sprayed lightlyfor the white sand.the root.

NACA TN 4010

with a flat black paint to serveElectrical resistance strain gages

9——

‘i”

Results and Discussion

The first three natural vibration modes and frequencies are shownin figures 8 to 17. methe root to the tip withwing semis~ans have beenand the wing deflections

The vibration modes,

modes are presented in perspective, looking fromthe leading edge on the viewer’s right. Theforeshortened slightly as an aid in perspective,have been normalized on the maximum deflection.

as drawn, provide a qtick qualitative repre- —

sentation of the manner & which the vibrating wing is deflecting. Fora more detailed ~alysis of the vibration modes, tales of the normalizeddeflections at the 0.1, 0.3, 0.5, 0.7, 0.9, and 1.0 semispan, and theO, 0,25, 0.50, 0.75, and 1.00 chord stations sre presented with eachdrawing.

The first three mode shapes and frequencies for a cantilevered rec-tangular 2- by 7- by 0.064-inch ragnesiumplite were measured and arepresented in figure I-8. Also presented in we figure are theoreticallyobtained mode shapes and frequencies. First torsion modes were calcu-lated by means of the plate theory of reference 5 (using the solutionof eq. (43)) and also by means of elementary torsion theory. (See, forexample, ref. 3.) Elementary beam theory was used to calculate thebending nmdes.

CONCLUDING REMAIU=

A unique acceleration technique for measuring the natural vibrationmode shapes of small wing madels is described. The first three naturalvibration mode shapes are presented for ten cantilever-wingmodels.

Langley Aeronautical Laboratory,National Advisory Conmittee for Aeronautics,

Langley Field, Vs., December 6, 19560

.

.

NACA TN ~10

REFERENCES

1. Herr, Robert W.: Preliminary Experimental Investigation of FlutterCharacteristics of M and W Wings. NACA RM L51E31, 1951.

2. BispMnghoff, Raymond L., Ashley, Eolt, and Halflnan,Robert L.:Aeroelasticity. Addison-Wesley Pd. Co., Inc. (Cs@ridge, Mass.),c ●1955.

3. Scanlan, Robert H., and Rosenbaum, Robeti: Introduction to the Studyof Aircraft Vibration and Flutter. The Macmillan Co., 1951.

4. Brsgg, Willis: An Instrument for Measuring Small Amplitudes of Vibra-tion. Jour. Sci. Instr., vol. VI, no 6, June 1929, pp. 196-197.

-.

5. Reissner, Eric, and Stein, Manuel: Torsion snd Transverse Bending ofCantilever Plates. NACA TN 2369, 1951.

NACA TN 4010

TABLE I

NACA 65A-IL ORDINATH FOR t/2b = 0.020 and t/2b = 0.040

100(g) NACA 65A002 NACA 65AO04*z yz

o 0 095 .156 ●311.75 .190 .378

1.25 .242 .481

● 329 .656;:2 .439 .8777.5 ;5& : 1.06210.0 1.21615 .731 1.46320 .824 1.64g25 .895 1 ● 79030 :% 1.8g435 1.96240 ●9* 1.99645

●997 1.99650 9977 1.952

g“ .936 L.867.874 1.742

65...796 1.584

70 .704 1.400.601 1.193

G .488 .96685 .369 .72890 .247 .490

95 .126 .249100 .005 ●U

Leading-edgeradius . . 0.0255 0.102

Trailing-edgers,di.US. . 0.0047 0 ● 010

.

w

Amplitude

or

Acceleroticm

I

Wing ompiitude

Figure l.-

C

-n,

IUmstration of the relatkm of

b

x

taelglillemt owing

o

ampki.tides .

JM,gure2.- Sample photograph showing baw the regions

contour khles .

a ‘

of mv-iug and stationsxy Smd

L-95907indicate

< .

A.

1

1

.790

2

.88Y

,435

Figure 3.- Sample of a

.122

.024 \ \ \1.00 .D .50 .=

Frmtlon of ch.nd

composite drawing of a 45° swept-taperedwing vibra+dngshowing the contour 1*6.

.0

-9

.3

.1

0

12 NhcA TN 4010

I.0

.8

.4

.!iE=

-8

I

.2

-.2

-.

Fradlanof sernispan,y/2

lllgure 4.- ~o=ized deflection of a 45° swept-taperedwing plottedagainst semispan for five chordwise locations.

.

d

.

L

4010

L2

I.0

Froc$n

.8 SDan/

1.0

.6

// ‘ /

..4 ‘.9

.

.2 //

c1, .1

.7

—

.5

-.4 -

‘.6o .25 .50 .75

Fractkaof chord,x/2b

Figure ~.- Normalized deflection of a 45° swept-tapered

[.00

wing plottedagainst chord for six span~se locati&s.

Reavy long dashedltnesindicateshape

[

of wing at maximumamflitudeof cycle.

1.0

.9

Fracticm

of

chord

0.00.25

●5O●75

1*CQ

Normalized deflection at y~ g

.1 .3 .5 .7 .9 1.0

-.010 -.082 -*243 -.180 .147 .K16-.020 -*U4 -.Z3 -.180 .239 .518-.033 -.167 -.2% -.1.11 .338 .639-.o46 -.193 -.288 -.069 .436 .787-.o56 -.220 -.282 -*@3 .50 1.000

.7——_

/.Hem-y solid linesMica@

.5 Whg at rest.

.— .—

.3—.— —

Figure 6.- TlxLrd natural vibration mode. fj = 220 cycles per aeconti.

P-F

. # , ●

. .

4 ●

bt pma-kemld&)g@. ww’hgmMm’ty● 0.064 L@:G. at 52% Cbdk . NY

LSD= .Mi Inch

Tmm

1

O-f2.2s’

A.

*4WJ

~ w--g WKlmwbwebdti#qlnahlkndty .0.C64 M’?:G A 50% dwdL.30-

\.If). .04 Ml

NA13 Gi3A mctim

~’m$b02klY=0070 M?

W2b = 0.020

8 .

d-f2,E#

J

3.9d’

m+ F4ute-lexw MMmlbq edw imwM W’

W.00E4W:.G.d 60% ChMdk .45”i. 1,0. .041Inch

0’

p

T.?.20”

A

*HAsxlimLmbmtcd _

mty . owe M+CG at 4S% chordh . 45-)..0.4Mb 8 0.040

lotplate.w cddb-q * !mekd L@-

ertsllys O,(M4 Mr’w:,6at50% dud

1.60,s IDm M Imh

WA 65 4 ncttm

C.& at 4S% ChadA.m.i . 0.2

V2b=aw

T60’

4

Fbt PlOtG-kdhl R@

bevded I/E?N@#lmbmOarAty. 0.0s4 CM#Caot SO% Mud

43* ddiat . ,034 Ml

CG. at 46% clwdAZ6PA = 0.4tnb . 0040

-e 7.- Mmsn&lons, density, and center-of-gravity lmation of * imtiels. Sweep angles are ~

measuxed from O. ~ chord.

Heavy long datied llms indlca~s-

\

of wfng at mximm amplitude of cycle.

— —.

———

Fraction Nm&LizeII deflection d y~ m \tiavy solid lines lntica~

of 1.0wing at rest.

Ohord .1 .3 .5 .7 .9 1.0

O.m .039 .m .316 .5L7 .m .935.9

.25 .0L3 .175 .338 .569 .817 .552

.50 .048 ●3.85 .363 .5W:% :%.75 $55 :XJ: .3E3 .W

, 1.03 ● ●m .631 .875 l.m\ “7

5s ——.

%

.3

.1.——

0

(a) First natural vibration mcde; fl .35 cycles per second.

~gure 8.- ITatursl vibration rides for magnesium flat-plate ndel. A = 150; A = 1.0; t/2b = 0.020;A = 5.35.

Pm

4 & , ? .

. ‘? 1

rE9avy long

of vlng at

.

dashed lines indicata sha~maxhIuM amplitude of cycle.

EFraction

orchord

0.00.25.50.75

1.!X

Normalized deflection at 9/2 = I

/“-Heavy solid lines Indicate

wing at rest.

(b) Second natural. vibration mode; f2 - 20~ cycl.m per second..

-e 8.- Continued.

kivy long daabed li~s Indicati shape

r

of wing at maxim amplitude OF cycle.

Heavy eolid lines itiIcati

wing at rest.

I-

Fraotion NormaliEad deflection at y~ =of

ohord .1 .3 .5 .7 .9 1.0

0.00 .W9 .106 .162 .401 .7= leo~

.25 .021 -.035 -.021 .176 .472 .654 :

.50 -.042 -.190 -.225 -.085 .190’ .352 ~\.?.5 -.m -.3.45 +; -.338 -.099

1.02.085 C+

-.155 -.556 -.7/!+6-.373 -.M2g

\3~

%~

‘% \

“

(c) Third natuxal vibration mode; f3 .238 cycles per second.

PCn

Figure 8.- coti@ti.

. ..

. . , , r

Fraction Normalized deflection at YIZ w 1 g

of Ychord .1 ●3 ●5 .7 .9 Lo

El

0.00 .011 .052 .126 .233 .383 .472

.25 .o15. .067 .148 .267 .420 ;&W&$

.50 .018 .080 .170 .296 .461 0

.75 .025 .Cp8 .195 .328 .518 .778

1.00 .030 .YJ.8 .226 .370 .6o4 Lm

1.0

.9

—— .— _Heavy long daahed UneB indicate shape

Of wing at nwdnrum wnplitude of cycle,AI .7

‘j

%——— .

‘* “5

*Q+

%

%——— . _

.3 Wavy solid llnes lndl~te

.1

0

wing at refit.

(a) First natural vibration mode; f~ = 39

Figure 9.- llatm%l tibra’clonmodes for magnesiun flat-plate

A- 4.15.

cycles per second.

m-l. A= ~“; A= 1.0; t/2b = 0.018;

G

I fiactio~ NormalizexI deflection at yj~ =1 .!?

of

chord .1 .3 .5 .7 .9 1.0

~\0.00 -.025 -.264 -.494 -.563 -.500 -.361.25 -.031 -.264 -.405 -.400 -.222 -.028

?\>\.

;~ ::am -.22s -.295 -*MO .11o .300-.125 -.IL7

1.00.3-47 .4% .630

-.o11 -.022 ●m .45$ .816 1.OCQ1.0

.9

+ .7

‘%50

+

‘+ .3%

*P

%‘%

.3

wing at rest..1 .——

0

dashed lh?s iUdiC8te shape

maximum amplitude of cycle.

~ Heavy solid lioes indicate

(b) Second natural titration @e; f~= 212 cycles per second.

Figure 9.- Continued.

, #I

,,

.

●

Fraction Normalized deflection at y~ s

of

chord .1 .3 .5 .7 ●9 1.0

0.00 ,010 .028 ●on .361 .7%’ I.@co

.25 -.C06 -.051 -*1.23 .150 .600 .630

.50 -.019 -.232 -.335 -.11o .3m .613

.75 -.0’74-.U5 -.600 -.360 ●W .3551.00 -.168 ,-.677 -.910 -.65$ -.193 .0%’

Heavy long dashed lines Indicate shape

of wing at maximum amplitude of cycle.

(c) Third natural vibration mode; f5 = 272

l!Y.gure9.- Concluded.

.EIeavy solid lines indicate

wing at rest.

cycles per second.

1.0

Fraction Normalized deflection at y/z :

ofchord ●1 .3 .5 .7 ●9 1*O

0.Q3 .0U7 .054 .1?4 .210 .350 .467

.017 .070 .155 .259 .4M ::;.25.191 ●313 .510

.238 .395 .@ .810

Loo .Q39 .159 ●3Q-7 .5(X3 .812 1.003i

\

.9 Heavy long dashed lines indicate shape

of wing at mmimum .9n@itude of cycle.

%% “7

f<.~——

00?

.5

I%+7

%——.

% .3 Heavy solld lines Indicate

wing at WEt..

.1

0

‘d-!=oPo

. . *I

,

,.

* * ‘ , ,

I F!raddm I Nozmelked deflection at y~ .

I of

Ch’d

L0.00.25.50

●75m

.007

:52.050

S!22L m\

~

Heavy long dashed lines indicate shtipc

Z o, wing at mex~m .empIlt.udeof cycle.

HeaW solid lines indicatewing at rest.

\ \

(b) Second natural vibration MIMe; f~ .184 cycles per second.

me m.- cmkhuea.

cFractionof

chord

I 0.00

I ;25.’50

——

l\ l\ U.S.

Normalized deflection at y~ =

.1 ●3 .5 .7 .9 1.0

-.017 -on .053 .622 l*CKXI

“:E -.0s -.tis -.029 .520 .@l+X8 -.201 -.374 -.200 .u6 .735

-.050 -.470 -.676 -.4X? .3M J&u

-*2U -L~ -L~ -.504 .250 .651

‘-o-ii-ofvingat-~-i.deo...Heavy long dashed lhes idicate sha~

m 1“

(c) T.Wd natural vibration mode; f~ = 263 cyd.m pm mmrd.

Figure 10. - Concluded.

* , ,

#

Eoav’ylong dashed lines indicate shap

of wing at mimum amplitude of cycle,

1.0

.9

-7

4*QQ%

0?%$

— —.— __,

‘% ‘3

o

traction I l?ormMzwl deflection at yfi .

f_

ofohord ,1

0.00 .012

.25 .015

.50 .022

.75 .0401.00 .@

-3-

.040

.058

.091

.135

.211

solid lines indlcatiwing at rc5t.

.51 .71.9 I 1.0

m(a) First natural vibration tie; fl Q 47 cycles per second.

for ma~eslum KLat-plate model. AU 60°; A= 1.0; t/2b = 0.010; ~Figure 11. - Natural tibratlonmcilm

A= 1.39.- iii

1.0

Heavy long daehad lines lndicata shapeof wing at maximum emplitu@e of cycle.

.9

.7

%%Z .5

%0? -——. — Heavy solid lines indicate

wing at rest.

.—— —

0

Fraotion Normali.zaddeflection at y/? sofchord .1 .3 .5 .7 .9 l*fJ

0.00 -.03S -.039 1% -.3a -..495 -*515.25 -.016 -.078 ::258 -.433 -.445 -.390.50 -.022 -.133 -.31.2 -.328 -.1% .047.75 -.034 -.I.59 -.lso .019 .350 .515

1.00 -.055 -JX9 .3-09 .484 .867 1.OCQ

(b) Second nalmxil. vibration IIKJd.e;fp . 207 cycles per second..

F&we IL- Continued.

. , .1

1

1

, , .

\_ _ .—— Heavy long dashed linss indicate shape

\

of wing at ❑axima amplitude of cycle.

1.0

.9

\

.5Heavy solid lines indicate

+%J04

wing at recit.

%——.— —4

Fraotlon Nmmallzed defleotlon at y~ nof

Ohord .1 .3 .5 .7

0.00 -.02.4 -.m -.347 -.6c%.25 -.045 -.125 -.202 -.303.50 -.m .Ooo .na .x.6

.75 .101 ● 545 .008 .6571.00 ●465 ●WI 3.JJ03 .657

(c) !i!hinlnatural tibration mode; f3 = 380 cycles Per second.

z.9 Lo

-.657 -.650-.505 -.707-.252 -.6%-.162 -.656

-*353 l.m

Heaw long deshed lines indicate shapeOf’~ng at maximum empliLude of cycle.

&avy solid lines Indica

wing at ~e~t.

Raution IkmwJ3zd deflectbn at y~ .

of

chord ,1 .3 .5 .7 .9 1.0

O.CKI .003 .061 .180 .360 .686 .97$

.25 ●U .065 J@ .368 .ml 1:~

.065 .172:; .356 .696

:% .Og .364 .341 .676 .954J ——

1*CO .Oo1 .039 .151 .324 .642 J%? .

.$’$

0% ——. —

9 .5$

@&

.3 — —

.1

0

(a) First natural tibration mode; fl = 1251 cycles per secoti.

F@ure 12. - Natural vibration ties for ma@esium DuIMl with NACA 65A section. A. OO; A = O.Xl;t/2b = 0.02; A= 4.o.

,I

, ‘ , .

II

w03

Heavy long dazhed lines indicate shape

\

of wing at maximum amplitude of cycle.

Heavy solid lines indicate

wing at rest.

Eraotion Normalized deflection at y~t =

of

chord .1 ●3 .5 .7 ●9 1.0

0.00 -.000 -.0s2 -*2L6 - .10s

.25

,540 1*WO-.o13 -*WI -.246 -.KI.9 ..494 .924

.~o ..OI.6 -.109 -.254 -*151 .457 .$36

.75 -.012 -.093 -.264 -.203 .436 .6%3

1.WJ -.002 -.LM5 -.277 -.259 9424 .%5

_——

&

,+.& .5

%+~ . ——

(b) 8eCOti natural.

.3—— —

.1

0

vibration mode; ~2 = 370 cycles per second.

IWWe 12. - ConlxW&i.

Heavy solid

wing

Iof

Chmd I .1 I ●3 .5 .7

m

wo

Heavy long dashed lines indicate shapeof wing at maxhnum smplitude of cycle.7lines indicate

at rest.

7

I Normalized deflection at y~ . 1.0

.9 1.0

.9.&l ,472.363 ●U5

-.22’7 -.363-.636 -.’7?9-.817 -1.CKKI .7

$,$

*5

$>

.3

.1 ———

(c) Tblrd natural vibration mode; fj = 73.0 aycles per second.

Figure M.- Concltied.

*

Heaw long dashed lines indicate shape

\

of wing at maximum amplitude of cycle.

“r--h

1.0

LFraation

of

chord

L0.00.25.50

.751*CO

Normalized deflection at y/I .

.1 .3 .5 ●7 .9 1.0

.010 .075 .175 .329 .6o6 .7S8

.023 .097 .210 .378 .674 .W4

.030 .m ●245 .@ .729 .896

.07 ●137 .273 .475 ●m .946

.044 .158 .305 .532 .830 1.OCO

CJJ——

\ “’%

.3

solid lines indicatewing at rest.

(a) First natural

Natural vibration modes

ribration mode; fl = 95 Wck Pm ~eco~.

for wood model with WA 65A section. A = 45°; A = 0.4; uP

t/2b = 0.04j A= 4.0.

Heavy long

\

of wing at

3dashed llnes indicate s&pe

maximum amplitude of cycle.

1=Fi-actimi

of

ohord

0.00

L:25

.50

.75100

,

(b) Second natural

II

.1 .3 .5 .7 .9

.O1o .092 .265 .474 .479

.020 .097 .255 ..gg .408

.015 .051 .082 .097 .102-.031 -.163 -.323 -.439 -.503. w -.37’? -.n9 r.%4 -.?95

lines indicate

d rest.

tibration UK@; f2 = 326 cycles per SeMM.

E@Ore 13. - Continued.

* .

31.0

.lgo

.393

●W-.5M.IJmo

1.0

.9

Heavy long daahed lines indicate sha~

\

of wing at maximum maplitude of cycle,

4 .*

5’

IFractionI Nm?ml~zeddeflaat~onaty/z -

.7

d

%~$

% .5 IIeavy solid lines Indicati

0 wing at rest,.

——— —

.1 — — — --—-

0

I..—of

chord .1 .3 .5 .7 .9 1.0

0.00 -.010 -.082 -.~ -.I.80 ●1.47 ●u.25 -.020 -.I.34 -.262 -.U7 ●2.39 ●m

.50 -.033 -.167 -.278 -.I.XL .338 ;$$

.75 -.046 -.193 %283 -.069 .436

Loo -.056 -.220 -.282 -.033 .W l.m

(c) Tblrdnaturel vibration rode; f~. ~ cycles per second.

F@re 13.d Concluded.

f

Heavy long dask@ lines indicata shape

of W@ at maximum amplitude of cycle.

YTaction Nmmalized bflectl.on at yfi =of

chord .1 .3 ●5 ●7 .9

O.m .m .m .IJ!J1 .363 .6L6

.25 .002 .047 .177 .@ .692

1.01.00 .017 .136 .3-LO .5~l .936

.9

.7

$+——

%7%

.1

0

Figure 14. -

r,l

(a) First natural

Natural tibration modes

1.0

,K!fl,&6

.*.%O

L.OXJ

vibration mode; fl = 43 Cycles R’ SeC@.

for wood model tith ItACA 65A section. A . 60°; A = 0.4;

t/2b = 0.04; A = 4.0.

4 . *

9 . ( ,

1.0

g

/

Heavy long d.uhed lines indicate shape

of wing at iuaximumEU@itude of cycle. s

P \ ‘L.

~

\Fraction Normalizedileflmtion at y/z .

Ohord .1 .3 .5 ●7 .9 1.0

O.oa -awl -.oSo -.220 -.242 .055 .550

.9 .50 -.OX5 -.155 -.745 -.MO .327 .760.75 -.030 -.168 -.252 -.095 .@?

lJM.878

-.050 -.220 -.25? .C%5 .640 I.olm

.-r

@**Qt

<04

‘+.5 ~=m~~~~~; ,indicate

—.— —

.—— — —

———

(b) &COIld. mtUl@.

o \ >

vibration mode; fa . 171 cyclm per fiecond.

~gure 14.- ~tinued.

rHeavy long dashed lines indica+~ shaps

of vlng at maximum amplitude of cycle,

Fraction I Normalized deflection at yk = II .+ I Ilil&dl.ll. 31.51 ”71”911”ol

\ P-,?kxl\ Z .w . . . . . .

Heavy mlid lines Indicate

\

o

(c) T!bifi natural vibration mode; f~ u 328 cycles per second.

F&ure 14.- Concluied.

‘ * . ,

r Heavy longof wing at

daahed lines indicate shape

maximum amplitude of cycle,

Fraction I Ncumallmsd Mleotion at Y[2 = Iof

chord .1 ●3 .5 .7 ●9 1*Q

0.00 .CKJ1 .003 .OI.3 .035 ,121 .322.25 .OQ1 .CX35 .018 .W+l ;~o .460.50 .002 J& .022 .048 .6I2.75 .033 .028 .057 .310 .7S.4

1.00 ●w .015 ● .4 .on .@o 1.COO

1.0

.9

“%+

%0+ .3 llea~ snlid lines indicate

%wing at rest.

+J@

%

.3

.1

o

(a) Elrst natural vibration mode; fl .74 cyiks per Eecond..

R@ure 15.- Natural vibrationmodes for mod model with NACA 6’jABectlon. A - 60°j X = O. 20;

t/2b= 0.04;A = 4.0. Y

T

FractionHeavy long dashed lines indlcati tip-e

Nomallmd defbction at y/? E

of vlnE at maximum amplitude of cycle.ciamd .1 .3 .5 ,7 .9 Lo

0.00 .002 J20 .47L .65o .(58JJ .680,25 xx)’? .158 .390 .4!% :$? X40.50 .O11 .K!8 .074 -0049 .526●75 -.004 -.128 -.&u -.583 -.090 .&

1.00 -.OQ -.5zo -.9261.0

-1.OM -.316 .263

*

‘%~o+

\?? Heavy solid lines indl cate

‘%wing at rmt.

%/

/

\

\

(b) Second natural vibration tie; f* . =8 cycles per second.

R@ure 15.- Continued.

● ☛

i* r

wco

1

1.0

\

Heavy long

of wing at

dashed Moes indicat-s shape

maximm amplitude of cycle.

action I lfamelhied defleoklon at y~ . Iof

Cbnrd .1 ●3 .5 .7 .9 M

I&al -.002 -.010 -JM3 -.179 ●W .472.25 -.002 -.OI.8 -.121 -.3.45 .132 .555

-*W -.Q29 -.I36 -*UI .255 .694

E -*W -.055 -W-3 -B~~ ●395 ●@81.00 -.03’7 -.3.02 +45 .~ .555 l.~

.9

.7

+ Heavy mlid lioes indicate@o

~?

wing at rest.

%

.3 ———

.1

0

(c) Thira natural tibration~d.e; f3 = 415 CYclea per Secoti.

F@yxe 15.- Concluded.wW2

N5

Wavy long

of wing at

‘\

dashed line B Jndlcate shdw

maximum emplitude of cycle.

Ifiaotion ] N~ed deflection at y/z -I

of J

ohord .1 .3 .5 .7 .9 Lo

0.02 .Ow ●I86 ●357 .702 1.0.25 :% .1!77 .21J+ .#J .742 1.0.50 ●033 .L21 .ZI’+7 .453 .m 1.0

.75 .037 .133 .2!78 .507 .795 1.01.00 .0?7 .IJ!+4 ●297 .5s .820 1.0

\ ‘\—Heavy lines indicate “

at reti.

solid

wing

—— —— —. . .

.— ——— .——

(a) FirBt natural. vibration mcde; f~ = ~ cycles per second.

-e I_6. - Natural vibration mcdes fox 45° magnesium flat-plate delta model. t/2b = 0.006.

* ,* 1

1

,

k Heavy long dashsd liks indicate shape

of wing at maxinuIM emplitude of cycle. Fraction l&maMzed def’1.ection at y/z a

of

chord .1 .3 .5 .7 .9 1.0

0.00 :W& -.319 -.729 -.350 .325 1.0.25 -3& -.;g -.173 .&Lo Lo.50 -.0S5 .490 1.0●75 -.029 -::; ::~2 % .58o 1.0

1.00 .m7 .@ .681 1.0

1.0

.9

.7

.5 -~.HeaW solid lines indicate

wing at rest.

.3 -.\

.—

.1 ———.

0

(b) Second nsttu%l vibration mode; f2 = la

Fiwre 16.- continued..

cycle6 per second.

,.

a

k Hewy long dsshed lines indicats ah.qe

Of ving at maximum amplitude of cycle.

20

\

‘N\“

%cl’-’Ed””ecti”na’

m/

/

///

Heavy edid lines indicate

wing at rest.

,/

// _~—/

//H

./———

.—~

(c) Third natural

,,

Y/z

.51 .71.911.0

Figure 16. - Concl@ed..

m

tibration mcde; f3 = 258 cycles per second.

. .

R

;

. .

1 , . ,

44=-

l-1.I&4 .750 1.000.502 ●WI 1.0CU3

.%4 .835 1.000

.570 .850 1.OIXI

.5Ea .860 1.003

Fraction NormaMzad deflection at Y/z ‘Sof

chord .1 .3 .5 ~

0.00 SW .025 .3.39.25 ●u+ .067 .232

●325

1.0.337

l.m .025 .16J3 .342

.9-

Heavy SOIM 1~ nes lndlcate

.7wing at re6t.

——Heavy long dashed lines indicate shape

.5

of wing at meximum amplituda of cycle.

1 — ——— .

(a) First natural

Elgure 17.- Natural vibrationties

vibration mode; fl = 66 eyclea per second.

for magnesium 600 flat-plate delta nmdel. t/2b = 0.004, baaed

on root chord.

!2

&

L Ci+.on I NormMzed tlefleotion at y~~ =

ohord .1 .3 ●5 ●7 .9 1=0

0.00 -.023 -.3$0 -Lox -.849 .302 1.000

.25 -.093-.47’7-.s72-.550 ●S 1.000

1.0 ● 50 -.I16 -*3L9 -.45~ -*cm .640 1.W

.9.75 -.035 -.OKL .023 .377! .7m Low

.7 Heavy solid lines indicate

wing at rest.

Heavy long dashed lines indicate shape

of wing at nwinmm emplitude of cycle.

.3\

——. ——— .— ——

(b) Second mitural vibration mode; fp= lb cycles per second.

me 17. - Mnthd.

J , ● ✎ , .

I 1s ,,

1.C

.9

.7

.51

\

-? \

?raotim I Nomallzed ckflecWmaty~.

of

Chord .1 .3 .5 ●7 .9

O*CXJ .006 .076 .059 .150 .745..25 .029 .O’n .029 .132 .722.50 -.CQ6 -.053 -.3.47 .070 .695

.75 -.053 -JJ7 “J@ -.ol+o .6671.00 -*U7 -.500 -.523 -.m .640

1.0

looml.ocaloom1.000Lam

Heavy solid lines indicate

wing at rest.

Heavy long dashed lines Indicata shape

of wing at maximum amplltude of cycle.

.3

.1

o) \

(c) Thirdnatura lvibratlonmmle; f3= 336 cycles per second.

Figure 17. - Concluded.

&

46 NACA TN 4010

1.0,

.8 4- {

— Elementary tmbn theory/ / y , i {

(ref.3) 384cpS</

.6/ < ‘/ A

\ / / /

/ A/ “

/ ‘. f/ i

.4 Plate theory , /

f

(ref.-5) 433 cps=/

//F / .’ p

t FKSt bending mode shcspe- .2 / }/T 83 CPS

~-Rrstbending mode shope/- /9 /

%/ using elementary beom theory f

E. 80.8 CPS kbz

L

f

-.2 \

-— //

-.4 . Seccmd bending mode shape A‘ —Theory‘ \\ 525 cps\ /

\ / -—-– Experiment\ \ >-

“\\ /“‘.6

/-

< / k%econd bendii mode shope

‘.8 using elementary beom theory, 507 Cps

-Lo..I .2 .3 .4 .5 .6 .7 .8 .9 1.0

Semispan stottan

Figure 18. - Comparison of theoretical and experimental vibration modesof 2- by 5- by 0.064-inch magnesium flat-plate cantilevered-wingmodel.

NACA - LangleyField,Va.