Vibration characteristics of a cantilever plate with swept-back leading edge

College Report No. 82

" ' -GESCHOOL

;A ...V/ICUNDE Kanaalstraat 10 - DELFT

Klu HS DELFT £3 AU6.1955

THE COLLEGE OF AERONAUTICS

CRANFIELD

VIBRATION CHARACTERISTICS OF A CANTILEVER

PLATE WITH SWEPT-BACK LEADING EDGE

by

OCTOBER 1954

T H E C O L L E G E O F A E R O N A U T I C S C R A N F I E L D

V i h r a t i o n C h a r a c t e r i s t i c s of a C a n t i l e v e r ELate w i t h Swept-back Leading Edge

b y

-A.E, Ifeiba, B . S c . ( E n g , ) , D.C.Ae.

SUMvIAEÏÏ

The paper presents an experimental study of natural frequencies and modes of vibration of imiform cantilever plates. Sixteen planforms are tested by combining aspect ratios of 0,8, 1 ,2, 1,6 and 2 v/ith leading edge sweep-back Eingles of 0°, '15°^ 30° and 45°• Experimental values of the natxiral frequencies for the first six modes are presented for each plate, together with the nodal patterns concerned. The results have been plotted in frequency curves of the same 'family shape',

The effect of sv/eep-back of the leading edge on the frequency of a particular mode, and on the position of the nodal line of that mode is more marked on high aspect ratio plates, The frequency of the flexural modes varies only slightly v/ith aspect ratio. No general remarks could be made on the frequency variation v/ith the tangent of angle of sv/eep, v/hich depended

greatly on the aspect ratio,

A formula has been derived by v/hich one could

calculate the frequency of a particular mode on a given plate, presuming that the plate is geometrically similar to one of the planforms tested, but with different thickness and different material, and provided that the values of Poisson's ratio for both materials are the same,

This Report is based en a thesis submitted in J'une 1954 in part fulfilment for the requirements for the Diploma of the College.

••2*"

TABLE OP CONTENTS

Page

1• Introduction 3

2, Test Equipment and Procedure 4

3, Results 5

4, Discussion 6

4.1. Accuracy of Results 6

4.2, Compounding of Modes 7

4,3» Nodal Pattern Analysis 9

4,4« Frequency Variation with Aspect Ratio 10

4,5• Frequency Variation with /V 12

5* Conclusions 13

Appendix A 15

References 18

Table I 19

1 • Introduction

Severa.1 recent papers, have been published investigating experimentally the vibration charaucteristics of rectangular ajid skev/ cantilever plates,''*^ and of triangular ones,3 This report introduces a more realistic planform and aspect ratio range

appropriate to guided missile design practice, viz, a low aspect ratio, trapezoidal planform with a sv/ept-back leading edge and a

straight trailing edge,

The two parameters, v/hich identify the planform of a wing, chosen for variations in this work are» the aspect ratio

and the angle of sv/eep-back of the leading edge. The usual definition as regards the aspect ratio has been adopted, but has been mentioned in Fig, 1. for clarity. Four different values

of sweep angle, ./\_ > ranging from 0° to 45° have been chosen,

For each, fovir values for the aspect ratio have been tested, ranging from 0,8 to 2. The nodal lines and the frequencies of the first six natural modes of vibrations v/ere obtained. By the first six modes is meant those modes that could be first excited in the order of increasing frequency. It v/as possible to excite modes higher than the sixth, but these higher modes are believed to be of little practical interest and have not been included,

The nodal patterns obtained have been analysed, attributing each mode to its proper 'family', Knovdjig the a.ctual form of the mode enabled the plotting of all modes in

families of frequency curves. These curves provide sufficient

material for the study of the effect of variation of the two paraineters chosen, namely jA. and A separately, on the frequeïicy of a cantilever plate,

A vibrating cantilever plate represents a system v/ith an infinitely large number of degrees of freeda-a. There is no exact analytical solution for the problem. Sane success has been achieved v/ith the Raylieh^Ritz method by Young^ for a square

cantilever plate taking an 18-term series of the function to represent the deflection. In an attempt to extend the analysis to skev/ cantilever plates. Barton^ solved for different planforms in the first two modes. Satisfactory results v/ere obtained for rectangular plates, but the error increased with y\, and amounted to 20 per cent for the second mode of a 45° skev/ plate. This shov/s the need for further basic developments in the analysis, to apply to different planforms, and the present need for the experimental study of such problems,

A formula has been derived and verified (Appendix A ) , by v/hich one could calculate the frequency of any particular mode of a plate, of similar planform to one of the plates tested, but with different thicloiess and of different material as long

as the Poisson's ratios of the tv/o materials are the same. Good approximation hov/ever will be provided v/here the Poisson's ratios of the tv/o plates are not very different,

-V-2 , T e s t E q uip nen t and P r o c e d u r e 2,1o T e s t Rig

The test rig consisted of a concrete base v/hich was 40in, X 24in, x 30in, high, A steel bed-plate, 40in, x lOin, x 1in, thick, v/as attached to the concrete base by means of eight ^in, dia, bolts, screwed into special cemented-in sockets. The

testpiece was then clamped betv/een the bed-plate and another 3/4in, thick steel clamping plate by l/4in« dia. bolts. There were two rov/s of bolts, the front rov/ of l-gin, pitch, and the back row of 2in. pitch, A torque of about 200 lb,in, v/ae applied manually for tightening the bolts evenly,

2,2, Test Specimens

The plates tested were cut fron ground, flat mild steel sheets of l/8in, thickness. The different planforms tested are shown in Pig, 2, v/here their designations are also given. For the purpose of the study, four different aspect ratios v/ere

chosen! A = 2,0, 1,6, 1,2, 0,8j and for each aspect ratio, four different angles of sv/eep-back of leading edge v/ere investi-gated, yV.= 0, 15°, 30°, 45°» In all cases, the overhang lengthy virtually the semi-span of the planform, was kept constant as lOin,, v/hile the tip and root chords were adjusted to give the required planform characteristics. The breadth of any plate tested v/as léin,, thus giving 6in, for clamping,

The smaller aspect ratio plates (D series) vrere tested first and v/ere then reduced in chord length to form the larger aspect series. The upper siorface of the plates was marked out in a lin, square mesh so that the nodal patterns formed by sand could be easily plotted on graph paper,

2,3» Excitation

The plate was excited by means of a U-shaped electro-magnet mounted on a pedestal, (Fig. 3) so that the exciting force would be perpendicular to the flat plate. The pedestal was

adjustable so that the position of excitation could be varied, This was found necessary since the best place at v/hich to drive the plate depended on the mode and the planform,

Alternating current v/as supplied to the magnet by a Goodman's Power Amplifier type D / 1 2 0 , v/ith a separate condenser tuning unit connected in series with it, A Goodman's stabilised power supply unit type DS/120, was connected to the amplifier, The amplifier was driven by means of a Goodman's Audio-Oscillator

type RCA>.1.

The electromagnet was placed beneath the plate to be excited, thereby producing a pulsating magnetic force on the plate. Since the magnet is normally pulling tv/ice on the plate

for each cycle of the alternating current supplied, the pulsating attraction exerted v/as numerically double that of the frequency of the current supplied. Thus the plate vibrated usually at double the oscillator frequency. However, in some cases, especially in the fundamentai modes of most of the plates, it was possible to excite a particular mode by setting the oscillator frequency at the same value of the vibrating plate,

2,4, Measurement

A microphone v/as mounted near the edge of the vibrating plate in a plane perpendicular to it and connected to a C,R,T, oscilloscope, f/hen connecting the oscillator siailtaneously to the oscilloscope, a lissajons pattern v/^s obtained v/hich indicated the ratio betv/een the vibrating plate frequency and that of the oscillator. Thus the coirrect plate frequency could be obtained,

The amplitude of the microphone signal transmitted depended upon the intensity of the sound, which in turn depended upon the amplitude of the vibrating plate. Thus the peak

amplitude of the oscilloscope trace was used to indicate wiien the plate v/as on resonance,

The nodal patterns (Pigs, 5 to 20) v/ere found by

sprinkling fine sand on the top surface of the plates. The sand settled only at the nodal lines as shov/n in Fig, 4, By reference to the grid marked on the plates it was a simple matter to transfer the pattern to graph paper,

3, Results

The values of the frequencies obtained for the sixteen planforms are tabulated in a simmary form in Table I, Tlie nodal patterns are plotted to scale and sbov/n in Figs, 4 to 20, These patterns are then cjialysed to their proper forms o The frequency and the form of each mode accompany the nodai pattern concerned, Modes of vibrations have been referred to as first, second, e t c ,

according to the order of appearance on the frequency scale, In order to ajialyse the modes obtained, it has been assumed that any given mode can be reduced to a corjbination of nodal lines running parallel to the cli'jmped edge, and others perpendicular to it. The former v/ill then represent the pure

'flexura.1' modes, v/hile the latter v/ill represent the pure 'torsional' modes. This assumption, hov/ever, applied strictly only to the rectangular plates. In higher modes, as the angle of sweep and the aspect ratio increased, it v/as observed that the 'torsional' noda.l lines, namely those which were perpendicular ,to the clamped edge, modified their course. This has been

Thus, generally speaking, if the number of 'torsional' nodal lines is assvtmed to be 'm', v/hile that of the 'flexural' nodal lines is assumed to be 'n', then any normal mode should be of the form (m/n). The fundamental mode, namely the funda-mental bcnfüng mode v/as of the form (o/o), as no nodal lines

existed v/ithin the plate boundaries except that along the

clamping edge. The first flexural overtone mode vdll be of the fon.i (0/1), The fundamental torsional mode is that of the form

1/0), v/hile the first torsional overtone is that of the form 2/0), and so ono

The results obtained were plotted in Pigs, 21 to 28, In order to expect sraooth behaviour from the curves representing

the variations of frequency ivith aspect ratio for each f\~ , the

frequency ciorves should be drawn only for tliose modes which belong to the saine 'family sha.pe' regardless of the order of

their appearance on the frequency scale. The sane argument applies to the curves drawn for the study of the effect of

variation of tangent of sweep-ba-ck angle on the frequency (Figs, 25 to 28),

4» Discussion

4,1• Accuracy of Results

The principal sources of error in the frequency neasixrement can be sumnarised in the f ollov/ing>

1^ Clamping conditions at the root,

2) The deteraiination of precisely the resonant peak of the vibrating plate at any particular mode,

3) The reading of the frequency fron the oscillator dial. The first source of error seems to be the one which should be given most attention. It was observed tha.t a

reduction in the frequency of abcut 5 per cent v/as obtained if some of the bolts v/ere not tightened enough, especially if those bolts v/ere near to the edges of the plate. The use of larger numbers of bolts and sr.ialler pitch gave better clac-iping conditions especially as the location of that clar.ïping hole nearest to the trailing edge of the plate varied after shaping that particular plate to the next planform to be tested. It v/as difficult to estimate the effect of such variation, but in any case, the pitch v/as small enough not to allow an appreciable difference in the frequency measurements. The variation in the torque applied for tightening the bolts on different plates seemed to be inevitable, as there was no direct v/ay to control it,

However, as a check for the effect of clamping conditions at the root, series 'A' planforms v/ere all tested for the second time after being removed from the clamping jig ajid then replaced,

The naximum variation betv/een any two readings v/as found to be 1,46 per cent of the mean. In most cases it was about 0,85 per cent of the mean,

The second source of error v/as dealt v/ith by repeating the test on each plate. Having obtained the six modes required on any plate, the magnet v/as set up once more and the six modes were excited once more in succession. The value of any frequency quoted in the paper is the mean of the two readitigs. The testing technique adopted, together with the extreme sharpness of the resonance peaks for the plates, precluded the possibility of

obtaining any large error in the judgment, Hov/ever, in all cases, the difference between any two readings recorded for a particular mode did not exceed 0,7 per cent of the mean,

The third source of error was checked by calibration against a laboratory standard decade oscillator and found to be vïithin the limit of reading accuracy,

4,2, Compounding of Modes

If the vibrating plate is perforr.iing one of its noraal modes, it should be possible to analyse it in the form (m/n) as

explained before. In some cases, 'complex' modes v/ere obtained v/hich could not be attributed directly to the normal forr.i expected, It has been suggested by Grinsted that, 'compounding' of two

normal modes belonging to two different 'family shapes', nay take place if those tv/o nodes should have the scjiie frequencyj and they v/ould then exist simultaneously in the vibrating plate, This phencmenon v/as observed in several cases in the study when

the frequencies of tv/o modes were approximately the same, 'JSio

of these cases are discuased be lev/, a) B,2, Planforrat (Pig,10)

In examining the modes obtained for B,1, plate (Fig, 6) and B.3. plate (Fig, 14), the third mode of the B,2, plate v/as expected to be tlie first flexurjxl overtone mode of the form

(0/l), while the fourth mode was to be the first torsional over-tone mode of the form (2/o), The frequencies were found to be 23e, 273,8 c,p,s, respectively. In observing the nodal patterns obtained, it was believed that canpounding between the two normal modes took place v/hile tryijig to excite either of them, as the two frequencies were near to each other. It v/as impossible to obtain either of the nodes in its normal form,

Consider the plate vibrating in Fig, 29, if the shaded area is considered to be 'dov/n', while the unsha.ded area to be

'up', a nodal line should exist betv/3en the tv/o regions. It is argued tha.t vibration of the mode (a) may occur v/ith approximately the sar.ie period as (b), ''^en the two modes are superimposed as in (c), the areas which are doubly shaded v/ill have their amplitude augmented 'doubly dov/n' v/hile the unshaded areas will be 'doubly up',

-8-The single shaded area hov/ever v/ill contain the points v/here the two amplitudes cancel. Consequently the nodal lines pass between the 'doubly-up' and the 'doubly down' areas and ^vill occur only in the single-shaded areas. Their exaict courses in these areas v/ill depend on the relative amplitude of the tv/o modes and on the uniformity of the plate. Thus the modes discussed are believed to be as shown in Fig, 29,

3rd MODE 236 c p . s , (a) (0/1) (b) (2/0) Nodal line Obtained (c) (0/1+2/0) 4th MODE 273«8c,p,s, ^ ^

8

i s i ^

^ ^ (a) (2/0) (b)(-0/l) (c) (2/0-0/1)

FIG, 29, COMPOUNDING OF MODES ON B.2, PLANFOmi

b) C.I, Planformg (Fig, 7)

The fifth mode on C,1 , planform was excited at 293 c,p,s, v/hile the sixth was excited at 3'14 c^,s. The former v/as expected to be of the form (I/1), while the latter normally of the form

(3/0) on that particular planform. As the frequencies were very close to one aoiother, the tv/o nodes compounded with each other as shov/n in Fig» 30,

5 th MODE 293 c,p,s, 6th MODE 314 c,p,s, (a) (1/1)

^-^M^l

(b) (-3/0) ^ *

1

^ ' (c) (1/1-3/0)

p!^

B

1

1

(a) (3/0) (b) (1/1) (c) (3/0+1/1)

FIG, 30. CO]\IPOUNDING OF MODES ON C.I. HANFORI/I

In fact, compounding did not necessarily take place whenever the frequencies of two normal modes were close to one another. In planform C,2,, (Pig, 11), the fifth mode (l/l) v/as excited at a frequency 318 c,p,s., and the sixth mode (3/0) v/as excited at 335 c,p,s. In spite of very close frequencies, the tv/o modes were obtained in the 'normal'form. All other cases of ccoipounding on modes have been analysed in the same manner, and the resultant form on each mode has been quoted under the corres-ponding planform,

4,3» Nodal Pattern Analysis

In studying the nodal patterns obtained, it was difficult in some cases to identify a normal mode ir. the usual manner. In fact the lack of symmetry in some of the plates seemed to be the major factor in this respect. This effect was more profound v/ith higher aspect ratio, higher angle of sv/eep plates,

The fundamental mode of all l6-plates, is primarily a 'flexural' mode, resembling the behaviour of a simple cantilever beam, Hov/ever, some chordv/ise variation in displacement was

-10-observed especially v/ith longer chord plates,

The second mode of all plates is the fundamental torsional mode (l/O), For the rectangular plates, the nodal line runs across between mid-points of the clamped edge and the opposite free edge, A small angle of sweep-back of the leading edge has no effect on the position of the line. In fact, the line then runs from the mid-point of the free edge, and perpen-dicular to it up to the clamped edgej thus behaving as a rec-tangular plate v/ithout the triangular part, until / V = 30° v/hen the line shifts slightly forward, but again only on low chord

plates, l^fhen y \ = 45°^ hov/ever, the forward shift on the nodal

lines becaaes more clear vidth the increase of aspect ratio,

The third mode on all the plates is the first torsional overtone (2/o), except on the 'A' series of plates v/here it is the first flexural overtone (o/l). This sliov/s the tendency of the plate to act as a cantilever beam as the chord gets narrower thus exhibiting the flexural modes earlier on the frequency scale, Neither the chord length, nor the leading edge angle of sv/eep seem to have any effect on the position of the nodal line on the (0/l) mode. It m m s roughly parallel to the clamped edge and at about 0,8 of the cantilever straight length. Some irregularity in that

line is sanetimes observed especially on the long chord plates which may be due to the clamping conditions. On the (2/o) mode,

the forward nodal line changes position forward as the angle of s\/eep increases} v/hile the rear line seems to be unaffected on lov/ aspect ratio plates, but changing its course completely on smaller chord plates,

The fourth mode is foimd to be the (2/o) mode on 'A' series, the (0/l) mode on 'B' and 'C' series, and the (5/o) mode on 'D' series plates. This shows that the torsional modes become more predominant as the chord of the plate becomes longerj thus

exhibiting those modes earlier on the frequency scale,

The fifth mode is the (l/l) mode on all plates except on the 'D' series v/hich then exhibit the first flexural overtone (O/I), The sixth mode is the (2/l) on 'A' series, (3/0) on 'B' and 'C' series and the (l/l) mode on the 'D' series of plates, On all these higher modes, the arigle of sweep and the aspect ratio have various effects on the shape and position of the nodal lines v/hich is again more marked on the 'A' series of plates,

4«4» Frequency Variation vith Aspect Ratio

A picture of the variation of the frequency with the aspect ratio for a particular sweep-back leading edge angle is afforded by Pigs, 21 to 24.

For the rectangular plate (Fig, 21), it is interesting to see the very slight variation in the frequency with the aspect ratio of the pure flexural modes, for both the fundamental (o/o) and the first flexural overtone (o/l). The fundamental frequency is about 40 c,p,s,, v/hile the first overtone is 250 c.p,s. The ratio being 1:6,25, For a uniform cantilever beam', the first

two flexural frequencies are of the ratio 1t6,27, This shov/s the tendency for the uniform rectangular plate, when performing pure flexural modes to behave in a manner that may be correlated

to the vibration of a cantilever beam,

The frequency of any other mode of vibration increases v/ith the increase of aspect ratio. Generally speaking, the rate of increase seems to be slightly higher in the range A = 1 ,2 to A = 1,6, For the torsional family of modes, namely of (m/o) foim, the higher the mode, the hi^er is the rate of increase of frequency v/ith aspect ratio,

The curves also afford a better picture for the order in which different modes are obtained on the frequency scale, and the frequency at which each mode is excited, for any given aspect ratio within the range surveyed. Intersection of any two curves of two different 'family shapes' indicates that at that particular aspect ratio, the two modes will be obtained simultaneously at the same frequency. Referring to Pig, 21 for a rectangular plate of A = 1,06, the fourth mode (3/0) and the fifth mods (O/I) v/ill be excited simultaneously at 25O c,p,s. Also, a rectangular plate of A = 1,16 will have its fifth mode (3/0) and sixth mode (1/I) excited simultaneously at 29O c,p,s. As the (3/0) mode on the

'A' series plates did not appear amongst the first six modes on these plates, the curve for (3/0) 'family shape' has been extra-polated a^ shown in Pig, 21,

For the sv/ept-back leading edge plates, the fundamental frequency is shov/n to increase slightly \7ith the aspect ratio,

but with a higher rate of increase as ,,\ increases. The (0/1)

mode frequency, hov/ever, is fairly constant over the aspect ratio rajige, but changing slightly and irregularly for the y V ~ 45° plates. The ratio betv/een the tv/o frequencies starts to depart slightly f^om that of a uniform cantilever beam as the angle of sweep increases, especially on the high aspect ratio plates,

The pinre torsional modes (I/O), (2/0) and (3/o), show an increase in the frequency v/ith the aspect ratio, v/ith a higher rate of increase as /V. gets larger. It is also clear that the rate of increase of the frequency is higher for the range A = 1 ,2 to A = 1,6, which is more marked on the -A. = 43° plates, (Pig, 24).

The (1/1) mode frequency increases regularly v/ith the a,spect ratio, -with a higher rate as ,/\, increases. Except on the -A.= 45° series, v/here the frequency shov7s a much larger rate of increase in the range A = 1 , 2 to A = 1,6, v/ith a very slight increase otherv/ise,

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To summarise, the pure flexural modes (O/o) and (0/l) frequencies shov/ a very slight variation over the aspect ratio range v/hich is more marked v/ith higher_/\,. , The pure torsional modes frequencies increase v/ith the aspect ratio, v/ith a higher rate of increase as „/L gets larger. The torsional frequencies also shov/ a higher rate of increase as the mode gets higher, On the sv/ept leading edge planforms, a higher rate of increase of frequency is marked in the range A = 1,2 to 1,6,

4,5 , Prequency Variation v/ith ./L

The graphs (Figs, 25-28) where frequency is plotted against tangent of sv/eep-back angle of the leading edge, shov/ clearly how the frequency changes as the plates are sv/ept back through increasing angle. As in the case of the previously discussed frequency-aspect ratio curves, one must be careful in plotting these curves to consider only mode shapes belonging to the same 'family'. It has been sha//n, however, that all modes of a given order also belong to the same 'family' as long as the aspect ratio is the same; except on the A = 1,6 series, v/here the third mode of the rectangular plate is (2/o) mode which

belongs to the same family on the fourth mode of other sv/ept-back plates. The point of interchange of frequencies occur at sweep

angle of approximately tan""^0,09, (Pig, 26),

Yfith the inspection of the curves, it could be generally concluded that the effect of sv/eep-back of the leading edge is more marked on the higher aspect ratio plates. As the chord

becomes longer, the increase ofVL has less effect on the frequency, It is difficult to generalise remarks as to the effect of varying yV. on the frequency, as that effect differed greatly for different

aspect ratios, Hbvrever, an attempt will be made here to discuss each set of curves obtained for a particular aspect ratio

separately,

Several general remarks ma^'- be made for the A = 2 series (Pig. 25), The fundamental frequency increases very slightly \/ith tan.A. , up to ./I = 150^ then it keeps almost constant up to y\. = 30°, after which a further increase takes place v/ith the

increase of J\. , A similar behaviour is noted for the pure

torsional modes (I/0) and (2/0), with liigher rate of increase outside the range Z^-, = 150 to y V = 30° v/here the frequency keeps constant. The (1/I) mode curve however shows an increase of the

frequency with tan A- up to J\u= 30°, before dropping off rapidly,

The (2/1) mode frequency shows a slight increase in the fïequency

vri-th tanyl. up to .J\.= 15°, then it starts to decrease regularly,

For A = 1,6 series planforms (Pig, 26), the frequency of the pure torsional modes (I/O) and (2/0) behaves similarly to the A = 2 plates described above, v/hile the (3/0) mode behaves in an entirely different fashion. The frequency of that mode

increases v/ith tan/\- up to ./l= 15 , then it decreases up to

-A- = 30°, then it keeps constant up to J'L= 45°. Also, the

(1/1) mode behaves differently fron that on A = 2, Its frequency increases v/ith tany^. up to yi,= 15°j keeps constant up to y V = 30°, and then increases once more up to Vl. = 45°.

Yihen inspecting the A = 1,2 series of plates (Fig, 27) the effect of varying^, on the frequency of a particular mode becomes less marked. The pure torsional modes (I/0) and (2/o) behave in a different manner from that described above for other plates. The frequency increases v/ith tan 7 ^ , then it drops

off after A. = 30°, V^Tiile the (3/o) mode frequency increases

up to yV = 15? then keeps constant between J\. = 15° to A = 30°,

after v/hich it again increases. The (1/1) mode frequency behaves similarly to the (3/0) one,

The longer chord plates A = 0,8 series, (Pig, 28) show a very slight effect of variation of yV on the frequency of a particular mode, 17ith the exception of the (3/0) and (1/1) modes, the frequency keeps constant v/ith increasing tan yN^,

The (3/0) frequency keeps constant betv/een A.= 0 to 15°^ then

it increases betv/een A = 15° - 30°, after \/hich it keeps constant

once more. The (1/I) mode frequency drops off at first up to - A = 15°> then it increases v/ith tan A - •

Thus it may be concluded that the torsional modes (m/o) frequency, generally spealcing, increases v/ith tan A - except v/ithin the range A ^ = 15° - 30° v/here it keeps constant. The fundamental fx-equency behaves similarly v/ith a lower rate of increase. Other modes behave differently on different aspect ratio plates,

5, Conclusions

When studying the results obtained for the series of

plates tested, the follov/ing concluding remarks may be nadei

5,i, ".Tien the natural frequencies of t\/o different modes of a certain vibrating plate are approximately similar,

'caTipounding' may take place bet\Teen the two nodes, naniely they may exist sinultaneously when trying to obtain one of them, The nodal pattern obtained then v/ill be a 'complex' one. For example the third and fourth modes on B,2, planform, (Fig, IO),

5,2, The effect of sv/eep-back of the leading edge on the position of the nodal lines of a particular mode is found to be more marked on high aspect ratio plates, especially on higher modes,

-14-5,3» The lower chord plates show the tendency of behaving as a beam thus exhibiting the flexural modes earlier on the

frequency scale, w*hile on the larger chord plates, the torsional modes become more predominant. On the 'D' series plates, the

second, third and fourth modes are pure torsional ones,

5,4, It has been observed that the rectangular plates v/hen perfomiing the pure flexural modes (O/o) and (0/l), tend to behave in a manner that nay be correlated to the vibration of a uniform cantilever bean. The ratio between the frequencies of those two nodes being 1»6,25j v/hile on a beam it is 1j6,27, The ratio hov/ever varied slightly v/ith the increase of A - ,

5»5» Generally speaking, the fundamental frequency is shown

to increase slightly v/ith aspect ratio for a particular Ys. , The

rate of increase gets liigher as A . increases. The (O/l) node frequency is fairly constant over the range of aspect ratio

STxrveyed except for the 45° plates v/here it varies in an irregular manner,

5ȏ, The frequency of the torsional modes (m/o) increases

with the aspect ratio increase. The rate of increase is higher for higher modes of vibrations. The frequency is also shov/n to increase at a higher rate v/ith the increase of the angle of sweep, It has a.lso been observed that, generally speaking, the rate of increase is higher for the range A = 1,2 to A = 1,6 v/liich is more marked on the A = 45° plates,

5.7. The (l/l) mode frequency increases regulajrly v/ith aspect

ratio, v.ith a higher rate of increase as J\ increases,

5.8, The effect of sv/eep-back of the leading edge on the frequency of a particular mode, for the saine aspect ratio plates, is more marked on the higher aspect ratio planforms. The effect is slight on longer chord plates. It is difficult to generalise remarks ats to hov/ the frequency varies v/ith tan A- , as this depends much on the aspect ratio. However, generally speaking, the torsional modes (n/o) frequency increa.ses "ivith tanyL except within the range A = 15° to A = 30° v/here it keeps constant, The fundamental frequency behaves in a similar manner,

Acknov/ledgements

The author i/ishes to aisknov/ledge the kind help offered by Professor ""^,3, Hemp, in the course of a theoretical investigation, He is very much indebted to the useful guidance and encouragement

offered by lie, C,K, Trotnan throughout the work, and 'wishes to

APEENDIX A Nomenclature

E

er

h

D

w

P

f

P

a

= = = = = = = = =

Young's Modulus of Elasticity Poisson's Ratio

Thickness of Plate

Flexural Rigidity of Plate

= E hVl2(l-a-^)

Lateral Deflection

Density of Plate Material Frequency

Angular Frequency = 27cf

Geometrical Similarity Factor,

lb/inch inch lb, inch inch lb/inch^ cycles/sec, Rad,/sec,

B

/ / / / /

^'1

r^>^

r*-i

_{Plate (I)} ^

1

y

» ; / *

/3a';

y > / f / / / ' ^

V

\ A .

A\,^^

^^v

^

a. ^

A'

Plate (II) (FIG. A.I.) PLATES CONSIDERED IN M/OZSIS

Consider plate (ll) with the dimensions shovvH,, where /? is some factor, v/ith x and y as coordinates of any point, In order to work non-dimensionally. consider plate (l) which, is geometrically similar to plate (ll), but v/ith a unit overhang,

Therefore

_{5 = f}

1 6

-Por p l a t e ( l l ) , vve g e t -,a ,^(^a-x t a n A ) I j / 6 w\ / d •w\ o_ d Y/ d w „ / . A /djw\'' i , , 2 ^

P = J h

2 1 .a (pa-x t a n A ) w dydx o ' / o

.(2)

T/here v/ = w ( x , y ) i s t h e d e f l e c t i o n a t any p o i n t f o r a v i b r a t i o n mode of p l a t e ( l l ) d i v i d e d b y t h e d e f l e c t i o n a t A ' , S i m i l a r l y w = w(^,ri) i s t h e d e f l e c t i o n a t any p o i n t f o r a v i b r a t i o n mode

of p l a t e ( l ) d i v i d e d by t h e d e f l e c t i o n a t A, I n t h e c o r r e s p o n d i n g n a t u r a l modes on b o t h p l a t e s , a s l o n g a s d i s p l a c e m e n t s ai^e v e r y s m a l l tlius h a v i n g no e f f e c t on t h e n a t i n r a l f r e q u e n c y , t h e n from knov/n t h e o r y t h e d i s p l a c e m e n t s w i l l be e q u a l , T h e r e f o r e w ( x , y ) = w(?,Ti)

_.(3)

Prom ( l ) vre g e t j dx = a d^ , From (3) v/e g e t s <3y = a dT), dw _ ÖW d ^ _ 1_ ÖW dx ~ a? • dx ~ a a^ /ö_w\ d /'dw\ dg 1 dx ~ 2 a . 2 d W ÖW _ 1_ ÖW , a y ~ a ÖT1 * 1 d W 2 2 2 a w _ 1 a v/ , a v/

, 2 - 2 - 2 ' dxdy " ^ apri^

ay a on ' ' a '/

.(4)

Ufeing r e l a t i o n s ( 4 ) , v/e can w r i t e e q u a t i o n (2) i n t h e formj

r

E h 1 P ' a ^ 1 2 ( 1 - 0 ^ ) ll

_(^gtanA)

o '•' (f .,2 v2 /•,2 . 2 ,2 ,2 i'd v/\ fa v/\ r,_ a w a w + ! — - j + 20" — ^ , — - + 3^ ari'

o c U v ^ U V ^ ' - ^ g ^ - ,2

>1 i ( / ? - g t a n ) <3€ «>< o '• o

v/ , dg.d-n

/Therefore ,,.

Therefore

2 Eh^ 1 ^

v/here K = constant for all plates of a planform geometrically similar to that of plate (II) and v/ith the same mode of vibration

(w), with different material and thickness as long as Baisson's ratios are the same,

Thus in order to find out p' for a given plate, of a planform similar to that of frequency pj if a is the

similarity factor betv/een the two, v/here primed symbols refer to the given platet

• ^|2 2 1 E',h'^,(

• • P' = P . - r — 2 '

sT Eh , p'

-^

• ~ •

V,

,/FTi

15;

-18-REFERENCES 1» Da H e y , J,'V7, and Rippergex', E,A. 2, Martin, H,C, and Gursahaney, H,J. 3, Gustafson, P,N,, Stokey, Yl,F, and Zorowski, C,P,

4. Young, D,

5# Barton, M,V,

6, Grinsted, B.

7» Den Har tog, J,P.

8. Timoshenko, S,

9» Lord Rayleigh

Experimental values of natural frequencies for skew and rectangular cantilever plates, Proceedings of the Society for Experimental Stress Analysis, Vol, 9 No, 2, 1952, p.51-59. On the deflection of sv/ept cantilcvered

surfaces.

Journal of Aeronautical Sciencas, Vol, 18, December 1951, pp,805-812.

ioi experimental study of natural frequencies of cantilevered triangular plates,

Journal of Aeronautical Sciences, Vol, 20, i.fey 1953, PP.331-337.

Vibration of rectangular plates by Ritz's method,

Journal of Applied Iilechanics, Vol, 17, December 1950, pp,448-453.

Vibration of rectangular and skev/ cantilever plates.

Journal of Applied Mechanics, Vol. 18, June

1951, pp.129-134.

Nodal pattern analysis,

Proceedings of the Institution of Mechanical Engineers, Vol. I66, No, 3, 1952, pp,509-326. Mechanical vibrations,

Third Ed, McGrav/-Hill Company I n c , 1947» Vibration problems in engineering,

2nd Ed, Van Nostrand Co, Ltd,, 1944. The theory of sound,

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ASPECT RATIO

FREQUENCY VARIATION WITH ASPECT RATIO FOR SERIES '2' PLANFORMS

FIG. 2 2 . 7 0 0 6 0 0 500 4 0 0 m CL o >• 3 0 0 ö 200lX 2 ^ ASPECT RATIO

FREQUENCY VARIATION WITH ASPECT RATIO FOR SERIES '3' PLANFORMS

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FREQUENCY VARIATION WITH ASPECT RATIO FOR SERIES '4' PLANFORMS

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(o/l) .(3/0) (2/0) > ^ ^ (o/o) o tan A

FREQUENCY VARIATION WITH TAN A

FOR SERIES "C' PLANFORMS

FREQUENCY VARIATION WITH TAN A

FOR SERIES "D* PLANFORMS