Journal of Sound and Vibration (1974) 32(4), 481--490

A N E X P E R I M E N T A L - T H E O R E T I C A L S T U D Y O F F R E E

V I B R A T I O N S O F P L A T E S O N E L A S T I C P O I N T S U P P O R T S

T. R. LEUNER t Department of Aerospace and Mechanical Sciences,

Princeton University, Princeton, New Jersey 08540, U.S.A.

(Received 12 May 1973, and in revised form 25 September 1973)

A theoretical and experimental study is made to investigate the effect on plate vibrations of varying the stiffness of corner elastic point supports. An experiment is conducted in which the bending stiffness of horizontal beams is used to support a square plate at its four corners. The stiffness of these supports can be varied over such a range that the plate fundamental frequency is lowered to 40% of the rigid support frequency. The variation with support stiffness of the frequencies of the first eight plate modes i s measured, and compared with results from a theoretical model in which a Rayleigh-Ritz analysis is used which approximates the plate mode shapes as products of free--free beam modes. The elastic point supports are modelled both as massless translational springs, and springs with tip masses, which are included to better represent the experimental supports. The plate mode shapes, for the rigid support condition are analyzed by using holographic interferometry. There is excellent agreement between the theoretical and experimental results, except for high plate modes where the theoretical model is considered to be inadequate.

1. INTRODUCTION

The purpose of this investigation is to analyze vibrational behaviox of plates with discrete elastic supports. This configuration has received relatively little analytical or experimental attention previously, although it is representative of some proposed thermal protection systems for high performance vehicles [1].

The only work known to the author relating the change in plate frequency to the variation in stiffness of point, corner supports is that of Dowell [2]. He has calculated the frequency variation of the fundamental plate mode with stiffness by using a Rayleigh-Ritz analysis with normal modes of the unconstrained plate. The support conditions are introduced into the analysis by means of Lagrange multipliers.

This paper is a summary of a study [3] intended to expand upon the theoretical work presented by Dowell [2], and to provide experimental data to compare to this theory. For a more complete discussion of spring-supported plate vibrations, the reader is referred to this study [3], which provides greater detail in the areas of theoretical analysis, experimental details, and holography.

A number of authors have investigated the problem of a plate with rigid corner supports. R e e d [4] has presented analytical solutions to plate vibrations, obtained by using both a

Rayleigh-Ritz analysis and a series solution to the plate equation. His recorded experimental frequencies agree remarkably well with those of the theoretical analysis for rigid supports,

t Now with McDonnell Douglas Astronautics Company, Huntington Beach, California, U.S.A. 481

4 82 T.R. LEUNER

md Reed's experimental apparatus was used as a basis in designing the experimental set-up tsed in this study.

Tso [5] also has presented experimental data for a point-supported plate, and this data las been compared with those from energy and finite difference analyses by Johns and qagaraj [6]. The correlation between theory and experiment is only fair, probably due to a ack of convergence in the numerical procedures employed in the theoretical work. Dowell [7] ias compared the experimental measurements of the plate fundamental frequency from both ources with results from a revised theory in which support conditions are modelled by .agrange multipliers. There is better agreement between theory and experiment for this nalysis of rigid point supports. Recently Johns and Nataraja [8] have published an improved nd extended version of the former's original analysis [6].

Most previous work has been concerned with the fundamental plate mode, with the xceptions of references [4] and [8]. For the most part, Rayleigh-Ritz analyses have been sed in these analyses. This research is intended to fill some of the gaps left by previous avestigators by providing theoretical-experimental correlation for the effects of flexible ,oint supports.

2. EXPERIMENTAL PROGRAM

.1. APPARATUS

The experimental plate was an 8 in by 8 in by 0.125 in aluminum (5052 alloy) plate, which ad a fundamental frequency of 146 Hz for rigid point corner supports. Thus, when the apport stiffness was decreased, the frequency of the fundamental mode could be reduced ) less than one-third of its rigid value without encountering non-linearities in the magnetic ~citer and capacitive pick-up used. The relatively small power of the magnetic exciter used, oupled with the large stiffness of this experimental plate, made Chladni patterns utilizing lass beads useless. Therefore, holographic interferometry was used to study the plate rode shapes. The method of approximating point supports was adapted from Reed [4]. The four

3rners of the plate were beveled top and bottom at 35 ~ angles, so that the total included ngle of the plate corners was 70 ~ The plate was supported by the bending stiffness of four orizontal beams, whose protruding ends were routed out to 90 ~ angles. The support beams were aligned along the diagonal axes of the plate. When the beveled

late corners were inserted in the routed beam edges, and the four beams fastened firmly into osition, the extreme corners of the plate were restricted to vibrate vertically with the beams, ut the plate corners were allowed to rotate about a horizontal axis because of the angular ifference between the beam and the plate. The beams were held clamped in support stanchions at a height of four inches over the

lrface of the base plate. The support stanchions were blocks of solid aluminum, with :ctangular channels housing the support beams, which were fastened through the 0.125 in lick cover plate by three set screws. The bottom of each stanchion was tapped to accept two olts which passed through the base plate and could be tightened from underneath the plate

clamp the support stanchion into position. The base plate which supported the experimental apparatus was a heavy aluminum plate

�9 ft by 2 ft by 0.5 in. Four quarter-inch wide slots were routed along the plate diagonals to ilow the support stanchions to be fastened at varying distances from the corners of the !perimental plate. The experimental apparatus is shown in Plate 1.

PLATES ON ELASTIC POINT SUPPORTS 483

2.2. PROCEDURE

The equivalent spring stiffness of the support system was measured directly. The stiffness parameter used in the experiment was the length, L, of the protrusion of the support beam beyond the front of the support stanchion. The support stiffness was determined by adjusting the support beams to hold the plate at length L, and then placing P lb of weight on the plate and measuring the vertical deflection of the plate corner with a travelling microscope. This procedure was repeated for varying loads and beam length until a calibration curve was developed relating the support beam length to the non-dimensional stiffness of the corner support, Ks. This stiffness was defined as

Ks = ka2/ D

where a is the plate length, D the plate stiffness and k the support stiffness (lbf/in). The plate was locked into the support system by placing the plate corners into two adjacent

support beams which were fastened into position in the support stanchions. The remaining pair of beams were pushed firmly against the free plate corners and held in position by tightening the set screws in the support stanchion cover plate. Although this method intro- duced some compressive stresses in the plate, Reed [4], who performed a similar experiment for rigid corner supports, assumed that the effect of the compressive force was negligible, since it approximated a point load which would produce a small average stress in the entire plate.

The plate was excited by a magnetic transducer, Brtiel and Kjaer (B& K) No. MM 0002, powered by a beat frequency oscillator, B & K Type 1022. Small pieces of steel foil were cemented to the plate at the probable locations of maximum displacement for the first eight plate modes for excitation. A B & K No. MM 0004 capacitive transducer was suspended over the plate to measure the displacement amplitude. The output of the transducer was displayed on an oscilloscope screen and plate modal frequencies were determined to be plate resonances by varying the frequency.

Since the plate mode shapes were not observable through the Chladni technique, each plate resonance was identified with a particular vibrational mode by examining the location of maximum plate deflection, the resonant frequency, and the number of lower frequency resonances detected. This information was then compared with predicted results to determine the frequencies of the first eight plate modes as the non-dimensional stiffness of the support system was varied from 2.5 to 95.

2.3. HOLOGRAPHY

The plate mode shapes were investigated through the use of time-average holographic interferometry. The experimental details follow. Holography was utilized to provide mode shape information unattainable through the normal application of the Chladni technique. Experimental difficulties involving coupled modes and unsatisfactory fringe definition due to the limited power of the laser source prevented the author from accomplishing his secondary objective: to qualitatively analyze the variation of mode shape with support stiffness. For additional information relating to the use of holography in this experiment, or the theory behind, and applications of, time-averaging holography, the original report [3] and references [9], [10] and [11] are recommended. Due to the relatively large size of the experimental plate, and the relatively low power of the eight-milliwatt Spectral Physics Model 120 continuous wave laser used, difficulties were anticipated in obtaining good holograms of the entire surface. This experiment was therefore designed to observe only one-quarter of the plate's

484 T.R. LEUNER

- b .

/ /

/ /

/

/

. . . . Loser beam

. . . . . Diverging beam

, Base plate

Z~ Prism

o Spatial filter

Beam splitter

( ) Laser

Hologram

Vibrating plate

Figure 1. Optical arrangement of holographic bench.

surface, and the symmetrical properties of the square plate are utilized in interpreting the entire plate motion.

The base plate of the experiment was modified slightly so that it could be clamped to the holographic bench in a vertical position. Figure 1 shows the position of the vibrating plate and the optical equipment on the pneumatically isolated, welded steel bench.

In setting up the experiment, the surface of the experimental plate was first lightly sand- blasted to provide a diffuse optical surface. Black draftsman's tape was used to mark the plate axes around the quadrant to be hologrammed. The plate supports were adjusted to provide maximum stiffness, Ks ~ 400, in the hope of obtaining rigid support mode shapes. The plate was vibrated by the same equipment as described in the previous section, but the capacitive pick-up and framework were removed so as not to interfere with the optics.

Resonant frequencies were tuned in by ear with the transducer operating at maximum power. The output voltage was then adjusted by trial and error to provide between six and ten fringe orders. Initially, filters were used so that the light intensities due to, respectively, the object and reference beams were equal over the entire surface of the photographic plate. Later it was found that brighter pictures were obtained when the filters were removed and the reference beam was approximately eight times more intense than the object beam.

The experiment was performed in a light and vibration baffled laboratory. During the course of experimentation, it was found that small changes in the experimental set-up could have significant effects on plate exposure times, object and reference beam intensities, and the darkness of the hologram. Although it is extremely difficult to recreate holographic results with different equipment, there follows a brief description of the experimental procedure used.

A typical hologram was produced in the following manner. The reference beam from the eight-milliwatt laser was adjusted to be approximately eight times more intense than the reflected object beam (readings of 12 and 9, respectively, on a GotTen Luna Pro light meter). The photographic plates (Kodak Type 649-F backed spectroscopic plates) were exposed to

! the laser light for 50 to 55 s, developed for 5 min in Kodak D-19 Developer, fixed for 5 min by using standard Kodak Fixing Agents, and washed in a solution of ethyl alcohol and water (1:1). Pictures of the holographic image were taken at F22, Polaroid Type 57 film

i (3200 ASA) being used. The exposure time varied from 2 to 5 min depending on the darkness of the developed photographic plate.

PLATES ON ELASTIC POINT SUPPORTS 485

3. EXPERIMENTAL RESULTS

3. I. FREQUENCY VARIATION WITH STIFFNESS

The data derived from the experimental investigation were compared with the results of a computer program utilizing a Rayleigh-Ritz analysis in which the plate modes were repre- sented by the products of free-free beam modes. The relevant equations are presented in reference [3], along with a detailed analysis of the computer program and its output. The plate modes were analyzed in four categories: doubly-symmetric, symmetric-antisymmetric, antisymmetric-symmetric, and doubly-antisymmetric. Two beam modes were used in each plane to represent the plate motion. Only symmetric beam modes were used in the symmetric plane and antisymmetric modes in the antisymmetric plane.

It should be noted that the computer formulation of the problem was demonstrated to be inadequate for very large stiffnesses and high frequency plate modes. The problem was traced to numerical difficulties in the eigenvalue routine involving small differences between large numbers. The numerical problem also prevented the solution from converging when additional beam modes were used to approximate plate motion. The difficulty was not corrected, but an analytical procedure was employed which gave acceptable results for all but the seventh and eighth plate modes (see reference [3]).

Figures 2 through 7 show comparisons of the measured values of plate frequency de- pendence on support stiffness with theoretical results for both massless spring supports and

i i i i l I I I I �9 .

/ c o . ~ O ~

t

t -t ~ Mode shape

I I 1 I I I I I I 0 I0 20 30 40 50 60 70 80 90 IO0

Ks Figure 2. Frequency ~'s. support stiffness, first mode. - - , Theory, plate with simple beam supports;

, theory, plate with massless spring supports; �9 experiment.

simple beam supports. In the first case the fundamental frequency of the cantilevered support beam is assumed to be much greater than the frequencies of the plate modes under con- sideration. If this condition holds, beam inertia or mass distribution will not affect the natural vibration of the plate and the support beams can be assumed to act as massless translational springs. This theory does not accurately predict the plate frequencies for the higher mode shapes, and the support beam mass was incorporated in the theoretical model. A second analysis was performed modelling the support beams as springs with point masses, such that the frequency of vibration of the point mass would be the same as that of the actual

486 T.R. LELINER

support beam. Results show this to be a superior analytical model over the stiffness range (95 > Ks > 2.5) for which this approximation is valid.

The frequency vs . stiffness variation of the first plate mode is illustrated in Figure 2. The largest quantitative difference between experiment and theory is 5Yo. For stiffnesses of /(3 < 20, the agreement between experiment and theory is excellent, although for larger stiffnesses the theoretical results appear to be conservative. The qualitative agreement is good, however, and energy methods like the Rayleigh-Ritz analysis used are expected to over-estimate system eigenvalues.

The experimental results presented in Figure 3 follow the same pattern as the fundamental mode frequencies. As the support stiffness is varied, the frequencies of both modes two and

io

io ~ s

6

4

2

o

J I I I I J I I I

~o//O ~ ~ z/e :~

/ -

/e / Mode 2 Mode 3

I I I I I I" 1 I I 10 2 0 3 0 4 0 5 0 6 0 7 0 8 0 9 0 100

Ks

Figure 3. Frequency v s . support stiffness, second and third modes. , Theory, plate with simple beam supports; - - - , theory, plate with massless spring supports; e , experiment.

three increase, qualitatively in accordance with the theoretical result based on the assumption that the support beams are stiff rods with tip masses. The maximum quantitative difference is 4 Yo, and the experimental and theoretical results are in better agreement for Ks < 30 than for higher stiffnesses, where the theoretical results over-estimate the experimental.

It shofild be noted that, for K, < 25, the experimental frequencies are bracketed between theoretical results for massless spring supports, and for simple beam with tip mass supports. This indicates �9 lumping the beam mass into one point mass at the plate corner may be a crude approximation over-estimating the correction needed to account for vibration of the support beams.

The experimental results presented in Figure 4 prove that the frequency of the fourth plate mode is independent of the support stiffness. This was predicted by theory, but there is a large discrepancy in the magnitude of the plate frequency. This, however, seems to be .an inaccuracy of the analysis, since Reed [4] predicts K = 19.6, and the measured frequency was found to be K = 19.3. Again, the theory is conservative. The non-dimensional frequency, K, is defined to be

I K = (.oa2 " V / ~ .

PLATES ON ELASTIC POINT SUPPORTS 487

28

24

20

16

12

8

4

0

O q l ~ J O 0 �9 �9 0 �9 �9 �9 �9 �9 �9 �9

~ M o d e shope

I 1 I I I I I I I I0 20 30 40 50 (50 70 80 90 I00

Ks

Figure 4. Frequency vs. support stiffness, fourth mode. - - , Theory ; o , experiment.

The fifth plate mode shows the best correlation between experiment and theory. The maximum differential between experimental frequency and the predicted value plotted on Figure 5 is 2~o. For most values of the support stiffness, the measured plate frequency is higher than that predicted by assuming the support beam to be a flexible rod with tip mass, but less than the theoretical frequency for a plate on massless spring supports. This confirms the assumption that lurlaping the beam mass is an over-estimation of the support vibration correction.

42

36

30

24

18

12

I I i I I I I i

... j~_...=jL.=...~ , r - f ~ . ~ ' ~ " ~ " -

~ M o d e shope

I I I I I I0 20 50 40 50

Figure 5. Frequency vs. support stiffness, fifth mode. - -

I I I I 60 70 80 90 I 0 0

, Theory, plate with simple beam supports; - - - , theory, plate with massless spring supports; e, experiment.

The experimental plate frequencies plotted in Figure 6 behave in a manner similar to that of Figure 5, but the discrepancy between theory and experiment is greater for the sixth mode as opposed to the fifth plate mode. The largest quantitative difference is 5 Yo.

Plate modes seven and eight exhibit the worst correlation between theoretical and experi- mental results as shown in Figure 7. The experimental plate frequencies plot as a smooth

4 8 8 T . R . L E U N E R

42

36

30

24

18

12

6

I I I I I 0 I0 20 30 40 ~0

Ks

Figure 6. Frequency vs. support stiffness, sixth mode. - -

I I I I I I I I I

p-

~ Mode s h a p e

I I I I GO 70 80 90 I O0

, Theory, plate with simple beam supports; .--, theory, plate with massless spring supports; �9 experiment.

oo

56

48

40

32

24

16

8

0

I I I I I I I I I

Mode 7 Mode 8

~ Mode s h a p e s

w i i t I I I I I I 0 20 50 4 0 50 60 70 130 90 1 0 0

Ks

Figure 7. Frequency vs. support stiffness, seventh and eighth modes. , Theory, plate with simple beam upports; - - - , theory, plate with massless spring supports; �9 experiment.

arve over the entire stiffness range, and are in reasonable agreement with theory for Ks < 20. 'or higher stiffnesses, however, results of theoretical analysis exceed the experimental results y about 1 0 ~ . As was previously mentioned, the discrepancy between theoretical and ~perimental results for the seventh and eighth plate modes is due to numerical error in the aeoretical solution. Note that the nodal pattern shown is for rigid corner supports. For exible supports, in particular the limiting case Of no support, the nodal lines do not pass arough the corners and hence the natural frequency changes with support stiffness. ! In general, there appears to be excellent agreement between the experimental results and ~e theoretical analysis in which the support beams are modelled as springs with point lasses at the plate corners. Discrepancies are observed at high stiffnesses due to the conserva- lve nature of the Rayleigh-Ritz analysis, and for high plate modes due to inaccuracies in ~e theoretical computations.

Plate 1. Plate and support system.

(facing p. 4~8)

A

Plate 2. Mode shape o f first mode . K~ = 400, toJ2r,= 152 Hz.

Plate 3. Mode shape o f second mode. K5 = 400, ~212~r = 306 Hz.

Plate 4. Mode shape o f four th mode . K, = 400, c94/2r: = 392 Hz.

Plate 5. Mode shape o f sixth mode. Ks = 400, to612z~ = 850 Hz.

PLATES ON ELASTIC POINT SUPPORTS 489

3.2. PLATE MODE StIAPES

Holographic analysis of the plate vibration generated well-defined mode shapes for the first, second, fourth, and sixth plate modes, and holographic representations of these mode shapes are included in this report as Plates 2 through 5. The nodal pattern of the second mode shape indicated that coupling existed between modes two and three and between modes seven and eight. Since pure mode shapes could not be obtained for these modes, they were eliminated from further study. Mode shapes of the fifth and seventh plate modes are presented in reference [3]. The fifth plate mode exhibited node lines along the quadrant axes as expected, but the dark area of maximum deflection in the center of the hologram obscured some of the interference patterns.

The mode shape of the first plate mode is shown in Plate 2. Reference [3] describes how, in holographic interferometry, nodallines of the plate vibration are distinguished by very bright interference fringes, and how, as the amplitude of the plate vibration increases, the intensity of the light fringes decreases. There are no node lines in the plate fundamental mode, and the hologram is consequently so dark that the center fringe orders are barely observable. To photograph the hologram, the center of the plate was artificially lightened to provide a better picture. In all the photographs, the plate support position is in the lower right-hand corner, and the plate center is marked by the intersection of the X and Y plate axes in the upper left-hand corner. The observed mode shape is as expected, increasing from zero amplitude (for rigid supports, Ks g 400) at the support position to maximum displacemefit at the plate center.

Plate 3 shows the observed mode shape of the second plate mode. This antisymmetric- symmetric mode should exhibit a node line along the entire Y axis. The observed node line originates in the negative X, negative Y quadrant of the plate, crosses the Y axis near the plate center, and continues into the positive X, positive Y quadrant. The mod e shape of the seventh mode was found to be similarly skewed. This behavior of the two pairs ofsymmetrir antisymmetric modes is attributed to coupling between the paired modes, possible asym- metry in the support stiffnesses, and the effect of driving a mode at a location other than the point of maximum displacement of the natural mode.

The mode shape of the fourth plate mode, Plate 4, is exactly as anticipated. The node line begins at the plate corner, and follows the diagonal to the plate center to join with the node lines from the other corners. The antinodes are located along the X and Y axes at the plate edges.

The sixth plate mode, Plate 5, exhibits the dearest node line. The measured node line is a circle, as is predicted by theory, with a diameter equal to 64 Yo of the plate width. The plate antinode is observable at the plate center, where the light intensity is low due to the relatively large fringe order at that point. A local maximum displacement is visible between the plate support and the nodal circle.

Holographic analysis of the plate mode shapes not only provided a clearer definition of nodal patterns than that provided by Chladni patterns but also defined the positions of antinodes and relative deflection ratios across the plate surface. Except for the coupled mode pairs two and three and seven and eight, the rigid mode shapes are as predicted by Reed [4]. An investigation of the apparent coupling between these modes was not undertaken.

4. CONCLUSIONS AND RECOMMENDATIONS

An experiment was performed to determine the frequency variation of the natural modes ff a corner-supported plate as the stiffness of these supports was varied. The experimental

T. R. LEUNER

Its were found to agree favorably with those o fa Rayleigh-Ritz analysis in which products ee-free beam modes were used, and the corner supports were approximated by trans- real springs with finite mass. Holographic interferometry was used to investigate the al patterns of plate vibration for the rigid support condition. It is recommended that ler compute~" analysis should be performed which would give accurate results for high .~ modes and large support stiffnesses. It is further recommended that the use of holo- ,hy be expanded to examine (a) the apparent coupling of symmetric-antisymmetric mode ~, (b) the effect of varying the support stiffness on the plate mode shapes and (c) the :t of unequal support stiffnesses on plate mode shapes.

ACKNOWLEDGMENTS

ork supported by NASA Grant N G R 31-001-146. The author wishes to acknowledge guidance provided by his faculty advisor, Professor E. H. Dowell. Holography was )rmed under the direction of Professor Robert Mark in the Photomechanics Laboratory .e Civil and Geological Engineering Department.

REFERENCES

~,. H. DOWELL 1972 (April) American Institute of Aeronautics and Astronauties Paper No. 72-350. Theoretical panel vibration and flutter studies relevant to space shuttle. !. H. DOWELL 1972 Journal of Applied Mechanics 39,727-732. Free vibrations0f an arbitrary tructure in terms of component modes. �9 R. LEtmER 1972 (July) Princeton Unit'ersity AMS Report No. 1052. An experimental- aeoretical study of free vibrations of plates on elastic point supports. !. R. REED, JR. 1965 (September) NASA TN D-3030. Comparison of methods in calculating .'equencies of corner-supported rectangular plates. V. K. Tso 1966 American Institute of Aeronautics and Astronautics Journal 4, 733-735. On the andamental frequency of four point-supported square elastic plate. ). J. JOHNS and V. T. NAGARAJ 1969 Journal of Sound and Vibration 10, 404--410. On the funda- lental frequency of a square plate symmetrically supported at four points. ',. H. DOWELL 1971 Journal of Applied Mechanics 38, 595-600. Free vibrations Of a linear :ructure with arbitrary support conditions. ). J. JOHNS and R. NATARAJA 1972 Journal of Sotmd and Vibration 25, 75-82. Vibration of a :luare plate symmetrically supported at four points. :. J. COLLIER, C. B. BURCKHARDT and L. H. LIN 1971 OpticalHolography. New York: Academic ress. ). A. EVENSEN and R. APRAHAMXAN 1970 (December) NASA CR-1671. Application of holo- raphy to vibrations, transient response, and wave propagation. '. V. HORVATH and J. WALLACH 1970 (October) General Electric Report No. 70-C-352. Holo- raphic interferometry applied to experimental mechanics.