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Proceedings of PACAM XICopyright c© 2009 by ABCM

11th Pan-American Congress of Applied Mechanics - PACAM XIJanuary 04-08, 2010, Foz do Iguaçu, PR, Brazil

FLEXURAL VIBRATIONS OF PLATES:THEORY AND EXPERIMENT

Rafael A. Méndez-Sánchez, [email protected] Morales, [email protected] Gutierrez, [email protected] de Ciencias Físicas, Universidad Nacional Autónoma de México. A.P. 48-3, 62251, Cuernavaca, Morelos, Mexico

Betsabé Manzanares, [email protected]ón de Ciencias e Ingeniería, Unidad Regional Sur de la Universidad de Sonora. Blvd. Lázaro Cárdenas 100, 85880 Navojoa,Sonora

Felipe Ramos, [email protected] de Investigación en Física, Universidad de Sonora, A.P. 5-088, 83000, Hermosillo, Sonora, Mexico

Abstract. Theoretical and experimental results for flexural waves of a rectangular plate with free ends are ob-tained. Both the frequency spectrum and the wave amplitudes are analyzed. The plane wave expansion method isapplied to the Kirchhoff-Love equation to obtain theoretically the normal-mode frequencies and wave amplitudesof the rectangular plate. The method of measuring the flexural vibrations of the metallic plate is based on the useof electromagnetic-acoustic transducers (EMATs). The EMATs, that are non-contact devices, were also used toexcite the out-of-plane waves. A point by point scanning to measure the wave amplitudes is performed. The theo-retical predictions of the 2-D Kirchhoff-Love model agree with the experimental measurements for an aluminumrectangular plate at frequencies below ≈ 2 kHz.

Keywords: rectangular plates, vibrations, electromagnetic acoustic transducers, Kirchhoff-Love equation, out-of-plane waves.

1. INTRODUCTION

The vibration of plates are very important in several engineering applications. Among others, they appear in piezo-electric transducers and as parts of buildings and structures. There are many theoretical and numerical studies on theflexural vibrations of plates (Leissa, 1993; Soedel, 1993; Szilard 1974; Gorman, 1999; Bardell, 1994) but only few areexperimental (Lee and Spencer, 1969; Low, Chai, Lim and Sue, 1998; Nieves, Gascón and Bayón, 2004; Ma and Lin,2001; Schaadt, Guhr, Ellegaard and Oxborrow, 2003; Ma and Lin, 2005).

In this work we show some theoretical and experimental results of a rectangular plate with free ends. The theory isbased on the plane wave expansion method while the experiment is performed using electromagnetic acoustic transducers(EMATs). In the next section we apply the wave expansion method to the Kirchhoff-Love wave equation. This equationyields the flexural displacement in plates for frequencies below 2 kHz. In section 3 a description of the experimental setupused to measure the normal modes frequencies and amplitudes is given. In section 4 we compare theory with experiment.Good agreement is found.

2. EXPANSION IN PLANE WAVES FOR FLEXURAL VIBRATIONS

The vibrations of thin plates of width h are governed, at low frequencies, by

∂2

∂x2

[D

(∂2W

∂x2+ ν

∂2W

∂y2

)]+ 2

∂2

∂x∂y

[D(1− ν)

∂2W

∂x∂y

]

+∂2

∂y2

[D

(∂2W

∂y2+ ν

∂2W

∂x2

)]= −ρh

∂2W

∂t2, (1)

known as the Kirchhoff-Love model. Here W is the flexural displacement in the z direction. E, ρ and ν are the Youngmodulus, density and Poisson ratio, respectively; D = Eh3/12(1− ν2) is the flexural rigidity.

The plane wave expansion is a method that has been applied to several undulatory periodic systems but here we willapply it to a non-periodic system. In this case a unit cell containing the system of interest is surrounded by a materialthat mimics the vacuum. This rectangular cell of side lengths a and b is repeated periodically. In the plane wave methodthe parameters D, Dν and ρh, appearing in Eq. (1) are expanded in a Fourier series with coefficients αG, βG and ηG,respectively. The reciprocal vector is denoted by G = (Gx, Gy) = ( 2π

a p, 2πb q) where p and q are integers. This procedure



yields the following generalized eigenvalue problem

MΦ = (2πf)2NΦ, (2)

where f is the frequency and Φ is a vector whose entries are the Fourier coefficients of the flexural displacement. M andN are infinite matrices with entries

M(G,G′) = αG−G′ [(k + G) · (k + G′)]2

+ βG−G′[(kx + Gx)

(ky + G′y

)− (kx + G′x) (ky + Gy)]2

, (3)

and

N(G, G′) = ηG−G′ . (4)

The wavenumber is denoted by k = (kx, ky). Eq. (2) is solved using standard numerical methods.

3. EXPERIMENTAL SETUP BASED ON ELECTROMAGNETIC ACOUSTIC TRANSDUCERS

The experimental setup, employed to measure the normal-mode frequencies and amplitudes, uses electromagneticacoustic transducers (EMATs). The EMATs are composed by a coil and by a permanent magnet that induce eddy currents(See Fig. 1). It can be used to detect or excite elastic waves. We have used these EMATs previously to study torsional,compressional and flexural vibrations of rods (See Morales, Gutiérrez and Flores 2001; Morales, Flores, Gutiérrez andMéndez-Sánchez 2002; Díaz-de-Anda et al, 2005 ).

��

��

��

��

magnet

coil

AMPLIFIERPOWER

AMPLIFIER

OSCILLATOR

PXI

LOCK−IN

DETECTOR

PLATE

ALUMINUM

X−Y PLOTTER

ALUMINUM

EXCITER

Figure 1. Experimental setup used to measure the normal modes of the rectangular plate. The dashed line between thedetector and the X-Y recorder indicates mechanical coupling. The dashed dotted lines indicate the threads supporting the

plate. The coupling between the PXI and the computer is done through an optical fiber.

A schematic drawing of the experimental setup is shown in Fig. 1. A signal of frequency f of an oscillator is amplifiedand sent to an EMAT. With the configuration of the EMAT, shown in Fig. 1, it is possible to excite and detect flexuralwaves. Another EMAT is used to detect the vibrations. The plates hang vertically from two thin nylon threads whichdo not disturb the measurements for the purposes of the present work; therefore the plate can be considered as freelysupported. To measure the normal-mode frequencies the detector and exciter are located at the borders of the plate. Tomeasure the normal-mode wave amplitudes the position of the detector is varied with an X-Y recorder, controlled witha computer through a PXI (PCI eXtensions for Intrumentation). The signal of the EMAT detector is sent to a lock-inamplifier and then to the PXI. The experiment is controlled by a computer working on Linux.

4. THEORETICAL VERSUS EXPERIMENTAL RESULTS

To compare theory with experiment we have chosen a rectangular plate. The normal-mode frequencies and amplitudesof the plate were calculated using the plane wave expansion method and measured using the experimental setup of Fig. 1.In Fig. 2 we show the comparison between the measured and predicted normal-mode frequencies. As can be seen thetheoretical predictions agree with the experimental results. In Fig. 3 we show an example of experimental wave amplitudesand the theoretical prediction. As can be seen the agreement is excellent.

5. CONCLUSIONS

An experimental setup to measure the normal-mode frequencies and amplitudes of flexural waves in thin plates basedon electromagnetic acoustic transducers was described. We have also applied the plane wave method to the Kirchhoff-Love equation to obtain theoretical predictions. The agreement between theory and experiment is excellent for, approxi-mately, the ten lowest modes.



___________

___________

0

0.5

1

1.5

2

freq

uenc

y (k

Hz)

Figure 2. Spectrum of a rectangular plate of 203 mm × 355 mm × 6.12 mm obtained with the plane-wave expansionmethod and measured with the EMATs. The left and right columns correspond to experimental and theoretical predictions,

respectively.

(a)

(b) (d)

(c)

Figure 3. Normal-mode wave amplitudes (in arbitrary units) of out-of-plane waves in the rectangular plate. The upperpart, i.e., (a) and (c), are experimental results. The lower part, i.e., (b) and (d), corresponds to theoretical amplitudes. The

modes at the left and right columns correspond to the frequencies 1350.8Hz and 1470.4Hz, respectively.

6. ACKNOWLEDGEMENTS

This work was supported by DGAPA-UNAM project PAPIIT IN111307, by CONACyT, Grants No. SEP-2004-C01-47636 and 50308-F and by Subsecretaría de Educación Superior e Investigación Científica, México, Programa deMejoramiento del Profesorado, special Grant “Redes de Cuerpos Académicos 2004”. We thank J. Flores and G. Monsivaisfor their important contribution on this work.

7. REFERENCES

Bardell, N. S., 1994, “Chladni figures for completely free parallelogram plates: an analytical study”, Journal of Soundand Vibration, Vol. 174, pp. 655-676.

Díaz-de-Anda, A., Pimentel, A., Flores, J., Morales, A., Gutiérrez, L., and Méndez-Sánchez, R. A., 2005, “Locallyperiodic Timoshenko rod: experiment and theory”, Journal of the Acoustical Society of America Vol. 117, 2814-2819.

Gorman, D. J., 1999, “Vibration Analysis of Plates by the Superposition Method”, Series on Stability Vibrations andControl of Systems, Vol. 3, World Scientific, Singapore.



Lee, P. C. Y. and Spencer, W. J. 1969, “Shear-Flexure-Twist Vibrations in Rectangular AT-Cut Quartz Plates with PartialElectrodes”, Journal of the Acoustical Society of America, Vol. 45, pp 637-645.

Leissa, A. W., 1993, “Vibration of Plates”, Acoustical Society of America, Woodbury, New York, Vol. 1, p. 115.Low, K. H., Chai, G. B., Lim, T. M. and Sue S. C., 1998, “Comparisons of experimental and theoretical frequencies for

rectangular plates with various boundary conditions and added masses”, International Journal of Mechanical SciencesVol. 40 pp. 1119-1131.

Ma, C. C. and Lin, C. C., 2001, “Experimental investigation of vibrating laminated composite plates by optical interfer-ometry method”, American Institute of Aeronautics and Astronautics Journal, Vol. 39 pp. 491-497.

Ma, C. C., and Lin, H. Y., 2005, “Experimental measurements on transverse vibration characteristics of piezoceramicrectangular plates by optical methods”, Journal of Sound and Vibration Vol. 286 pp. 587-600.

Morales, A., Gutiérrez, L. , and Flores, J., 2001, “Improved eddy current driver-detector for elastic vibrations”, AmericanJournal of Physics Vol. 69 517-522.

Morales, A., Flores, J., Gutiérrez, L. and Méndez-Sánchez, R. A., 2002, “Compressional and torsional wave amplitudesin rods with periodic structures”, Journal of the Acoustical Society of America Vol. 112, 1961-1967.

Nieves, F.J., Gascón, F. and Bayón, A., 2004, “Natural frequencies and mode shapes of flexural vibration of plates:laser-interferometry detection and solutions by Ritz’s method”, Journal of Sound and Vibration Vol. 278 pp. 637-655.

Schaadt, K., Guhr, T., Ellegaard, C., and Oxborrow, M., 2003, “Experiments on elastomechanical wave functions inchaotic plates and their statistical features”, Physical Review E Vol. 68 p. 036205.

Soedel, W., 1993, Vibrations of Shells and Plates, Marcel Dekker, New York, 1993.Szilard, R., 1974, Theory and Analysis of Plates: Classical and Numerical Methods, Prentice-Hall, Englewood Cliffs,

New Jersey, 1974.

8. RESPONSIBILITY NOTICE

The authors are the only responsible for the printed material included in this paper

flexural vibrations of plates: theory and · pdf fileflexural vibrations of plates: theory and...

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