Flexural vibrations of a rectangular plate for the lower normal modes B. Manzanares-Martı ´nez a , J. Flores b , L. Gutie ´ rrez c , R.A. Me ´ ndez-Sa ´ nchez c,Ã , G. Monsivais b , A. Morales c , F. Ramos-Mendieta d a Divisio ´n de Ciencias e Ingenierı ´a, Universidad de Sonora. Blvd. Lazaro Cardenas #100, 85880 Navojoa, Sonora, Mexico b Instituto de Fı ´sica, Universidad Nacional Auto ´noma de Me´xico, A.P. 20-364, 01000 Me´xico, D.F., Mexico c Instituto de Ciencias Fı ´sicas, Universidad Nacional Auto ´noma de Me´xico, A.P. 48-3, 62251 Cuernavaca, Morelos, Mexico d Departamento de Investigacio ´n en Fı ´sica, Universidad de Sonora, A.P. 5-088, 83000 Hermosillo, Sonora, Mexico article info Article history: Received 30 March 2009 Received in revised form 11 May 2010 Accepted 10 June 2010 Handling Editor: M.P. Cartmell Available online 14 July 2010 abstract Theoretical and experimental results for flexural waves of a rectangular plate with free ends are obtained. Both the natural frequencies and mode shapes are analyzed for the lower normal modes. To take into account the boundary conditions, a plane wave expansion method is used to solve the thin plate theory also known as the 2D Kirchhoff–Love equation. The excitation and detection of the normal modes of the out- of-plane waves are performed using non-contact electromagnetic-acoustic transducers. We conclude that this experimental technique is highly reliable due to the good agreement between theory and experiment. & 2010 Elsevier Ltd. All rights reserved. 1. Introduction Vibrating plates are important in several engineering applications. Among many others, they are structural components of great importance; second, piezoelectric plates are used as oscillators in some electronic systems, and furthermore vibrating plates are also of importance in the analysis and construction of musical instruments. Therefore the normal modes of plates are of wide interest. For these reasons, extensive studies of the normal modes of plates of various shapes and for several boundary conditions have been performed. These studies are, however, mainly theoretical or numerical [1–5], and experimental studies are scarce [6–11]. The latter can be classified by the method used: X-ray diffraction topography [6], measurements with accelerometers or mechanical transducers [7,8], using TV-holographic systems, laser interferometry or speckle pattern interferometry [9–11]. Among these, the interferometric methods have the advantage of being non-contact techniques. However, additional work is needed to establish them as precise and controllable techniques [12]. We therefore perform the measurements with an electromagnetic-acoustic transducer (EMAT) which is very selective, made with commercial components and inexpensive. Compared with the interferometric methods these transducers have some advantages. For instance, to study in-plane oscillations our method requires only a slight change of the EMAT configuration used previously [13–17], whereas in the interferometric method a completely different configuration, i.e. the speckle configuration, is needed. Furthermore, our non-contact device can be used both as an exciter or as a detector, whereas in interferometric methods it is only the detection that is performed without any mechanical contact. Our method also gives directly the phases of the wave amplitudes. We therefore conclude that using the electromagnetic-acoustic transducers that we have developed is more efficient than the methods previously used in the study of vibrations of plates. We should point out, however, that our method requires a longer time for each measurement since a point by point Contents lists available at ScienceDirect journal homepage: www.elsevier.com/locate/jsvi Journal of Sound and Vibration 0022-460X/$ - see front matter & 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.jsv.2010.06.010 Ã Corresponding author. Tel.: + 52 55 56 22 77 88; fax: + 52 55 56 22 77 75. E-mail address: mendez@fis.unam.mx (R.A. Me ´ ndez-Sa ´ nchez). Journal of Sound and Vibration 329 (2010) 5105–5115