Research Papers

A Perturbation Solution of the Thickness Natural Vibrations of a Piezoelectric Plate

[+] Author and Article Information
Piotr Cupiał

Department of Process Control,
AGH University of Science and Technology,
Al. Mickiewicza 30,
Krakow 30-059, Poland
e-mail: pcupial@agh.edu.pl

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF VIBRATION AND ACOUSTICS. Manuscript received January 11, 2013; final manuscript received September 10, 2013; published online October 23, 2013. Assoc. Editor: Thomas J. Royston.

J. Vib. Acoust 136(1), 011008 (Oct 23, 2013) (8 pages) Paper No: VIB-13-1014; doi: 10.1115/1.4025440 History: Received January 11, 2013; Revised September 10, 2013

This paper discusses a perturbation approach to the calculation of the natural frequencies and mode shapes for both the displacement and the electrostatic potential through-thickness vibration of an infinite piezoelectric plate. The problem is formulated within the coupled theory of linear piezoelectricity. It is described by a set of two coupled differential equations with unknown thickness displacement, the electrostatic potential and a general form of boundary conditions. A consistent perturbation solution to the natural vibration problem is described. An important element not present in the classical eigenvalue perturbation solution is that the small parameter appears in the boundary conditions; a way to handle this complication has been discussed. The natural frequencies and mode shapes obtained using the perturbation approach are compared to exact solutions, demonstrating the effectiveness of the proposed method. The advantage of the perturbation method derives from the fact that coupled piezoelectric results can be obtained from the elastic solution during the postprocessing stage.

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Grahic Jump Location
Fig. 1

A cross section through the plate with a general form of boundary conditions: (a) mechanical and (b) electrical. k1,k2 are the spring constants per unit area, C1,C2 stand for capacitances per unit area.

Grahic Jump Location
Fig. 2

Mode shapes (displacement and electrostatic potential) of a fixed-free plate with electrically shorted faces for (a) first mode, (b) second mode, (c) third mode, and (d) fourth mode

Grahic Jump Location
Fig. 3

Mode shapes of a free-free plate with electrically shorted faces for (a) first mode, (b) second mode, (c) third mode, and (d) fourth mode



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