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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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References

Tiersten, H. F., 1969, Linear Piezoelectric Plate Vibrations, Plenum Press, New York.
Auld, B. A., 1973, Acoustic Fields and Waves in Solids, Wiley, New York.
Nowacki, W., 1983, Electromagnetic Effects in Deformable Solids, PWN, Warsaw.
Royer, D., and Dieulesaint, E., 2000, Elastic Waves in Solids, Springer, Berlin.
Tiersten, H. F., 1963, “Thickness Vibrations of Piezoelectric Plates,” J. Acoust. Soc. Am., 35, pp. 53–58. [CrossRef]
Mindlin, R. D., 1972, “High Frequency Vibrations of Piezoelectric Crystal Plates,” Int. J. Solids Struct., 8, pp. 895–906. [CrossRef]
Mindlin, R. D., 1974, “Coupled Piezoelectric Vibrations of Quartz Plates,” Int. J. Solids Struct., 10, pp. 453–459. [CrossRef]
Mitchell, J. A., and Reddy, J. N., 1995, “A Refined Hybrid Plate Theory for Composite Laminates With Piezoelectric Laminae,” Int. J. Solids Struct., 32, pp. 2345–2367. [CrossRef]
Dash, P., and Singh, B. N., 2012, “Geometrically Nonlinear Free Vibration of Laminated Composite Plate Embedded With Piezoelectric Layers Having Uncertain Material Properties,” ASME J. Vibr. Acoust., 134(6), pp. 1–13. [CrossRef]
Cupial, P., 2005, “Three-Dimensional Natural Vibration Analysis and Energy Considerations for a Piezoelectric Rectangular Plate,” J. Sound Vib., 283, pp. 1093–1113. [CrossRef]
Heyliger, P., and Saravanos, D. A., 1995, “Exact Free-Vibration Analysis of Laminated Plates With Embedded Piezoelectric Layers,” J. Acoust. Soc. Am., 98, pp. 1547–1557. [CrossRef]
Courant, R., and Hilbert, D., 1953, Methods of Mathematical Physics, Vol. 1, Interscience, New York.
Wigner, E. P., 1959, Group Theory and Its Application to the Quantum Mechanics of Atomic Spectra, Academic Press, New York.
Kato, T., 1966, Perturbation Theory for Linear Operators, Springer, Berlin.
Nayfeh, A. H., 1973, Perturbation Methods, Wiley, New York.
Landau, L. D., and Lifshitz, E. M., 1974, Quantum Mechanics: Non-Relativistic Theory, 3rd ed., Nauka, Moscow.
Friedman, B., 1956, Principles and Techniques of Applied Mathematics, Wiley, New York.
Cupial, P., 2008, Coupled Electromechanical Vibration Problems for Piezoelectric Distributed-Parameter Systems, Monograph 362, Series Mechanics, Politechnika Krakowska, Krakow.

Figures

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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