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Research Papers

Nonlinear Forced Vibrations of Laminated Piezoelectric Plates

[+] Author and Article Information
Muhammad Tanveer

Department of Mechanical and Materials Engineering, University of Western Ontario, London, ON, N6A 5B9, Canada

Anand V. Singh1

Department of Mechanical and Materials Engineering, University of Western Ontario, London, ON, N6A 5B9, Canadaavsingh@eng.uwo.ca

1

Corresponding author.

J. Vib. Acoust 132(2), 021005 (Mar 16, 2010) (13 pages) doi:10.1115/1.4000768 History: Received August 11, 2008; Revised November 26, 2009; Published March 16, 2010; Online March 16, 2010

A numerical approach is presented for linear and geometrically nonlinear forced vibrations of laminated composite plates with piezoelectric materials. The displacement fields are defined generally by high degree polynomials and the convergence of the results is achieved by increasing the degrees of polynomials. The nonlinearity is retained with the in-plane strain components only and the transverse shear strains are kept linear. The electric potential is approximated layerwise along the thickness direction of the piezoelectric layers. In-plane electric fields at the top and bottom surfaces of each piezoelectric sublayer are defined by the same shape functions as those used for displacement fields. The equation of motion is obtained by the Hamilton’s principle and solved by the Newmark’s method along with the Newton–Raphson iterative technique. Numerical procedure presented herein is validated by successfully comparing the present results with the data published in the literature. Additional numerical examples are presented for forced vibration of piezoelectric sandwich simply supported plates with either a homogeneous material or laminated composite as core. Both linear and nonlinear responses are examined for mechanical load only, electrical load only, and the combined mechanical and electrical loads. Displacement time histories with uniformly distributed load on the plate surface, electric volts applied on the top and bottom surfaces of the piezoelectric plates, and mechanical and electrical loads applied together are presented in this paper. The nonlinearity due to large deformations is seen to produce stiffening effects, which reduces the amplitude of vibrations and increases the frequency. On the contrary, antisymmetric electric loading on the nonlinear response of piezoelectric sandwich plates shows increased amplitude of vibrations.

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Copyright © 2010 by American Society of Mechanical Engineers
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Figures

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Figure 1

Piezoelectric sublayers

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Figure 2

Front view of rectangular bimorph plate

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Figure 3

Front view of rectangular sandwich piezoelectric plate

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Figure 4

Front view of square laminated composite piezoelectric plate

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Figure 5

Nonlinear central displacements versus electric potentials for simply supported sandwich square plate with PZT5 as facings. Aluminum as core: (solid line) present method and (x) Ref. 17; cross-ply [02/902]s: (dashed line) present method and (+) Ref. 17.

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Figure 6

Displacement and electric potential time histories for PZT5/[020/9020]s/PZT5 subjected to two values of uniformly distributed load: 1.7 kN/m2 (solid line) linear and (solid line with o) nonlinear; 3.0 kN/m2 (dotted line) linear and (dotted line with +) nonlinear. (a) Linear and nonlinear displacement time histories. (b) Linear and nonlinear electric potential time histories.

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Figure 7

Electric potentials along the thickness of PZT5/[020/9020]s/PZT5 subjected to uniformly distributed load of 3.0 kN/m2: at time 0.00037 s (solid line), linear and at time 0.0003 s (dashed line), nonlinear

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Figure 8

Displacement time histories for PZT5/[020/9020]s/PZT5 subjected to electric potential of two values: 120 V (solid line) linear and (solid line with o) nonlinear; 200 V (dotted line) linear and (dotted line with +) nonlinear

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Figure 9

Electric potential time histories for PZT5/[020/9020]s/PZT5 subjected to electric potential of 200 V: (solid line) linear and (dotted line) nonlinear

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Figure 10

Electric potentials along the thickness of PZT5/[020/9020]s/PZT5 subjected to electric potential of 200 V at time 0.00037 s (solid line), linear and at time 0.0003 s (dashed line), nonlinear

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Figure 11

Displacement time histories for square plate subjected to suddenly applied uniformly distributed load of 2.2 kN/m2 and a uniform electric potential of 140 V for the duration of 0.02 s from 0.02 s to 0.04 s: (solid line) PZT5/[00/900/00/900]s/PZT5 and (dashed line) PZT5/[450/−450]4/PZT5. (a) Linear displacement time histories. (b) Nonlinear displacement time histories.

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Figure 12

Nonlinear electric potential time histories for square plate subjected to suddenly applied uniformly distributed load of 2.2 kN/m2 and a uniform electric potential of 140 V for the duration of 0.02 s from 0.02 s to 0.04 s: (solid line) PZT5/[00/900/00/900]s/PZT5 and (dotted line) PZT5/[450/−450]4/PZT5

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Figure 13

Displacement time histories for square plate PZT5/[00/900/00/900]s/PZT5 subjected to uniform electric potential of 700 V: (dotted line) linear response, (solid line) nonlinear response with [Kme]NL, and (dashed line) nonlinear response without [Kme]NL

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