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Research Papers

Nonsymmetric Nonlinear Dynamics of Piezoelectrically Actuated Beams

[+] Author and Article Information
Mergen H. Ghayesh

School of Mechanical Engineering,
University of Adelaide,
South Australia 5005, Australia
e-mail: mergen.ghayesh@adelaide.edu.au

Hamed Farokhi

Department of Mechanical and Construction Engineering,
Northumbria University,
Newcastle upon Tyne NE1 8ST, UK
e-mail: hamed.farokhi@northumbria.ac.uk

1Corresponding author.

Contributed by the Technical Committee on Vibration and Sound of ASME for publication in the Journal of Vibration and Acoustics. Manuscript received August 14, 2018; final manuscript received May 5, 2019; published online June 14, 2019. Assoc. Editor: Mohammed Daqaq.

J. Vib. Acoust 141(5), 051012 (Jun 14, 2019) (11 pages) Paper No: VIB-18-1350; doi: 10.1115/1.4043716 History: Received August 14, 2018; Accepted May 06, 2019

The nonlinear behavior of a piezoelectrically actuated clamped–clamped beam has been examined numerically while highlighting the nonsymmetric response of the system. The nonlinearly coupled electromechanical model of the piezoelectric beam system is developed employing the Bernoulli–Euler theory along with the piezoelectric stress–voltage equations. A general nonsymmetric configuration is considered with a piezoelectric patch partially covering the beam. The geometric nonlinearities of stretching type are taken into account for both the piezoelectric patch and the beam. Through use of the generalized Hamilton's principle, the nonlinearly coupled electromechanical equations of transverse and longitudinal motions of the piezoelectrically actuated beam are derived. A high-dimensional Galerkin scheme is utilized to recast the equations of partial differential type into ordinary differential type. For comparison and benchmark purposes, a three-dimensional finite element model is developed using abaqus/cae to verify the model developed in this study. It is shown that the response of the system is strongly nonsymmetric and that it is essential to retain many degrees-of-freedom to ensure converged results.

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Figures

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Fig. 1

Schematic of a piezoelectrically actuated clamped–clamped beam

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Fig. 2

Nonlinear static deflection of the piezoelectrically actuated beam obtained via 3D FEM (symbols) and the beam model developed in this study (solid line), at z = tb/2: (a) and (b) the transverse displacement under 5 kV and 10 kV actuations, respectively; (c) and (d) the longitudinal displacement under 5 kV and 10 kV actuations, respectively

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Fig. 3

The nonlinear static transverse displacement at x/L = 0.3 obtained via 3D FEM (symbols) and the beam model developed in this study (solid line)

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Fig. 4

FE analysis results: (a) and (b) the contour plots of the transverse displacement

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Fig. 5

Primary resonance frequency–amplitude plots of the piezoelectrically actuated beam when Vd = 13.0 V: (a) maximum transverse displacement at x/L = 0.56 and (b) maximum longitudinal displacement at x/L = 0.66

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Fig. 6

Frequency–amplitude plots of the system of Fig. 5, showing maximum amplitudes of the generalized coordinates q1, q2, q3, and r1

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Fig. 7

Motion of the system of Fig. 5 at Ωp/ω1 = 1.0373, with (a) and (b) showing the time trace and phase-plane portrait of the transverse displacement and (c) and (d) showing those of the longitudinal displacement

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Fig. 8

Motion of the system of Fig. 5 at Ωp/ω1 = 1.1558, with (a) and (b) showing the time trace and phase-plane portrait of the transverse displacement and (c) and (d) showing those of the longitudinal displacement

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Fig. 9

Force–amplitude plots of the piezoelectrically actuated beam when Ωp/ω1 = 1.04: (a) maximum transverse displacement at x/L = 0.56 and (b) maximum longitudinal displacement at x/L = 0.66

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Fig. 10

Force–amplitude plots of the system of Fig. 9, showing maximum amplitudes of the generalized coordinates q1, q2, q3, and r1

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Fig. 11

Secondary resonance frequency–amplitude plots of the piezoelectrically actuated beam when Vd = 13.0 V: (a) maximum transverse displacement at x/L = 0.81 and (b) maximum longitudinal displacement at x/L = 0.34

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Fig. 12

Frequency–amplitude plots of the system of Fig. 11, showing maximum amplitudes of the generalized coordinates q1, q2, q3, and r1

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Fig. 13

Motion of the system of Fig. 11 at Ωp/ω2 = 1.1164, with (a) and (b) showing the time trace and phase-plane portrait of the transverse displacement and (c) and (d) showing those of the longitudinal displacement

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Fig. 14

Nonlinear transverse deformation of the system obtained via 3D FEM model and various discretized models

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