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Nonlinear vibrations of a sandwich piezo-beam system under piezoelectric actuation

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Abstract

The paper describes the use of active structures technology for deformation and nonlinear free vibrations control of a simply supported curved beam with upper and lower surface-bonded piezoelectric layers, when the curvature is a result of the electric field application. Each of the active layers behaves as a single actuator, but simultaneously the whole system may be treated as a piezoelectric composite bender. Controlled application of the voltage across piezoelectric layers leads to elongation of one layer and to shortening of another one, which results in the beam deflection. Both the Euler–Bernoulli and von Karman moderately large deformation theories are the basis for derivation of the nonlinear equations of motion. Approximate analytical solutions are found by using the Lindstedt–Poincaré method which belongs to perturbation techniques. The method makes possible to decompose the governing equations into a pair of differential equations for the static deflection and a set of differential equations for the transversal vibration of the beam. The static response of the system under the electric field is investigated initially. Then, the free vibrations of such deformed sandwich beams are studied to prove that statically pre-stressed beams have higher natural frequencies in regard to the straight ones and that this effect is stronger for the lower eigenfrequencies. The numerical analysis provides also a spectrum of the amplitude-dependent nonlinear frequencies and mode shapes for different geometrical configurations. It is demonstrated that the amplitude–frequency relation, which is of the hardening type for straight beams, may change from hard to soft for deformed beams, as it happens for the symmetric vibration modes. The hardening-type nonlinear behaviour is exhibited for the antisymmetric vibration modes, independently from the system stiffness and dimensions.

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Acknowledgements

This article is an extended version of the paper originally presented at 16th International Conference “Dynamical Systems—Theory and Applications”, DSTA 2021, Łódź, Poland.

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The authors declare that no funds, grants, or other support was received during the preparation of this manuscript.

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Authors and Affiliations

  1. Częstochowa University of Technology, Dąbrowskiego 73, 42-201, Częstochowa, Poland

    Jacek Przybylski

  2. Division of Mechanics and Machine Design Fundamentals, Faculty of Mechanical Engineering and Computer Science, Częstochowa University of Technology, Dabrowskiego 71, 42-201, Częstochowa, Poland

    Jacek Przybylski

  3. Department of Civil Engineering, Faculty of Civil Engineering, Częstochowa University of Technology, Akademicka 3 Street, 42-201, Częstochowa, Poland

    Krzysztof Kuliński

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  1. Jacek Przybylski
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Jacek Przybylski and Krzysztof Kuliński contributed to methodology, investigation and numerical calculations, provided resources, supervised the study, wrote the original draft and was involved in editing.

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Correspondence to Jacek Przybylski.

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Przybylski, J., Kuliński, K. Nonlinear vibrations of a sandwich piezo-beam system under piezoelectric actuation. Nonlinear Dyn 109, 689–706 (2022). https://doi.org/10.1007/s11071-022-07477-5

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