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Dynamics of a Longitudinal Piezoelectric Beam

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Piezoelectric materials, actuated by voltage [1, 2], charge or current [3], can exhibit electric responses to mechanical stress and vice versa. Piezoelectric materials can be used as actuators/sensors or be integrated into a mother host structure. Some known piezoelectric materials are quartz (), lead zirconate titanate (), barium titanate ) and bismuth ferrite () [3].

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Consider a beam of piezoelectric material, clamped on one side and free to oscillate on the other, covered by electrodes at the top and bottom surfaces. These electrodes are connected through a circuit. The beam has length and thickness . Assume that the transverse oscillations of the beam are negligible, so that longitudinal vibrations, in the form of expansions and compressions of the center line of the beam, are the only oscillations to consider. By the dynamic electromagnetic implied by Maxwell's equations [1, 2], the total charge accumulated on the electrodes of the beam is strongly coupled to the longitudinal vibrational dynamics of a voltage-actuated piezoelectric beam. This can be represented by the following PDE model

(1)

where , , , denote mass density, magnetic permeability, elastic stiffness, piezoelectric coefficient and permittivity, respectively, is the viscous damping coefficient, and are strain and voltage actuators and specify the initial position, total charge, velocity and total current of the piezoelectric beam for any . To suppress all the vibrations on the beam, state feedback is chosen where are the sensor signal amplifiers for the tip velocity and total current accumulated at the electrodes of the beam.

Since , there is a huge difference in the wave propagation speed of the elastic vibrations and the speed of the electrical vibrations (close to the speed of light). For realistic material parameters [4, 5], it is not computationally feasible to simulate the dynamics of (1). Instead, electrostatic and quasi-static approaches are used to approximate the overall vibrational dynamics [1, 4, 5]. This is equivalent to discarding the term and in (1) so that (1) reduces to the electrostatic/quasi-static model

(2)

where , refer to the total current amplifier with the signal amplifier . For simplicity, and are taken unity.

The total energy of solutions of (2) decays to the equilibrium state exponentially fast (see [1, 3] and the references therein).

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Authored by: Jacob Walterman, Ahmet Kaan Aydin, Samuel Leveridge and Ahmet Özkan Özer (June 13)
Open content licensed under CC BY-NC-SA

Details

The yellow strips on the top and bottom surfaces of the gray beam are perfectly bonded electrodes. The electrodes are attached through an electrical circuit in which the voltage is controlled. The centerline of the beam is compressing and stretching during the motion, depending on the applied voltage. The total current data accumulated at the electrodes is collected and fed back to the strain actuator. Indeed, the amount of total charge accumulated at the electrodes is proportional to the strains through the piezoelectric coefficient . For simplicity, it is taken as [4].

For this Demonstration, the length is taken and a finite-difference based algorithm is implemented with the equal meshing of [0, 1]:

with and .

Two types of initial conditions for both and are considered, linear and oscillatory, respectively, and where , and is the ratio of applied stretching or compression initially proportional to the beam length. Therefore, the maximum amount of stretching or compression can be set to 15% of the beam length. The amount of voltage accumulated at the electrodes, sensor and actuator data, and the total energy (normalized by the energy of the initial conditions) are also shown for .

This material is based upon the work supported by the National Science Foundation under Cooperative Agreement No. 1849213. Any opinions, findings and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation.

References

[1] K. A. Morris and A. Ö. Özer, "Modeling and Stabilizability of Voltage-Actuated Piezoelectric Beams with Magnetic Effects," SIAM Journal on Control and Optimization, 52(4), 2014 pp. 2371–2398. doi:10.1137/130918319

[2] A. Ö. Özer, "Further Stabilization and Exact Observability Results for Voltage-Actuated Piezoelectric Beams with Magnetic Effects," Mathematics of Control, Signals, and Systems, 27(2), 2015 pp. 219–244. doi:10.1007/s00245-020-09665-4.

[3] A. Ö. Özer, "Stabilization Results for Well-Posed Potential Formulations of a Current-Controlled Piezoelectric Beam and Their Approximations," Applied Mathematics and Optimization, 84(1), 2021 pp. 877–914. doi:10.1007/s00245-020-09665-4.

[4] A. Erturk and D. J. Inman, Piezoelectric Energy Harvesting, Chichester: Wiley, 2011.

[5] J. Yang, An Introduction to the Theory of Piezoelectricity, New York: Springer, 2005.

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