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Topology optimization of 2DOF piezoelectric plate energy harvester under external in-plane force

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Abstract

In this paper, the goal is to design a two degrees of freedom piezoelectric plate energy harvester which can harvest the energy from external in-plane harmonic force coming from different directions. The most challenging problem in this case is the charge cancellation due to combination of tension and compression in different parts of the plate. Therefore, topology optimization method is utilized to find the best possible layout and polarization profile of the piezoelectric plate to maximize the electrical output and to overcome the problem of charge cancellation. To do so, a detailed two dimensional finite element modelling of the piezoelectric material suitable for topology optimization is presented primarily. The topology optimization algorithm is established based on the finite element model to have minimum amount of numerical instabilities. To follow the optimized polarization profile, the electrode in top surface of the piezoelectric plate is separated to two sections that can have potentials with different sign on the same surface. Numerical simulation by COMSOL Multiphysics finite element software and experimental investigation on the fabricated designs demonstrated that the optimized design is highly superior to the classical full plate in terms of produced voltage and electrical power while having less volume of piezoelectric material.

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Acknowledgments

This work has been supported by the national CODE-TRACK project (ANR-17-CE05-0014-01, Control theory tools for optimal design of piezoelectric energy harvesters devoted to birds tracking devices). This work has also been partially supported by the Bourgogne Franche-Comté region project COMPACT.

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Authors and Affiliations

  1. Laboratoire Génie de Production, Nationale School of Engineering in Tarbes (ENIT), Toulouse INP, University of Toulouse, 47, Avenue d’Azereix, Tarbes, France

    Abbas Homayouni-Amlashi & Micky Rakotondrabe

  2. FEMTO-ST Institute, Université Bourgogne Franche-Comté, CNRS, Besançon, 25000, France

    Abbas Homayouni-Amlashi & Abdenbi Mohand-Ousaid

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  1. Abbas Homayouni-Amlashi
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  2. Abdenbi Mohand-Ousaid
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  3. Micky Rakotondrabe
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Correspondence to Abbas Homayouni-Amlashi.

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Appendix: Plane-stress assumption for piezoelectric plate

Appendix: Plane-stress assumption for piezoelectric plate

Based on the full 3D piezoelectric constitutive equation [17] the mechanical stiffness matrix, piezoelectric matrix and permittivity matrix for a transverse isotropic material like PZT class can be written as

$$ \begin{array}{@{}rcl@{}} c^{E}=\left[\begin{array}{cccccc} c_{11}^{E} & c_{12}^{E} & c_{13}^{E} & 0 & 0 & 0\\ c_{12}^{E} & c_{11}^{E} & c_{13}^{E} & 0 & 0 & 0\\ c_{13}^{E} & c_{13}^{E} & c_{33}^{E} & 0 & 0 & 0\\ 0 & 0 & 0 & c_{44}^{E} & 0 & 0\\ 0 & 0 & 0 & 0 & c_{44}^{E} & 0\\ 0 & 0 & 0 & 0 & 0 & c_{66}^{E} \end{array}\right] \\ e=\left[\begin{array}{cccccc} 0 & 0 & 0 & 0 & e_{15} & 0\\ 0 & 0 & 0 & e_{15} & 0 & 0\\ e_{31} & e_{31} & e_{33} & 0 & 0 & 0 \end{array}\right] \\ \varepsilon^{S}=\left[\begin{array}{ccc} \varepsilon_{11}^{S} & 0 & 0\\ 0 & \varepsilon_{11}^{S} & 0\\ 0 & 0 & \varepsilon_{33}^{S} \end{array}\right] \end{array} $$
(27)

Now by considering the plane-stress assumption all the nominal stresses perpendicular to the xy plane is zero [11]. Therefore, the elements of the reduced order matrices mentioned in Eq. 2 can be derived as [12]

$$ \begin{array}{@{}rcl@{}} c_{11}^{*E}=c_{11}^{E}-\frac{(c_{13}^{E})^{2}}{c_{33}^{E}}, \qquad c_{12}^{E*}=c_{12}^{E}-\frac{(c_{13}^{E})^{2}}{c_{33}^{E}} \\ c_{22}^{*E}=c_{22}^{E}-\frac{(c_{13}^{E})^{2}}{c_{33}^{E}}\qquad c_{66}^{*E}=c_{66}^{E} \\ e_{31}^{*}=e_{31}-\frac{c_{13}^{E}e_{33}}{c_{33}^{E}}, \qquad \varepsilon_{33}^{*S}=\varepsilon_{33}^{S}+\frac{e_{33}^{2}}{c_{33}^{E}} \end{array} $$
(28)

The coefficients of the PZT-5H which is fabricated for experimental investigation and the calculated plane-stress assumption is mentioned in Table 2.

Table 2 Piezoelectric coefficients
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Homayouni-Amlashi, A., Mohand-Ousaid, A. & Rakotondrabe, M. Topology optimization of 2DOF piezoelectric plate energy harvester under external in-plane force. J Micro-Bio Robot 16, 65–77 (2020). https://doi.org/10.1007/s12213-020-00129-0

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