Analysis of an optimization problem for a piezoelectric energy harvester

Abstract

The problem of optimal energy harvesting for a piezoelectric element driven by mechanical vibrations is stated in terms of an ODE system with hysteresis under the time derivative coupling a mechanical oscillator with an electric circuit with or without inductance. In the piezoelectric constitutive law, both the self-similar piezoelectric butterfly character of the hysteresis curves and feedback effects are taken into account in a thermodynamically consistent way. The physical parameters of the harvester are chosen to be the control variable, and the goal is to maximize the harvested energy for a given mechanical load and a given time interval. If hysteresis is modeled by the Preisach operator, the system is shown to be well-posed with continuous data dependence. For the special case of the play operator, we derive first-order necessary optimality conditions and an explicit form of the gradient of the total harvested energy functional in terms of solutions to the adjoint system.

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Acknowledgements

The authors thank both reviewers for their valuable comments. Moreover, financial support by the GAČR Grant GA15-12227S, RVO: 67985840, and FWF Grant P30054, is gratefully acknowledged.

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Correspondence to Barbara Kaltenbacher.

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Supported by the GAČR Grant GA15-12227S, RVO: 67985840, and FWF Grant P30054.

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Kaltenbacher, B., Krejčí, P. Analysis of an optimization problem for a piezoelectric energy harvester. Arch Appl Mech 89, 1103–1122 (2019). https://doi.org/10.1007/s00419-018-1459-6

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Keywords

  • Hysteresis
  • Piezoelectricity
  • Differential equations
  • Optimal control