# Asymptotic and Spectral Analysis of a Model of the Piezoelectric Energy Harvester with the Timoshenko Beam as a Substructure: A Recent Study

The well-known model of a piezoelectric energy harvester is mathematically analysed. The harvester is a cantilever Timoshenko beam with piezoelectric layers on the top and bottom faces. The top and bottom faces of the piezoelectric layers are covered with thin, completely conductive electrodes. A resistive load is linked to these electrodes. A system of three partial differential equations governs the model. The equations for the Timoshenko beam model are the first two, while Kirchhoff’s law for the electric circuit is the third. Due to the piezoelectric effect, all equations are intertwined. In the Hilbert state space of the system, we express the system as a single operator evolution equation. A non-selfad joint matrix differential operator with compact resolvent serves as the dynamics generator for this evolution equation. There are two key conclusions in the paper. Both results are explicit asymptotic formulations for this operator’s eigenvalues, i.e., the electrically loaded system’s modal analysis is conducted. The residual terms in the first set of asymptotic formulas are of the order O(1/n), where n is the number of eigenvalues. For the model with varying physical parameters, these formulas are derived. The second set of asymptotic formulas is obtained for a less general model with constant parameters and has remaining terms of the order O(1/n2). This second set of formulas includes additional terms that are dependent on the model’s electromechanical parameters. The spectrum is demonstrated to break asymptotically into two distinct subsets, the -branch eigenvalues and the -branch eigenvalues. The set of the system’s vibrational modes is obtained by multiplying these eigenvalues by “i.” The asymptotically placed -branch vibrational modes are on the left half of the complex plane, and the -branch is asymptotically near to the imaginary axis. With such spectral and asymptotic results, the asymptotic representation for mode shapes and voltage output may be derived. The analysis of control problems for the harvester relies heavily on the asymptotics of vibrational modes and mode shapes.

**Author (s) Details**

**Marianna Shubov
**Department of Mathematics and Statistics, University of New Hampshire, 33 Academic Way, Durham, NH 03824, USA.

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