###### Abstract  
Mathematical analysis of the well known model of a piezoelectric energy harvester is presented. The harvester is designed as a cantilever Timoshenko beam with piezoelectric layers attached to its top and bottom faces. Thin, perfectly conductive electrodes are covering the top and bottom faces of the piezoelectric layers. These electrodes are connected to a resistive load. The model is governed by a system of three partial differential equations. The first two of them are the equations of the Timoshenko beam model and the third one represents [PERSON]'s law for the electric circuit. All equations are coupled due to the piezoelectric effect. We represent the system as a single operator evolution equation in the Hilbert state space of the system. The dynamics generator of this evolution equation is a non-selfadjoint matrix differential operator with compact resolvent. The paper has two main results. Both results are explicit asymptotic formulas for eigenvalues of this operator, i.e., the modal analysis for the electrically loaded system is performed. The first set of the asymptotic formulas has remainder terms of the order \(O(\frac{1}{n})\), where \(n\) is the number of an eigenvalue. These formulas are derived for the model with variable physical parameters. The second set of the asymptotic formulas has remainder terms of the order \(O(\frac{1}{n^{2}})\), and is derived for a less general model with constant parameters. This second set of formulas contains extra term depending on the electromechanical parameters of the model. It is shown that the spectrum asymptotically splits into two disjoint subsets, which we call the \(\alpha\)-branch eigenvalues and the \(\theta\)-branch eigenvalues. These eigenvalues being multiplied by \"\(i\)\" produce the set of the vibrational modes of the system. The \(\alpha\)-branch vibrational modes are asymptotically located on certain vertical line in the left half of the complex plane and the \(\theta\)-branch is asymptotically close to the imaginary axis. By having such spectral and asymptotic results, one can derive the asymptotic representation for the mode shapes and for voltage output. Asymptotics of vibrational modes and mode shapes is instrumental in the analysis of control problems for the harvester. partial differential equation; boundary--value problem; differential operator; eigenvalues; right reflection matrix; left reflection matrix +
Footnote †: journal: Applied Soft  
counterparts. Piezoelectric materials create an electric charge in response to a mechanical stress (the _direct_ piezoelectric effect) and, in the reciprocal process, these materials develop mechanical strain in response to the electrical potential (the _inverse_ piezoelectric effect). Since the charge generated in the piezoelectric material is proportional to the applied stress, an energy harvester is designed to maximize stress under a certain mechanical load. The most commonly used geometric structure is the cantilever beam, since this structure provides the highest average strain for a given input force. Ambient vibrations are coupled to the cantilever-mass system through the base of the cantilever causing the structure to oscillate. The alternating bending strains, due to oscillations, are converted to an alternating voltage by the piezoelectric material. To maximize the voltage output, ambient vibration frequencies should be close to the natural oscillation frequency of the beam. In the present work, we analyze a mathematical model of the harvester. This model consists of a system of three coupled partial differential equations describing the dynamics of a cantilever Timoshenko beam whose top and bottom faces are covered by very thin piezoelectric layers. The system of equations is equipped with the standard sets of the boundary and initial conditions (see Equations (1)-(5) of Section 2).