Vibrations of a piezoelectric Timoshenko beam with resistive-inductive electrodes

This paper presents a one-dimensional theory for moderately thick piezoelectric beam-type structures with imperfect resistive electrodes. For practical applications, a special goal is also the finite element discretization of the electromechanically coupled partial differential equations, which combine the Telegrapher's equations with the elastic properties of a Timoshenko beam. Unlike ideal electrodes, which satisfy the equipotential area condition, the voltage distribution in resistive electrodes is governed by the diffusion equation. For the electrical domain, Kirchhoff's voltage and current rules are applied to derive the parabolic differential equation, which is driven by the time derivative of the axial strain. It is demonstrated that the current flow through the electrodes of the piezoelectric layer depends on the electrode resistance and the capacitance. For the mechanical domain, d'Alembert's principle is combined with the piezoelectric constitutive equations to derive an extended version of the Timoshenko beam equations, incorporating the x-dependent voltage drop across the electrodes. A one-dimensional finite element is then formulated using Timoshenko shape functions for the deflection and the rotation angle, along with linear shape functions for the voltage drop along the beam segment. For the validation of the model a clamped-hinged piezoelectric beam is used as a benchmark example to compare the results of the one-dimensional discretization with two-dimensional finite element (FE) simulations. Various types of resistive electrodes are considered, including static deflections, dynamic vibrations, and eigenfrequency analyses. The results demonstrate that the derived piezoelectric beam model also includes the case of ideal electrodes (short- and open-circuited), when the sheet resistance is very low, and the case of a non-electroded piezoelectric beam, when the sheet resistance is very high.