Modeling of slender laminated piezoelastic beams with resistive electrodes-comparison of analytical results with three-dimensional finite element calculations

In this work a theory for a slender piezoelectric laminated beam taking into account lossy electrodes is developed. For the modeling of the bending behavior of the beam with conductivity, the kinematical assumptions of Bernoulli-Euler and a simplified form of the Telegraph equations are used. Applying d'Alembert's principle, Gauss' law of electrostatics and Kirchhoff's voltage and current rules, the partial differential equations of motion are derived, describing the bending vibrations of the beam and the voltage distribution and current flow along the resistive electrodes. The theory is valid for applications that are used for actuation and for sensing. In the first case the voltage at a certain location on the electrodes is prescribed and the beam is deformed, whereas in the second case the structure is excited by a distributed external load and the voltage distribution is a result of the structural deformation. For a bimorph with constant width and constant material properties the beam is governed by two coupled partial differential equations for the elastic deformation and for the voltage distribution: the first one is an extension of the Bernoulli-Euler equation of an elastic beam, the second one is a diffusion equation for the voltage. The analytical results of the developed theory are validated by means of three-dimensional electromechanically coupled finite element simulations with ANSYS 11.0. Different mechanical and electrical boundary conditions and resistances of the electrodes are considered in the numerical case study. Eigenfrequencies are compared and the frequency responses of the mechanical and electrical quantities show a good agreement between the proposed beam theory and FE results.