A comparative study of smart passive piezoelectric structures interacting with electric networks: Timoshenko beam theory versus finite element plane stress calculations

The present paper is devoted to the coupling between mechanical, piezoelectric and electric fields in piezoelastic beams whose electrodes are connected by electric circuits with arbitrary impedances. Taking advantage of the linear piezoelectric constitutive relations, the kinematic assumptions according to Timoshenko, the Gauss law of electrostatics as well as Kirchhoff's voltage rule, the equations of motion for lateral vibrations are derived in the Laplace domain considering the direct and converse piezoelectric effects. For beams with layerwise constant width, the governing equations are similar to the well-known Timoshenko beam equations, but the bending moment is extended by a so-called induced electric moment, which is influenced by the electric circuits. In order to demonstrate the interaction of mechanical and electrical fields, the theory that we develop is compared against plane stress finite element computations using ANSYS. For flexural vibrations with various mechanical boundary conditions and electric impedances, an eigenfrequency analysis is performed and mechanical and electrical field variables are compared for different harmonic excitations.