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.

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The present paper is devoted to the development of an extended Bernoulli-Euler beam theory for passive piezoelectric composite structures which takes into account the presence of electric networks. The theory considers electromechanical coupling between the beam deformation and the electric circuit due to the piezoelectric effect that relates mechanical properties like displacement, strain and stress to electrical properties like electric field, voltage and current. Thereby, kinematic relations within the Bernoulli-Euler theory, a one-dimensional form of the constitutive relations for piezoelastic structures and a linear electric network, are presumed.Eventually, an adjusted one-dimensional formulation of a beam theory is obtained. It is shown that this formulation can be used for both power harvesting and passive shunt damping applications. Within the presented theory it is possible to analyse the influence of geometrical dimensions, piezoelectric constants and impedances of electric networks on the displacement field and on the energy flow between the mechanical and electrical parts. The second part of the paper is devoted to the concept of shape control and its application to passive damping and exact annihilation of vibrations of beams using shaped piezoelectric layers and tuned inductive networks. As a main result of the present paper, it is shown that, under certain conditions, concerning the shape of the piezoelastic material and the impedances of the electric circuits, exact annihilation of vibrations for a cantilever beam is possible.

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