• Smart Structures and Systems
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• v.18 no.2
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• pp.335-354
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• 2016

This contribution presents an extended one-dimensional theory for piezoelectric beam-type structures with non-ideal electrodes. For these types of electrodes the equipotential area condition is not satisfied. The main motivation of our research is originated from passive vibration control: when an elastic structure is covered by several piezoelectric patches that are linked via resistances and inductances, vibrational energy is efficiently dissipated if the electric network is properly designed. Assuming infinitely small piezoelectric patches that are connected by an infinite number of electrical, in particular resistive and inductive elements, one obtains the Telegrapher's equation for the voltage across the piezoelectric transducer. Embedding this outcome into the framework of Bernoulli-Euler, the final equations are coupled to the wave equations for the longitudinal motion of a bar and to the partial differential equations for the lateral motion of the beam. We present results for the wave propagation of a longitudinal bar for several types of electrode properties. The frequency spectra are computed (phase angle, wave number, wave speed), which point out the effect of resistive and inductive electrodes on wave characteristics. Our results show that electrical damping due to the resistivity of the electrodes is different from internal (=strain velocity dependent) or external (=velocity dependent) mechanical damping. Finally, results are presented, when the structure is excited by a harmonic single force, yielding that resistive-inductive electrodes are suitable candidates for passive vibration control that might be of great interest for practical applications in the future.

• Smart Structures and Systems
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• v.18 no.2
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• pp.355-374
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• 2016

This paper presents and compares a one-dimensional (1D) bending theory for piezoelectric thin beam-type structures with resistive-inductive electrodes to ANSYS$^{(R)}$ three-dimensional (3D) finite element (FE) analysis. In particular, the lateral deflections and vibrations of slender piezoelectric beams are considered. The peculiarity of the piezoelectric beam model is the modeling of electrodes in such a manner that is does not fulfill the equipotential area condition. The case of ideal, perfectly conductive electrodes is a special case of our 1D model. Two-coupled partial differential equations are obtained for the lateral deflection and for the voltage distribution along the electrodes: the first one is an extended Bernoulli-Euler beam equation (second-order in time, forth order in space) and the second one the so-called Telegrapher's equation (second-order in time and space). Analytical results of our theory are validated by 3D electromechanically coupled FE simulations with ANSYS$^{(R)}$. A clamped-hinged beam is considered with various types of electrodes for the piezoelectric layers, which can be either resistive and/or inductive. A natural frequency analysis as well as quasi-static and dynamic simulations are performed. A good agreement between the extended beam theory and the FE results is found. Finally, the practical relevance of this type of electrodes is shown. It is found that the damping capability of properly tuned resistive or resistive-inductive electrodes exceeds the damping performance of beams, where the electrodes are simply linked to an optimized impedance.