Modal Piezoelectric Transducers with Shaped Electrodes for Improved Passive Shunt Vibration Control of Smart Piezo-Elastic Beams

Modal control and spatial filtering technologies for vibration and/or structural acoustics radiation mitigation may be implemented through the use of distributed modal piezoelectric transducers with properly shaped electrodes which, in order to increase the robustness and stability of the controlled structural system, turn undesirable mode’s contributions unobservable and uncontrollable over the bandwidth of interest. In addition, distributed modal piezoelectric transducers may also yield a higher generalized modal electromechanical coupling coefficient which is an important design parameter to take into account for a proper and efficient passive shunt damping design. The improvements in passive shunt damping performance when using modal piezoelectric transducers with shaped electrodes are investigated in this article for a two-layered resonant-shunted piezo-elastic smart beam structure. The damping performance of the shunted smart beam with shaped electrodes is investigated and assessed in terms of the generalized modal electromechanical coupling coefficients and frequency responses obtained when considering uniform and modally shaped electrodes, and the underlying improved performance and advantages are assessed and discussed. INTRODUCTION The high applicability and interest on the use of piezoelectric materials for the control of vibration and/or structural acoustics radiation of flexible structures are nowadays well recognized. Piezoelectric transducers are more commonly used under active vibration control frameworks. However, piezoelectric transducers may also be connected to passive electrical circuitry with electrical impedance defined by the resistive, capacitive and inductive elements. In this process energy is converted from mechanical to electrical form in the piezoelectric transducer and by coupling it with a passive electrical network, with properly designed electrical impedance, the converted César M. A. Vasques, INEGI, University of Porto, Campus da FEUP, Rua Dr. Roberto Frias 400, 4200-465 Porto, Portugal. E-mail: [email protected] 6th European Workshop on Structural Health Monitoring Tu.3.C.4 License: http://creativecommons.org/licenses/by/3.0/ 1 energy is conveniently tackled so that stiffness and damping modifications of the structural system occur. Thus, it comes as no surprise that the electromechanical energy conversion efficiency is of paramount importance in this process for a wellsucceeded and damping-efficient approach. This latter strategy was coined as piezoelectric shunt damping [6] and due to its passive nature it strongly differs in simplicity of use and implementation when compared with active control strategies, somewhat avoiding bulky amplifying, signal conditioning and control logic systems [8]. Concepts of modal control and shaped transducers have gained increased popularity. The investigation on design techniques and optimized polarization profiles for distributed shaped sensors in beam [7], plate [2] and shell [3] structures, and the experimental application and investigation of these concepts, for the vibration control of beam [1] and plate [11] structures, have been presented recently. Applications of these technologies may include not only vibration and/or structural acoustics control, but also structural health monitoring [5] and energy harvesting applications, among others. In the context of shunt damping, the vibration suppression of a hard disk driver actuator arm using a topology-optimized piezoelectric transducer was investigated in [10] and the passive damping and exact annihilation of vibrations of beams using shaped piezoelectric layers and tuned inductive networks was investigated in [9]. From the analysis of the open literature, it stands out that sufficient attention has not been given neither to the actuation efficiency and utility of modal actuators nor to the underlying physics of the electromechanical coupling efficiency of piezoelectric transducers with shaped electrodes. Properly designed spatially shaped sensors may yield a sensing behaviour with improved modal electromechanical coupling coefficients, which may be used to increase the shunted damping performance. This last aspect, while addressed to some extent in a few of works [9, 10], was not properly explored in the context of shunt damping applications using modal piezoelectric transducers with shaped electrodes. It represents the main motivation and novelty of this article and is investigated here for beam structures with spatially shaped uniform and single-mode electrode designs of resonant shunted piezoelectric transducers, the main aim being to demonstrate the improved damping efficiency resulting from the use of modally shaped electrode designs for vibration control purposes. An electromechanical one-dimensional equivalent single-layer Euler-Bernoulli analytical model of twolayered smart piezo-elastic beams with arbitrary spatially shaped electrodes is used (see [15] for further details), the damping performance of a shunted smart beam with shaped electrodes is investigated and assessed in terms of the generalized electromechanical coupling coefficient, and frequency responses obtained when considering uniform and modally shaped electrodes and the underlying improved performance and advantages are assessed and discussed. SPATIALLY SHAPED SMART PIEZO-ELASTIC BEAM Let us consider a generic beam structure of total length l with a distributed piezoelectric transducer/layer mounted on one side of the beam (Figure 1). The piezoelectric layer is assumed to be perfectly bonded onto the beam and the electrodes are spatially shaped so that they have an arbitrary width, e 2 ( ) b x , which is non-uniform along the length coordinate of the beam, . x The subscripts b () ⋅ , p () ⋅ and e ( ) ⋅ are used here to denote quantities associated with material and geometric properties of the elastic and piezoelectric layers and with the electrodes, respectively. The cross-section