IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. 54, NO. 5, OCTOBER 2005 1881 Finite-Element Analysis of the Frequency Response of a Metallic Cantilever Coupled With a Piezoelectric Transducer Luca Dalessandro, Student Member, IEEE, and Daniele Rosato, Student Member, IEEE Abstract—This paper is devoted to modeling the dynamic re- sponse of an electromechanical system consisting of a piezoelectric transducer glued on part of the upper surface of a metallic can- tilever. The piezo works both as vibration sensor and as actuator, and the system is the basis of many vibration-control devices of current interest. A three–dimensional (3-D) finite-element method (FEM) model that reproduces the physical system is proposed, and its advantages with respect to an analytical approach and to one-di- mensional (1-D) and two-dimensional (2-D) FEM models are dis- cussed. In sensor mode, the frequency response in terms of the voltage at the electrodes is drawn; while in actuator mode, the fre- quency response of acceleration and displacement at the free end of the cantilever is calculated. The 3-D model has been verified through the comparison with the results from the experiment car- ried out at the University of L’Aquila, Italy. Furthermore, exper- imental inaccessible quantities such as stresses at the piezo–can- tilever interface are computed in both modes as a preliminary step in the study of delamination phenomena and their impact on the performance of the system in vibration-control applications. Index Terms—Debonding, electromechanical system three-di- mensional (3-D) modeling, finite elements, frequency analysis, piezoelectric transducer. I. INTRODUCTION P IEZOELECTRIC materials possess intrinsic electro- mechanical coupling effects, by virtue of which they have found extensive applications in smart devices such as electromechanical actuators and transducers. They are largely used in active vibration control and noise suppression of sensors in structures of different scale: rockets, weapon sys- tems, smart skin systems of submarines, and so on [1]–[3]. Piezoelectric plates appropriately assembled with structural elements can make up proper micromachines, for instance microelectromechanical systems (MEMS) or microsystems technology (MST), that have applications ranging from recent technologies of biomedical engineering to silicon technology [4]–[7]. Adhesive bonding of two piezoelectric bars (with one Manuscript received August 26, 2004; revised May 5, 2005. This work was supported by the Max Planck Society, Germany. L. Dalessandro was with the Max Planck Institute for Mathematics in the Sci- ences, D-04103 Leipzig, Germany. He is now with the Power Electronic Sys- tems Laboratory, Swiss Federal Institute of Technology (ETH) Zurich, CH-8092 Zurich, Switzerland (e-mail: dalessandro@lem.ee.ethz.ch). D. Rosato was with the Max Planck Institute for Mathematics in the Sci- ences, D-04103 Leipzig, Germany. He is now with the Electrical and Electronic Engineering Department (DEE), Politecnico di Bari, I-70125 Bari, Italy, and is also with the Institute of Applied Mechanics, University of Stuttgart, D-70500 Stuttgart, Germany (e-mail: rosato@mechbau.uni-stuttgart.de). Digital Object Identifier 10.1109/TIM.2005.853677 of them transversally and the other longitudinally polarized) allows one to realize an alternating voltage transformer, which has good features to work for high voltages and low currents. These are the fundamental requirements to supply devices such as cathode ray tubes (CRTs) or particle accelerators. In these applications, where piezoelectric devices are glued to the oscillating support, it is often necessary to assess ex- perimentally inaccessible quantities, such as interlaminar me- chanical stresses, electric potential, and field within the mate- rial, the knowledge of which is crucial to optimize the perfor- mance of the device and in order to assure correct working of the system. Sun et al. [8], Holnicki–Szulc, and Marzec [9] address the technological problem of debonding, i.e., the disjunction of the piezoelectric lamina from the host structure. Debonding compromises the system’s working capacity and can decrease the efficiency of the control and measurement apparatus. Since the typical dimension of debonding phenomena is of microm- eters or tens of micrometers in magnitude, a pointwise knowl- edge of interfacial mechanical stresses is necessary. For this aim, it is opportune and highly appropriate to use the finite-element method (FEM) [10]. The analytical solution of the differential equations providing the fields at the interface can be easily made for a one-dimensional (1-D) model of the system [11], [12], or for simple geometries [3], [13], but is very complicated for higher dimensional models and for complex geometry. How- ever, only the resultant forces and deformations can be calcu- lated through an analytical approach, and this does not provide any information on the points of the structure that are highly stressed. Moreover, if the coupling between longitudinal, flex- ural, and torsional modes is not included through a 3-D model [14], by adopting instead only a 1-D or two-dimensional (2-D) model, then the quantities calculated present large errors [10], [15]. For instance, under the assumption of a 1-D model, the interlaminar tangential stresses, which are the main stresses re- sponsible for the disjunction of the two laminas, are zero. Be- sides the pointwise determination of the interfacial mechanical quantities, FE analysis also allows one to compute the pointwise values of the electric potential and field within the piezoelectric plate. This paper analyzes the frequency response of the electro- mechanical system shown in Fig. 1, made up of a metallic can- tilever on which a piezotransducer is bonded. A 3-D FE model is proposed and discussed that reproduces with high accuracy the real physical system. This model allows us to compute the interfacial mechanical stresses and the pointwise values of the electric potential and field within the piezoelectric plate, all of 0018-9456/$20.00 © 2005 IEEE