Estimation of piezoelastic and viscoelastic properties in laminated structures A.L. Araújo a , C.M. Mota Soares b, * , J. Herskovits c , P. Pedersen d a ESTIG – Polytechnic Institute of Bragança, Campus de Sta, Apolónia, Apartado 134, 5301-857 Bragança, Portugal b IDMEC/IST – Instituto Superior Técnico, Technical University of Lisbon, Av. Rovisco Pais, 1049-001 Lisboa, Portugal c COPPE/UFRJ – Federal University of Rio de Janeiro, Caixa Postal 68503, 21945-970 Rio de Janeiro, Brazil d Department of Mechanical Engineering, Technical University of Denmark, DK-2800 Kgs. Lyngby, Denmark article info Article history: Available online 23 May 2008 Keywords: Laminated composite plates Piezoelectric patches Viscoelastic materials Finite element method Inverse problems Parameter estimation abstract An inverse method for material parameter estimation of elastic, piezoelectric and viscoelastic laminated plate structures is presented. The method uses a gradient based optimization technique in order to solve the inverse problem, through minimization of an error functional which expresses the difference between experimental free vibration data and corresponding numerical data produced by a finite ele- ment model. The complex modulus approach is used to model the viscoelastic material behavior, assum- ing hysteretic type damping. Applications that illustrate the influence of adhesive material interfaces and viscoelastic parameter identification are presented and a few simulated test cases aid the interpretation of results. Ó 2008 Elsevier Ltd. All rights reserved. 1. Introduction The need for accurate mechanical and piezoelectric parameters in modeling and analysis of active laminated plate type structures has become a major concern in active control applications. Tradi- tional estimates based upon engineering tables provided by manu- facturers are not always reliable for these applications, due to substantial variability among samples, dynamic ranges of interest and, more important, when different materials are combined as components in an active composite material configuration, the effective values of material parameters are usually found to be quite different. In fact, the use of most commercially available 2D numerical models along with the elastic, piezoelectric and dielec- tric properties provided by manufacturers does not guaranty suffi- cient accuracy for active control of noise and vibration in advanced applications such as those frequently encountered in the aerospace industry. During the last years several numerical models have been developed for simulation of active structures with piezoelectric sensors and actuators, both surface or embedded ones [1,2]. The ef- fect of the adhesive materials in the static or dynamic response in this type of structures has been studied experimentally [3] and numerical models that take into account the behavior and proper- ties of the adhesive materials have been proposed [4–6]. For applications where the use of 2D plate or shell equivalent single layer numerical models is required, it becomes necessary to determine the properties of the different constituent materials that best predict the real behavior of the structure, including damping characteristics. A non-destructive inverse method for estimation of material parameters of laminated active plates with surface bonded piezo- electric sensors and actuators was developed. A finite element higher-order equivalent single layer numerical model which includes the piezoelectric effect [7] is used. The esti- mation of the material parameters is made by adjusting the re- sponse of the numerical model to the experimental response of the structure, consisting in a set of free vibration natural frequen- cies and modal loss factors. Several techniques for the estimation of mechanical properties of structures have already been presented by several authors. An assessment of the different approaches to the identification of mechanical properties based on free vibration response methods and optimization techniques in laminated plates is presented in [8]. Some methods that also use natural fre- quencies of vibration for estimation of elastic constants in compos- ite material structures are based on surface response methods [9] and model updating techniques [10]. Another class of inverse methods combines wave propagation measurements with optimi- zation techniques and, more recently, with genetic algorithms [11,12]. Artificial neural networks were also used to solve this type of problems, based on wave propagation measurements [13] and also natural frequencies of free vibration for identification of elastic and piezoelectric parameters [14,15]. 0263-8223/$ - see front matter Ó 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.compstruct.2008.05.009 * Corresponding author. Tel.: +351 218 417 455; fax: +351 218 419 634. E-mail addresses: aaraujo@ipb.pt (A.L. Araújo), cristovao.mota.soares@ist.utl.pt (C.M. Mota Soares), jose@optimize.ufrj.br (J. Herskovits), pauli@mek.dtu.dk (P. Pedersen). Composite Structures 87 (2009) 168–174 Contents lists available at ScienceDirect Composite Structures journal homepage: www.elsevier.com/locate/compstruct