Parameter estimation in active plate structures A.L. Arau ´jo a , H.M.R. Lopes a , M.A.P. Vaz b , C.M. Mota Soares c, * , J. Herskovits d , P. Pedersen e a ESTIG – Polytechnic Institute of Braganc ¸a, Campus de Sta, Apolo ´ nia, Apartado 134, 5301-857 Braganc ¸a, Portugal b DEMEGI – Faculty of Engineering, University of Porto, Rua Dr. Roberto Frias, 4200-465 Porto, Portugal c IDMEC – Instituto Superior Te ´ cnico, Av. Rovisco Pais, 1049-001 Lisboa, Portugal d COPPE/UFRJ-Federal University of Rio de Janeiro, Caixa Postal 68503, 21945-970 Rio de Janeiro, Brazil e Department of Mechanical Engineering, Technical University of Denmark, DK-2800 Kgs. Lyngby, Denmark Received 16 June 2005; accepted 14 January 2006 Available online 12 May 2006 Abstract In this paper two non-destructive methods for elastic and piezoelectric parameter estimation in active plate structures with surface bonded piezoelectric patches are presented. These methods rely on experimental undamped natural frequencies of free vibration. The first solves the inverse problem through gradient based optimization techniques, while the second is based on a metamodel of the inverse problem, using artificial neural networks. A numerical higher order finite element laminated plate model is used in both methods and results are compared and discussed through a simulated and an experimental test case. Ó 2006 Civil-Comp Ltd. and Elsevier Ltd. All rights reserved. Keywords: Piezoelectric parameters; Gradient optimization; Neural networks; Finite element method 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. Traditional estimates based upon engineering tables provided by manufacturers are not always reliable for these applications, due to substantial variability among samples, dynamic ranges of interest and, more important, when different materials are com- bined as components in an active composite material con- figuration, the effective values of material parameters are usually found to be quite different. To address these issues we present two non-destructive methods for parameter estimation in active plate structures with surface bonded piezoelectric patches and a comparison is established between the two approaches to the inverse problem: gradient based optimization techniques and meta- modeling techniques using artificial neural networks. In both techniques, the system response used was a set of undamped experimental natural frequencies of free vibration. Thus, a numerical model capable of accurately predicting the response of the physical system is of crucial importance for both approaches. Modeling and simulation of adaptive structures with surface bonded or embedded piezoelectric patches or layers has been conducted exten- sively during the last years as described in [1,2]. In this work, the numerical model is a finite element higher order laminated plate model that includes the piezoelectric effect [3]. Elastic parameter estimation techniques have been proposed by several authors. An assessment of different approaches to eigenfrequency and optimization-based iden- tification methods for estimation of mechanical properties on composite laminated plates is available in [4]. Other eigenfrequency based methods for identification of elastic constants in laminated composite materials include 0045-7949/$ - see front matter Ó 2006 Civil-Comp Ltd. and Elsevier Ltd. All rights reserved. doi:10.1016/j.compstruc.2006.01.017 * Corresponding author. Tel.: +351 21 841 7455; fax: +351 21 841 7915. E-mail address: cmmsoares@alfa.ist.utl.pt (C.M. Mota Soares). www.elsevier.com/locate/compstruc Computers and Structures 84 (2006) 1471–1479