A new method for finding the first‐ and second‐order eigenderivatives of asymmetric non‐conservative systems with application to an FGM plate actively controlled by piezoelectric sensor/actuators

Abstract

AbstractIn this paper, a new method for computing eigenvalue and eigenvector derivatives of asymmetric non‐conservative systems with distinct eigenvalues is presented. Several approaches have been proposed for eigenderivative analysis of systems with asymmetric and non‐positive‐definite mass, damping and stiffness matrices. The proposed formulation that is developed by combining the modal and algebraic methods neither have the complications of modal methods in calculating the complex left and right eigenvector derivatives nor suffer from numerical instability problems usually associated with algebraic methods. The method is applied to a functionally graded material (FGM) plate actively controlled by piezoelectric sensor/actuators. In this system, the feedback signal applied to each actuator patch is implemented as a function of the electric potential in its corresponding sensor patch. The use of this closed‐loop controlling system leads to a non‐self‐adjoint system with complex eigenvalues and eigenvectors. A finite element model is developed for static and dynamic analysis of closed‐loop controlled FGM plate. The first‐ and second‐order approximations of Taylor expansion are used to estimate the corresponding changes in the plate modal properties due to change in design parameters (the displacement feedback gains and the piezoelectric layer thickness in each S/A pair). Copyright © 2008 John Wiley & Sons, Ltd.