© Faculty of Mechanical Engineering, Belgrade. All rights reserved FME Transactions (2012) 40, 1-9 1 Received: September 2011, Accepted: November 2011 Correspondence to: Nemanja Zorić Faculty of Mechanical Engineering, Kraljice Marije 16, 11120 Belgrade 35, Serbia E-mail: nzoric@mas.bg.ac.rs Nemanja D. Zorić Teaching Assistant University of Belgrade Faculty of Mechanical Engineering Aleksandar M. Simonović Assistant Professor University of Belgrade Faculty of Mechanical Engineering Zoran S. Mitrović Full Professor University of Belgrade Faculty of Mechanical Engineering Slobodan N. Stupar Full Professor University of Belgrade Faculty of Mechanical Engineering Multi-Objective Fuzzy Optimization of Sizing and Location of Piezoelectric Actuators and Sensors This paper presents the multi-objective fuzzy optimization of sizing and location of piezoelectric actuators and sensors on the thin-walled composite beam for active vibration control, using the degree of controllability (DC) for controlled modes as optimization criteria. The optimization process is performed constraining the original dynamics properties change including the limitation of increase of the mass, using or neglecting the limitation in degrees of controllability for residual modes for reduction spillover effect. Pseudogoal functions derived on the fuzzy set theory gives a unique expression for global objective functions eliminating the use of penalty functions. The problem is formulated using the finite element method based on the third-order shear deformation theory. The particle swarm optimization technique is used to find optimal configuration. Several numerical examples are presented for the cantilever beam. Keywords: multi-objective optimization, fuzzy logic, piezoelectric sensor, actuator, particle swarm. 1. INTRODUCTION Piezoelectric actuators and sensors have a wide range of applications in vibration suppression and shape control. The optimization of sizing and location of actuators and sensors for active vibration control of flexible structures has been shown as the one of the most important issues in the design of active structures since these parameters have a major influence on the performance of the control system [1,2]. The optimization problem can be divided into two approaches. The first approach consists of the combination of optimal location and size of sensors and actuators and controller parameters [3,4]. The second approach deals with optimal location and size of sensors and actuators independently of controller definition [5-7]. Many times, an active structure is discretized into a finite number of elements for vibration analysis and control. For practical implementation, this model needs to be truncated, where only the first few modes are taken into account. However, the state feedback control law, based on a reduced model, may excite the residual modes resulting with spillover instability for even simple beam problem [8]. Considering that fact, some authors involved residual modes in the optimization problem [6,9]. Due to the complexity of the problem, classical optimization methods that apply gradient-based search techniques are not convenient for use. A good solution for such optimization problems relies on heuristic optimization algorithms. Some studies have used genetic algorithm (GA) [6,10], and Simulated annealing (SA) algorithm [11] for finding out the optimal sizing and location of sensors and actuators and other parameters related to the control performances. This paper deals with the multi-objective fuzzy optimization of placement and sizing of collocated piezoelectric actuators and sensors on a composite beam for maximum active vibration control effectiveness. The optimization problem is formulated independently of controller definition using the DC to measure control effectiveness for the vibration in the controlled modes. Optimization criteria are used ensuring a good DC for the controlled modes. The optimization process is performed constraining the original dynamic properties change including the limitation of increase of the mass, using or neglecting the limitation in DCs for residual modes for spillover effect reduction. To the best of our knowledge, the multi-objective fuzzy optimization based on the particle swarm optimization technique will present an innovative approach for solving the problem of sizing and location of piezoelectric actuators and sensors. Both objective functions and constraints are evaluated by the membership function. In that way, the use of weighting coefficients and penalty functions are avoided. 2. FINITE ELEMENT MODEL A laminated composite beam with integrated piezoelectric sensors and actuators is considered (Fig. 1). The coordinate x is coincident with the beam axis, the x – y plane coincides with a mid-plane of the beam and the z axis is defined as normal to the mid-plane according to the right-hand rule. Both elastic and piezoelectric layers are supposed to be thin, such that a plane stress state can be assumed. The sensors and actuators are perfectly bonded on the upper and lower surfaces at different locations along the length of the beam. It is assumed that they span the entire width of the beam. Elastic layers are obtained by setting their