AIAA JOURNAL Vol. 40, No. 6, June 2002 Piezoelectric Sensor and Actuator Spatial Design for Shape Control of Piezolaminated Plates Abhijit Mukherjee ¤ and Shailendra Joshi † Indian Institute of Technology Bombay, Mumbai 400 076, India Controlled actuation and sensing of structures by spatially distributing the piezoelectric material has been a topic of interest in recent years. We present an iterative technique to design the shape of piezoelectric actuators in order to achieve the desired shape of the structure. A gradientless shape design procedure based on the residual voltages is developed. It aims at minimizing the quadratic measure of global displacement residual error between the desired and the current structural con guration. The actuators gradually adapt to a shape that is most ef cient in resisting the external excitation. The present technique can be well suited for any static and time-varying excitation. In vibration control it is often necessary to create modal sensors and actuators in order to observe or excite some speci c modes. Such modal sensors and actuators alleviate spillover problems, and thus they avoid exhaustive signal processing. Several numerical examples for static as well as dynamic cases are presented to demonstrate the ef cacy of the present technique. Nomenclature B = strain-displacement matrix C = capacitance of the piezoelectric material D = dielectric displacement vector E = electric eld vector e = matrix of piezoelectric stress constants N p , M p = actuator force and moment resultants Q = matrix of elastic constants q = closed-circuit charge V S = sensor voltage ® = quadratic measure of residual error between desired shape and actual shape ± e = nodal displacement vector N ² = matrix of dielectric constants " = mechanical strain vector ¾ = stress vector induced by mechanical and electrical effects I. Introduction P IEZOELECTRIC materials are widely used in the develop- ment of high-performance structures that are energy ef cient and autonomous, known as smart structures. These materials are capable of sensing and controlling the changes in the structural characteristics. Optimal distribution of the piezoelectric material in the structure to induce controlled actuation has been a subject of interest in recent years. In many practical situations (for example, re ector antennas, deformable mirrors, aircraft wings, etc.) this is of utmost importance in order to exercise precise control over the shape of the structure. In dynamic conditions it is often desirable to observe or excite speci c modes of interest. Optimal spatial distri- bution of piezoelectric material allows observation or excitation of desired modes without exhaustive signal processing. In this paper we demonstrate a technique for optimum shape design of sensors and actuators in controlling plate structures. Some reviews on controlled actuation are available. 1;2 Several re- search works have been carried out on developing techniques for the placement of discrete number of actuators to control the re- Received 19 May 2001; revision received 2 October 2001; accepted for publication 19 November 2001. Copyright c ° 2001 by the American In- stitute of Aeronautics and Astronautics, Inc. All rights reserved. Copies of this paper may be made for personal or internal use, on condition that the copier pay the $10.00 per-copy fee to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923; include the code 0001-1452/02 $10.00 in correspondenc e with the CCC. ¤ Professor, Department of Civil Engineering. † Research Student, Department of Civil Engineering. sponse of the structure. Lim 3 proposed a method of determining the optimal subset of actuators and sensors based on a weighted mea- sure of modal participation of each sensor/actuator combination. Gawronski and Lim 4 employed a placement index based on the Hankel singular values (HSV) to determine the sensor and actu- ator locations that maximize observability and controllability cri- teria. Clark and Cox 5 experimentally demonstrated a band-limited method based on HSV decomposition for selecting actuators and sensors to lter out the modes outside the desired bandwidth for structural acoustic control. Gaudenzi and Barboni 6 determined the size and location of the discrete actuators for Euler– Bernoulli beams by applying a number of constraints that uniquely de ne the design variables. However, this method requires arti cial constraints to generate unique solutions. Barboni et al. 7 employed the pin-force model and the modal approach to obtain closed-form solutions for optimal size and location of actuators on beams under dynamic con- ditions. Seeley and Chattopadhyay 8 solved a multiobjective opti- mization problem that includes discrete actuator location, vibration reduction, reduction in power consumption, and maximization of fundamental frequency in the objective function. A nonlinear con- strained programming approach based on the method of feasible directions was employed in the solution of cantilever box beams. Kapania et al. 9 used a heuristic integer programming approach to determine optimal locations of actuators in the control of thermal deformations of spherical mirror segments. Onoda and Hanawa 10 used genetic algorithms (GA) and simulated annealing algorithms for optimal placement of actuators in shape control of space trusses. Zhang et al. 11 employed a GA for sensor and actuator locations and the feedback gains simultaneously. Kang et al. 12 presented a gradient-based numerical scheme based on the damping character- istics to optimize the sensor and actuator placement for vibration control of plates. Some investigators have used optimal voltage distribution in the control of the structures. Lin and Hsu 13 used sine-shaped sensors in the static control of beams. The voltage distribution in sensors and actuators are written in terms of the displacements, and the so- lution is obtained using Fourier sine series. Agrawal and Treanor 14 presented analytical and experimental results on optimal placement of actuators and optimal voltages for beams. They developed an al- gorithm that optimized actuator placement and voltages for a given shape function. The authors reported that simultaneous optimization of actuator position and input voltages were unreliable because of the differences in the order of actuator locations and voltage terms in the optimization of cost function. Soares et al. 2 used gradient-based optimization techniques for optimal design of piezolaminated struc- tures. In the static shape control of plates, the voltages were found by minimizing the mean-squared error between the desired and actual 1204