Optimal Shape Control of Composite Thin Plates with Piezoelectric Actuators

Emerging technologies in microsensing, microactuation, active airfoils, turbine blades, and large, lightweight, flexible space structures can benefit from piezoelectric actuators for active shape control. Piezoelectric materials develop strain under applied voltage which induces structural deflection. This article presents analytical models, FEM solution, and optimal shape control of composite thin plates with piezoelectric actuators surface embedded or bonded in a biomorph arrangement. A 2D FEM approach is developed which is accurate, plus simpler and more efficient than existing 3D solutions. Three optimal shape problems are presented: applied voltage, actuator layout, and actuator number optimizations. For the latter two problems, a novel method is introduced using a vector of binary variables. 1 Graduate Research Assistant 2 Assistant Professor, Corresponding Author 3 Associate Professor 3 INTRODUCTION The integration of composite materials with piezoelectric actuators will significantly improve the performance of aircraft and space structures. The feasibility of such integrated smart structures has been demonstrated by various analyses and numerical models (e.g. Donthireddy and Chandrashekhara, 1996; Ghosh and Batra, 1995; Crawley and Lazarus, 1991; Lee and Moon, 1990). An overview of smart structure technology is presented by Gandhi and Thompson (1992); few problems have extended to practical designs (Jia and Rogers, 1989; Sepulveda and Schmit, 1991). Shape optimization for such structures is of great importance, especially in low-weight aerospace applications. The effectiveness of the control system strongly depends on the active element locations (Fanson and Caughey, 1987). In those 3D models, the problems are large and complex. For example, a plate with thin sensors and actuators is modeled with the isoparametric hexahedron solid element (Sepulveda and Schmit, 1991), requiring Guyan reduction to reduce the total degrees of freedom (dof). When the plate is very thin, there are problems of excessive shear strain energies and higher stiffness coefficients in the thickness direction. Another example is applying an 8-node, 32-dof brick element to a thin plate example (Sepulveda and Schmidt, 1991); 3D incompatible modes were necessary for predicting the deflection. Presented in a previous article (Agrawal, Tong, and Nagaraja, 1994) is a 1D model, finite difference solution, and optimal input voltage estimation for a cantilevered composite beam. The current article extends this work to formal optimization problems for a thin plate, while maintaining computational efficiency for practical application. This article is organized as follows. First, a simpler (relative to existing 3D models) 2D mathematical model for the deflection of a composite thin plate with piezoelectric actuators surface embedded or bonded is presented. Then an efficient FEM solution is presented using a 4-node, 12dof thin plate discrete Kirchhoff quadrilateral (DKQ) bending element. Shape optimization is then developed, for the optimal input voltages, optimal actuators layout, and optimal number of actuators problems. These optimization problems may be used either off-line for optimal design or for real-time shape control. Last, two examples are given.