<title>Linear and nonlinear actuations in shape control of beams</title>

Abstract

Recently, there have been extensive investigations on the shape control of beams using piezoelectric actuators. Most studies, however, assumed linear piezoelectric constitutive equations, which are not valid for actuations involving high electric fields. This paper examines the effect of nonlinear behavior of the piezoelectric materials in the shape control of piezoelectric beams. The governing coupled electromechanical equations for nonlinear material behavior is developed for a general three-dimensional structural element using the Gibbs free energy formulation. Nonlinearity is accounted in the analysis by incorporating an adequate number of nonlinear terms in the Gibbs free energy expression. Since many of the higher order material properties are not available for most of the piezoelectric materials, experimental data that are available for the nonlinear relationship between the electric field and the electric displacement are used. A finite element model is developed for the beam with piezoelectric actuators using a modified bilinear four-node quadratic element. The expression for the actuation voltages required for shape control is then obtained by minimizing an error function, defined as the area between the achieved and desired shape. The final system of coupled equations is solved by an iterative finite element procedure. Comparison of numerical results obtained for both linear and nonlinear piezoelectric behaviors showed the importance of incorporating nonlinear effects in the shape control of piezoelectric beams.