Shape control of coupled nonlinear piezoelectric beams

This paper deals with the shape control of composite laminated beams with nonlinear piezoelectric patch actuators. The governing electromechanical coupled equations for nonlinear material behaviour are developed for a general three-dimensional structural element using Gibbs free energy formulation. In the analysis, nonlinearity is accounted for by incorporating enough 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, the 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-noded quadratic element. The expression for the actuation voltages required for shape control is then obtained by minimizing an error function, which is a measure of the area between the achieved and desired shape. The final system of coupled equations is solved by an iterative finite element procedure. Numerical results are obtained for several piezoelectric patch configurations on beams with various boundary conditions. The results show the significance of nonlinear effects in the shape control of beams with piezoelectric actuators, which has hitherto not been accounted.