Dynamic shape control of small vibrations superposed on large static deformations of thin plates

Abstract

Control of continuous structures requires suitably distributed actuation and sensing. In the present contribution, we consider thin composite plates with piezoelastic layers under the action of a given set of imposed forces. Our goal is to suppress the force induced plate vibrations by means of distributed (shaped) piezoelectric actuation. This problem is referred to as the Shape Control Problem. The present contribution investigates the dynamic shape control problem in the special but practically important case of small vibrations superposed on large deformations of a quasi-static intermediate state. Moderately large deformations are taken into account by means of the kinematic approximation of von Karman. Linearization of the non-linear electromechanical field equations, with respect to the static intermediate state, results in a set of linear partial differential equations for the superposed vibrations. These equations are cast into convolution integral formulations for both the transient piezoelectric actuation and the transient external forces. Comparing the kernels of the convolution integrals, a distributed piezoelectric actuation is found, which exactly eliminates the forced vibrations. The distribution (shape) of the actuation coincides with the distribution of the statically admissible stress due to the transient external forces.