Dynamic shape control of beam-type structures by piezoelectric actuation and sensing

Abstract

In the present paper the concept of dynamic shape control of structures is addressed. The term “shape control”, not to be confused with automatic control, means to identify the spatial distribution (or shape) of an actuating control agency, such that a structural displacement field with a desired spatial distribution (or shape) is reached. This field aspect requires a spatially distributed control actuation, which, in the present paper, is performed by means of assigned piezoelectric eigenstrains. The goal of our proposed dynamic shape control procedure is to shape the piezoelectric actuation such as to obtain a displacement field coinciding with a dynamic displacement field induced by external forces. Equivalently, we may eliminate the force induced dynamic displacements. Bending vibrations of straight composite piezoelectric beams are studied in more detail. First, the coupled electro-mechanical field equations are developed and then the dynamic shape control problem is solved in closed form for this coupled formulation. It turns out that exact elimination of force induced vibrations is possible when the shape of the piezoelectric actuation coincides with a statically admissible bending moment distribution of the force loaded beam. Distributions characterizing non-unique solutions are discussed, and aspects of collocated sensing are addressed.