Collocative Control of Beam Vibrations with Piezoelectric Self-Sensing Layers

Abstract

In the first part of the paper, we present a short summary of a simple but accurate electromechanically coupled theory for plane flexural vibrations of slender smart beams. The beams under consideration are assumed to be composed of electroded piezoelastic layers perfectly bonded to substrate layers. For a detailed derivation, see Krommer and Irschik [1], Irschik et. al. [2] and Krommer and Irschik [3]. In the present paper, spatially distributed self-sensing layers with an axially varying intensity of piezoelectric activity are considered within the theory of Refs. [1] – [3]. Self-sensing piezoelectric layers are single piezoelectric layers applicable for both, actuator and sensor applications. As an amazing fact from the point of control theory, perfect collocation between sensors and actuators is automatically provided by self-sensing piezoelectric layers. For details of the self-sensing sensor/actuator concept, see for example Dosch and Inman [4], Tzou and Hollkamp [5], Vipperman and Clark [6] and Oshima et. al. [7]. The main purpose of our derivations is the solution of a dynamic shape control problem, namely to find shape functions for a piezoelectric self-sensing layer such that vibrations due to known external forces can be exactly annihilated by piezoelectric actuation. It is shown that shape functions corresponding to the quasi-static bending moment distributions due to these external forces do represent solutions of this shape control problem. Previous investigations concerning this problem in the context of an electromechanically decoupled, not self-sensing theory have been presented in Irschik et. al., Refs. [8] – [10]. For Finite Element calculations in the context of the coupled theory without reference to self-sensing, see [11].