Control of Bending Vibrations Within Subdomains of Thin Plates—Part II: Piezoelectric Actuation and Approximate Solution

Abstract

In the first part of this paper, we presented the theoretical basics of a new method to control the bending motion of a subdomain of a thin plate. We used continuously distributed sources of self-stress, applied within the subdomain, to exactly achieve the desired result. From a practical point of view, continuously distributed self-stresses cannot be realized. Therefore, we discuss the application of discretely placed piezoelectric actuators to approximate the continuous distribution in this part. Using piezoelectric patch actuators requires the consideration of electrostatic equations as well. However, if the patches are relatively thin, the electromechanical coupling can be incorporated by means of piezoelastic (instead of elastic) stiffness (piezoelastically stiffended elastic constants). The placement of the patches is based on the discretization of the exact continuous distribution by means of piece-wise constant functions. These are calculated from a convolution integral representing the deviation of the bending motion in the controlled case from the desired one. A proper choice of test loadings allows us to eliminate representative mechanical quantities exactly and to make the resulting bending motion to match the desired one very closely; hence, to find a suboptimal approximate solution. In Part I of this paper we presented exact solutions for the axisymmetric bending of circular plates; it is also considered in Part II. For axisymmetric bending, only the radial coordinate is discretized. Hence, ring-shaped piezoelectric patch actuators are considered in this paper.