Control of Bending Vibrations Within Subdomains of Thin Plates—Part I: Theory and Exact Solution

Control of the surface profile or shape of structures that deform under externally applied dynamical loads is important in many applications where no control can be exercised on the applied loads. The only recourse is to make the structure adaptive by the action of smart actuators that can null or nearly null the resulting deformation. The class of problems, to which shape control may be applied, is huge and in this paper a theoretical approach is presented for a special subset of such problems, wherein, suitable actuation can be applied in order to keep a subdomain of the structure in its nondeformed state under the action of external dynamical loads. A suitable actuation to achieve this goal is the complement of the self-stress. An appropriate distribution of the self-stress should result in an elimination of the motion of the subdomain of the structure. Moreover, we seek a solution of the problem, which only requires the application of the self-stress in the subdomains or in a slightly larger domain. This is also a practical approach to such problems where it would be prohibitively expensive to design and power actuators to control the entire domain. We choose a linear, thin elastic plate to present the basics of our methodology. The main part of the paper is devoted to the theoretical foundation of the method; however, to show its validity, we also present exact results for the simple case of a circular plate in axisymmetric bending.