Modeling and Controllability of Interconnected Membranes

Abstract

We shall consider in this Chapter the motion of an elastic body consisting of several interconnected elastic membranes. Our goals are to obtain the dynamic equations of motion of such a configuration, especially to elucidate the geometric and dynamic conditions which must hold in junction regions where two or more such elastic elements are joined together, and to study the controllability properites of the resulting (linearized) system of equations. The modeling proceeds through two stages. In Section 1 the equations of motion and boundary conditions of a general nonlinear membrane and the corresponding Hamilton’s principle will be derived, assuming a linear stress-strain relation. The dynamic equations of a system of interconnected membranes are derived in Section 2 by minimizing an appropriate Lagrangian over the class of deformations which satisfies certain geometric constraints. In particular, the geometric constraint imposed in the junction regions is the very natural one that the initially connected system of membranes remains connected throughout the deformation process. When this constraint is imposed on the admissible deformations, the optimal motion is forced to satisfy a second, dynamic constaint in the junction regions that has the interpretation of a balance of forces law there. Both the equations of motion and the dynamic junction conditions involve nonlinear couplings among the components of the displacement vector. When these are linearized around the trivial equilibrium the familiar equations of motion of a single elastic membrane, in which in-plane displacements are not coupled to transverse motion, axe obtained for the displacement of each individual element. However, all displacements remain coupled through the geometric and (linearized) dynamic junction conditions. The question of exact controllability of this linearized coupled system is then considered in Section 3.