Compensation of Deformations in Elastic Solids and Structures in the Presence of Rigid-Body Motions

The present Lecture is concerned with vibrations of linear elastic solids and structures. Some part of the boundary of the structure is suffering a prescribed large rigid-body motion, while an imposed external traction is acting at the remaining part of the boundary, together with given body forces in the interior. Due to this combined loading, vibrations take place. The latter are assumed to remain small, such that the linear theory of elasticity can be applied. As an illustrative example for the type of problems in hand, we mention the flexible wing of an aircraft in flight. In this example, the rigid-body motion is defined through the motion of the comparatively stiff fuselage to which a part of the boundary of the wing is attached. The goal of the present paper is to derive a time-dependent distribution of actuating stresses produced by additional eigenstrains, such that the deformations produced by the imposed forces and the rigid-body motion are exactly compensated. This is called a shape control problem, or a deformation compensation problem. We show that the distribution of the actuating stresses for shape control must be equal to a quasi-static stress distribution that is in temporal equilibrium with the imposed forces and the inertia forces due to the rigid-body motion. Our solution thus explicitly reflects the non-uniqueness of the inverse problem under consideration. The present Lecture extends previous results by Irschik and Pichler (2001, 2004) for problems without rigid-body degrees of freedom. As a computational example, we present results for a rectangular domain in a state of plane strain under the action of a translatory support motion.