Paper detail

Control and stabilization of geometrically exact beams

We study well-posedness, stabilization and control problems involving freely vibrating beams that may undergo motions of large magnitude -- i.e. large displacements of the reference line and large rotations of the cross sections. Such beams, shearable and very flexible, are often called geometrically exact beams and are especially needed in modern highly flexible light-weight structures, where one cannot neglect these large motions. We view these beams from two perspectives. The first perspective is one in which the beam is described in terms of the position of its reference line and the orientation of its cross sections (expressed in some fixed coordinate system). This is the generally encountered model, due to Eric Reissner and Juan C. Simo. Of second order in time and space, it is a quasilinear system of six equations. The second perspective is one in which the beam is rather described by intrinsic variables -- velocities and strains or internal forces and moments -- which are moreover expressed in a moving coordinate system attached to the beam. This system, proposed in its most general form by Dewey H. Hodges, consists of twice as many equations, but is of first order in time and space, hyperbolic and only semilinear (quadratic). From the definition of the state of the latter model, one can see that both perspectives are linked by a nonlinear transformation. The questions of well-posedness, stabilization and control are addressed for beams governed by the intrinsic model, while by using the transformation we also prove that the existence and uniqueness of a classical solution to the intrinsic model implies that of a classical solution to the model written in terms of positions and rotations. In particular, this enables us to deduce corresponding results for the latter model. We also also address these questions for networks of beams attached to each other by means of rigid joints.