Boundary control of a Timoshenko beam attached to a rigid body: planar motion

Abstract

A flexible spacecraft modelled as a rigid body which rotates in an inertial space is considered; a light flexible beam is clamped to the rigid body at one end and free at the other end. The equations of motion are obtained by using the geometrically exact beam model for the flexible beam, and it is then shown that under planar motion assumption, linearization of this model yields the Timoshenko beam model. It is shown that suitable boundary controls applied to the free end of the beam and a control torque applied to the rigid body stabilize the system. The proof is obtained by using a Lyapunov functional based on the energy of the system.