On the Modeling, and Open-Loop Control of a Rotating Thin Flexible Beam

A set of governing differential equations is derived for the inplane motion of a rotating thin flexible beam. The beam is assumed to be linearly elastic and is connected to a rigid hub driven by a torque motor. Both flexural and extensional effects are included in the derivation. This coupling due to flexure and extension is usually neglected in studies dealing with the control of such a system. Models for typical control studies are often derived by utilizing an assumed mode approach where the mode shapes are obtained by solving the Euler-Bernoulli beam equation for flexural vibrations, with clamped-free or pinned-free boundary conditions. The coupled equations developed in this paper are used to demonstrate that typical models in control studies give satisfactory results up to a critical rotational speed. For the case where these coupled equations are specialized to simple flexure only, valid for low angular speeds, a unique feedforward control strategy can be derived. This is an open-loop control strategy that enables total elimination of an a priori specified vibratory mode from the gross motion in a finite critical time.