Linearized models for the control of rotating beams

A geometrically exact dynamical model for the rotation of a planar rigid body with an elastic beam attachment is presented. This model is essentially nonlinear and involves a partial differential-integral equation. The author proposes consistent finite-dimensional approximations of the model in terms of the dynamics of carefully chosen kinematic chains in which the one degree-of-freedom rotary joint motions are governed by idealized torsional springs. For purposes of local analysis or to support implementation of rotational control, the chain models can be linearized about equilibrium rotations. In the case of one particular body-beam system, it is shown that the linearized chain models provide consistent approximations to fourth-order partial differential equations. The form of these equations is crucially dependent on the underlying equilibrium rotation