Modeling of a large class of undamped flexible structures experiencing large angular excursions

The analysis of an arbitrarily constrained, nonuniform beam in a non-Newtonian reference frame requires special attention to centrifugal and coriolis effects caused by the rotation. In addition to distributed displacement and velocity coordinates of each beam cross-section within the non-Newtonian frame, the angular velocity components of the reference frame itself are treated as coordinates to be determined, resulting in redundant coordinates. Such a formulation requires the imposition of an arbitrary constraint. The equations of motion are derived from Hamilton's principle using Lagrange multipliers to preserve explicitly the constraint relationship. The resulting equations of motion are valid for large angular velocities and accelerations, but restricted by small flexible motion assumptions. Characteristics of the internal dynamics, including stiffening and softening effects, are examined, and shape and attitude control problems associated with this rudimentary spacecraft model are presented and discussed. In particular, a method will be presented which allows us to treat the non-linear equations of motion as time-varying linear equations. A numerical simulation of the response of an unconstrained beam subject to disturbance and control inputs verifies the analytically derived conclusions.