Constrained relative motions in rotational mechanics

Abstract

The dynamical effects of imposing constraints on the relative motions of component parts in a rotating mechanical system or structure are explored. It is noted that various simplifying assumptions in modeling the dynamics of elastic beams imply strain constraints, i.e., that the structure being modeled is rigid in certain directions. In a number of cases, such assumptions predict features in both the equilibrium and dynamic behavior which are qualitatively different from what is seen if the assumptions are relaxed. It is argued that many pitfalls may be avoided by adopting so-called geometrically exact models, and examples from the recent literature are cited to demonstrate the consequences of not doing this. These remarks are brought into focus by a detailed discussion of the nonlinear, nonlocal model of a shear-free, inextensible beam attached to a rotating rigid body. Here it is shown that linearization of the equations of motion about certain relative equilibrium configurations leads to a partial differential equation. Such spatially localized models are not obtained in general, however, and this leaves open questions regarding the way in which the geometry of a complex structure influences computational requirements and the possibility of exploiting parallelism in performing simulations. A general treatment of linearization about implicit solutions to equilibrium equations is presented and it is shown that this approach avoids unintended imposition of constraints on relative motions in the models. Finally, the example of a rotating kinematic chain shows how constraining the relative motions in a rotating mechanical system may destabilize uniformly rotating states.