On the dynamics of a massless beam with end mass rotating in the three-dimensional space

Abstract

Many elements of machines, like shafts, blades, connecting rods, etc., are often modeled using the so-called beam theory and, when these elements rotate with respect to an inertial reference frame, this rotation can deeply affect their dynamic behavior. The position of the beam with respect to the rotation axis and the possibility that the rotation is more complicated than a simple constant-rate rotation about a fixed axis influence this effect, and different models are usually employed. As a result, different phenomena, like gyroscopic effect, centrifugal stiffening or softening, and instability due to rotation are often mentioned in reference to the different cases. The aim of this article is that of building a much simplified beam model, and to subject it to a compound, nonconstant rate rotation. Since the model can be solved in closed form, at least in several cases, a general discussion of the revant phenomena can be done to shed some light on some aspects, like the instability ranges due to rotation and to the damping of the system.