BIFURCATION AND STABILITY ANALYSIS OF A ROTATING BEAM

We discuss small oscillations of an elastic beam clamped radially to t he interior of a rotating ring. It is well known that if the speed of rotation is sufficiently high, the trivial equilibrium of the beam may lose stability and the beam buckles. Earlier papers on this subject h ave, with a few exceptions, dealt with linear models and thus only con sidered the trivial equilibrium of the beam. In the present work, a no nlinear model including third-order terms is proposed. Using a Ritz-Ga lerkin procedure, the model equation is reduced to a nonlinear ordinar y differential equation (ODE) of second order, which contains no Corio lis terms. Previous nonlinear models in the literature have Coriolis t erms as the governing nonlinear effects, but this has been shown to be erroneous. This nonlinear ODE is investigated by phase plane techniqu es. Bifurcation and stability analysis reveals three different types o f behaviour, depending on the length of the beam relative to the ring radius. Bifurcation diagrams and associated phase portraits are given. The theoretical basis for the applied phase plane techniques is the f act that the reduced equation is conservative, i.e., possesses a first integral. An appendix briefly outlines the most important consequence s of this property.