Bifurcations of an exactly solvable model of rotordynamics

Diverse cyclic responses for an effective and useful nonlinear rotor-bearing-seal system model are studied. In this work two typical cases of self-excited vibrations in turbomachinery are considered: steamwhirl and oilwhip. The first one is an aerodynamic phenomenon induced by the flow of the working fluid at the disc power stages, this excitation being directly proportional to the turbomachine out/input power. The second excitation, oilwhip, is produced by a rotating motion of the lubricating oil film inside the bearings; its magnitude is proportional to the operating speed of the rotor. The studied rotodynamic system model exhibits several mathematical bifurcations which correspond to actual machine behavior, and the analysis for each autoinduced and forced excitation cases, together with its associated bifurcation, is performed. For the autonomous case, as well for the forced excitation exact periodic solutions are obtained. The resulting exact periodic solutions enable the study of multiparameter bifurcation. Completeness of the derived exact solutions for small amplitudes follows from the Hopf bifurcation theorem, which also gives an explanation for the onset of cycles stemming from the equilibrium rest point of the system. Global stability of the rest point for the present rotordynamic model without excitations is proved. The obtained solutions are of engineering importance and they can be used as a tool within a methodology for the preliminary dynamic design nowadays for high speed, or high-performance turbomachinery, as well as for design parameter sensibility analysis. Moreover, it is interesting to notice that for steamwhirl excitation, the onset of the Hopf bifurcation is also the same as the threshold limit for unbounded amplitude responses; whereas for oilwhip, beyond the onset of the Hopf bifurcation there exist a region of stable response limit cycles. The studied model shows a rich and complex behavior.