Nonlinear analysis of resonance between the blade and rotor by the bifurcation equation and numerical method

It is known from the previous studies that blades can cause resonance in bladed rotors under specific conditions. In this paper, the behavior of a nonlinear rotor, which faces with this kind of resonance, is investigated. In order to reach this goal, a bladed rotor, in which the disk and shaft are rigid, is considered. The blades are modeled by flexible beams. It is assumed that the disk is supported by elastic and nonlinear bearings. The nonlinear term of the bearing stiffness function is cubic. The rotating system vibrations include both cylindrical and conical whirling. The bifurcation equation is obtained by the method of multiple time scales method. The nonlinear effects are studied by the bifurcation equation. It is revealed that the system behavior, when it encounters Hamiltonian Hopf bifurcation, is dependent on the bearing stiffness nonlinearity type. Accordingly, both subcritical and supercritical Hopf bifurcation is possible. A numerical simulation is performed in order to study the effects of damping coefficients on the path of rotating disk center. The results and methods, which are used in this paper, are applicable for studying Hamiltonian Hopf bifurcation in other fields.