Hopf bifurcation analysis of a rotor/seal system

The Hopf bifurcation behaviour of a symmetric rotor/seal system was investigated using Muszynska's non-linear seal fluid dynamic force model. For a perfectly balanced system, the instability from certain critical equilibrium positions is proved to be the result of Hopf bifurcation and only the supercritical type is found for a specific rotor system using Poore's algebraic criteria. Hence, a stable periodic orbit bifurcates from the critical equilibrium position after the threshold speed is exceeded. Due to the dimensionless whirl frequency being found to be close to 1/2 over quite a large range of system parameters, the periodically perturbed Hopf bifurcation in 1/2 subharmonic resonance is dealt with for an imbalanced rotor system. The bifurcation of the averaged system, obtained using the centre manifold procedure and averaging method, is analyzed. The results show that non-synchronized whirl of the imbalanced rotor can either be a quasi-periodic motion resulting from a Hopf bifurcation, or a half-frequency whirl from period doubling bifurcation, determined by the structure parameters of the system and operation conditions. Numerical simulation verifies the analytical results.