Buckling and Vibration of a Long Shaft Rotor System with a Stabilized Bearing

Based on the dynamic theory of flexible multi-body systems, a nonlinear dynamic model is established of a long shaft rotor system with a stabilized elastic bearing located along the shaft at an arbitrary position by the generalized Hamilton’s principle. Lower-order polynomials are adopted to construct the Ritz basis for investigating the first two vibration frequencies and the critical rotational velocity, as well as the optimal stabilized position of the elastic bearing. The critical rotational velocity and optimal stabilized position of the elastic bearing are also verified by the finite element method. The results obtained by the two methods agree well. Then, the post-buckling and the nonlinear vibration responses of the system fluctuating at the rotational velocity are investigated. The present study indicates that critical rotational velocity exists for the long shaft rotor system with stabilized elastic bearing. Once the rotational velocity exceeds the critical value, the system’s trivial equilibrium state losses its stability by pitchfork bifurcation. The optimal stabilized position of the elastic bearing for the long shaft rotor system is located at 73% length of the shaft from the pinned end. When in the supercritical state, the system changes to different vibration modes with local period-1 solutions and global period-2 solutions around the post-buckling position due to fluctuation of the rotational velocity. Furthermore, the global period-2 vibration may evolve to the almost periodic motion as the fluctuation frequency varies. This study provides a theoretical basis for understanding the geometrical nonlinear effect and vibration modes of the long shaft rotor system working in the supercritical state.