Stability and whirl speeds of rotating shaft under axial loads

The stability behavior and whirl speeds of a rotating shaft subjected to an axial compressive load are studied by the finite element method. The governing equations for such a gyroscopic system are formulated based on Timoshenko beam theory. The effects of translational and rotational inertia, gyroscopic moments, bending and shear deformation are included in the mathematical model. In order to facilitate calculation of the whirl speeds, a rotating frame of reference is employed in the formulation. Numerical results show that the rotating shaft develops unstable behavior of the divergence type when the lowest backward whirl speed approaches the value of zero, and as the load is increased slightly, the lowest backward whirl speed and the lowest forward whirl speed become complex conjugate and the instability behavior is immediately shifted from the divergence type to the flutter type. In addition, the whirl speeds decrease as the axial compressive load is increased.