In this research, the stable regions of an unbalanced rotor are specified. Krein’s theorem is applied to this system for the first time. In this case, the disk has six degrees of freedom and there might be coupling between all dynamic modes. The effect of the skew angle is observable in equations of motion. The equations of motion are derived using Lagrange’s equations. Krein’s signature of each mode is calculated in order to find possibility of frequency coalescence. Campbell diagrams are used to verify Krein’s theorem. The damping and unbalancing effect on the system stability are studied. For all effective parameters, the stable and unstable zones are computed. Numerical analysis for the nonlinear governing equations is applied to compare the results. It was observed that, growth of the unbalancing makes the coupling more powerful and increase instability. Cylindrical rotor has more instability than disk-shaped rotor. When the polar moment of inertia is equal to the diametral moment of inertia, the maximum instability occurs. When rotor moves toward bearings, unstable regions shift toward larger velocities. Increasing the rotor mass shifts unstable velocities toward lower velocities.

Keywords: Rotor, Krein, stability, frequency coalescence, unbalance

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