Parametric instability of a Jeffcott rotor with rotationally asymmetric inertia and transverse crack

Both the rotationally asymmetric inertia and transverse crack frequently appear in the rotor system. The parametric excitations induced by this two features cause instability and severe vibration under certain operating conditions. Thus, the parametric instability of a Jeffcott rotor with asymmetric disk and open transverse crack is studied analytically. The vibration equations of four degrees-of-freedom of the system are established, and the stiffness coefficients of cracked rotor shaft are derived based upon the compliance method and strain energy release rate method. Then, utilizing the harmonic balance method and Taylor expansion technique, the unstable widths of simple and combination instability regions (SIR and CIR) are solved approximately. For a practical rotor system, the approximate unstable widths are verified by the Floquet numerical analysis. The effects of crack depth and position upon the unstable widths are discussed, and the conditions for zero unstable points (ZUPs) are given: Besides the asymmetric angle should be pi/2 (for SIR) or 0 (for CIR), the relationships between the inertia asymmetry and crack parameters (depth and position) are also presented analytically. These results would be useful for crack detection and instability control of the asymmetric rotor-bearing system.