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Gradual faults and abrupt faults of mechanical systems are differentiated by variation patterns of state var ibles.After the reasons of abrupt faults are analyzed by nonlinear dynamics,two methods establishing the mathematical models for abrupt faults are presented.As an example,a cusp catastrophe model of the vibration amplitudes of a Jeffcott rotor with nonlinear stiffness and its parameter condition of the vibration state catastrophe are obtained.