Damping by parametric excitation in a set of reduced-order cracked rotor systems

Abstract A common tool utilized for the stability analysis of parametrically excited linear systems, such as rotors with cracked shafts, is Floquet׳s method. The disadvantage is a long calculation time needed to evaluate the monodromy matrix and instability zones. An efficient alternative is the generalized Bolotin׳s method, where the instability zones are evaluated quickly, yet the matrices that must be calculated are of large dimensions. In the present paper, the stability analysis is conducted with both Floquet׳s method and the generalized Bolotin׳s method. However, the order of the model is reduced to two modes only and stability analyses are performed for the second-order systems obtained with various combinations of the reducing modes. Then, the results of such analyses are collected in an overall stability map. The stability map obtained in this way closely reconstructs the stability map calculated with the full-order model of the rotor, yet the calculation time needed to generate the collected map as well as the dimension of the problem are considerably reduced. The approach is demonstrated with a mathematical model of the machine with the breathing crack modeled using the rigid finite element method. The rotor is not rotating, yet the stiffness of the shaft is varied periodically to simulate the parametric excitation. An interesting indication of the developing shaft crack observed in the generated stability maps is the presence of anti-resonant zones, where the rotor vibration amplitudes quickly decay. It is anticipated that this phenomenon of increased damping at specific excitation frequencies may have potential application for shaft crack detection.

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