Stability characteristics of multi-frequency parametrically-excited rotor-oscillator systems

Abstract

The effect of multi-frequency parametric excitation on the stability of an oscillator is examined in the context of a rotating system where the rotor and the stator have comparable masses and hence the ‘stator’ cannot be considered to be mechanically fixed. The differential equations governing the dynamics of such systems have time-varying terms which are periodic in the rotor frequencies. Floquet theory is employed to numerically develop stability diagrams which distinguish stable from unstable parameter combinations. Reasons for the occurrence of instabilities are explained using the two time scale perturbation method. It was found that the most significant resonance in the single rotor case occurred when the rotor speed equaled the sum of two unforced natural frequencies. In addition, harmonic resonances were found to arise when the rotor speed matched the unforced natural frequencies. Furthermore, the excitation from a second rotor caused instabilities to arise around regimes where multiple combination resonances occurred together. The occurrence of certain experimentally-observed resonances and regimes of low amplitude oscillations were correlated with the model, and the corresponding rotor trajectories were explored. This investigation is useful for identifying operating regimes so as to avoid catastrophic failure in multi-frequency rotating machines such as multi-rotor helicopters and gas turbines with contra-rotating stages.