Analysis is presented which explains earlier hypotheses and experimental results that circumferential stiffness variations induced by rubbing over a portion of a rotor’s orbit can lead to parametric excitation of half-speed whirl at a rotor’s natural frequency. The results do not explain cited occurrence of 1/3 and 1/4 running-speed whirl associated with the operation of rotating machinery. A polar-coordinate version of the Jeffcott model is employed to model the effect of rubbing over a portion of a rotor’s precessional orbit, with the rub-induced restoring force defined in terms of a Fourier series expansion. Rubbing contact between a rotor and its housing results in both an increased radial stiffness and a tangential force due to Coulomb friction. Hence, there are two mathematically distinct sources for rotor parametric excitation, and the results concern the influence of viscous damping on the ranges of unstable speeds induced by these two sources of parametric excitation. The analysis based on Hsu’s method yields the following results: (a) For zero Coulomb and external viscous damping, parametric excitation yields a band of unstable speeds about a running speed that is twice the natural frequency (ω = 2λ). (b) For finite Coulomb damping and no viscous damping, a rotor in a partial rubbing condition is predicted to be unstable at all running speeds. (c) External damping reduces the range of unstable running speeds centered about the ω = 2λ parametric excitation frequency, and sufficient damping completely eliminates the instability.

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