This paper presents an investigation into the vibration of rotor-bearing systems with time-dependent spin rates. Due to this spin rate, parametric instability may take place in certain situations. In this work, the Galerkin method is used to eliminate the dependence on the spatial coordinates, and then the method of multiple scales is applied to derive periodic solutions and expressions for the boundaries of unstable regions analytically. Numerical results are given for the case where the spin rate is characterized as a small, harmonic perturbation superimposed on a constant rate. The effects of system parameters on the changes of the boundaries of unstable regions are shown.

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