Equations of motion for a four-degree-of-freedom dynamical system describing the vibrations of a one-disc elastic rotor taking into account gyroscopic moments on a non-linear flexural base are derived. A new version of the multiple scales method is developed and applied to analyse the non-linear dynamics of such a system for different resonances. The steady motions of the rotor are analysed. From the asymptotic analysis, it is shown that out-of-plane motions of the disc exist in the symmetric rotor.
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