Mean square stability of a rotating shaft under combined harmonic and stochastic excitations

The dynamic stability of a rotating shaft under parametric excitation consisting of a combination of harmonic terms and stationary stochastic processes is considered. The intensities of both harmonic and stochastic excitations and correlation time of stochastic excitations are assumed to be small in order to obtain approximate analytical results. Explicit stability conditions are derived for the first and second moments of a two-degree-of-freedom rotating shaft. When the stochastic excitation is a white noise excitation, the first moment stability conditions reduce to that of the deterministic case. It is shown that addition of non-white noise excitation has a stabilizing effect on the parametric instability of harmonically excited rotating shafts. Finally, the stability conditions of a symmetric shaft along with their numerical results are presented.