Krylov-Bogoliubov Approach to Non-Linear Hysteretic Instability in Rotordynamics

The internal friction due to the shaft hysteresis or the shrink fitting release exerts a destabilizing effect on the overcritical rotor whirl, but may be counteracted by other external dissipative sources and/or by proper anisotropy of the support stiffness. The internal friction effect may be treated by either dry or viscous models, obtaining similar results in the hypothesis of small dissipation levels, provided that proper equivalence criteria are defined between the two approaches. The equivalence is here stated by imposing the same energy dissipation over a large number of shaft revolutions. Approximate closed-form autonomous solutions for a symmetric rotor arrangement subject to Coulombian non-linear friction are derived by an averaging approach of the Krylov-Bogoliubov type, in order to ascertain the result similarity between the two dissipative assumptions. Summing up, the viscous equivalent linear assumption appears conservative in general and may be accepted for a straightforward analysis of the overall rotor dynamics in the whole speed range.