Stability analysis of rotor whirl under nonlinear internal friction by a general averaging approach

The two main sources of internal friction in a rotor-shaft system are the shaft structural hysteresis and the possible shrink-fit release of the assembly. The internal friction tends to destabilize the over-critical rotor running, but a remedy against this effect may be provided by a proper combination of some external damping in the supports and an anisotropic arrangement of the support stiffness, or at most by the support damping alone, depending on the system geometry. The present analysis reported here considers a general asymmetric rotor-shaft system, where the rotor is perfectly rigid and is constrained by viscous–flexible supports having different stiffnesses on two orthogonal planes. The internal friction is modelled by nonlinear Coulombian forces, which counteract the translational motion of the rotor relative to a frame rotating with the shaft ends. The nonlinear equations of motion are dealt with using an averaging approach based on the Krylov-Bogoliubov method with some adaptation to address the multi-degree-of-freedom nature of the problem. Stable limit cycles may be attained by the overcritical whirling motions, whose amplitudes are inversely proportional to the external dissipation applied by the supports. A noteworthy result is that the stiffness anisotropy of the supports is recognized as beneficial in reducing the natural whirl amplitudes, albeit mainly in the symmetric configuration of the rotor at the mid span and, to a rather lesser extent, in the asymmetric configuration, which then requires a stronger damping action in the supports.