Nonlinear vibrations of a rotating shaft with broadband random variations of internal damping

A simple Jeffcott rotor is considered with broadband temporal random variations of internal damping which are described using the theory of Markov processes. Transverse response of the rotor with stiffening nonlinearity either in external damping or in restoring force is studied by stochastic averaging method. This method reduces the problems to stochastic differential equations (SDEs) for which analytical solutions are obtained for the Fokker–Planck–Kolmogorov (FPK) equations for stationary probability density functions (PDFs) of the squared whirl radius of the shaft. These PDFs do exist beyond the dynamic instability threshold and they correspond to forward whirl of the rotor. At rotation speeds just slightly above the instability threshold, the response PDF has integrable singularity at zero which corresponds to intermittency in the response.