Transverse vibrations of rotating shafts: Probability density and first-passage time of whirl radius

Transverse vibrations are considered for a single mass/two-degrees-of-freedom rotating shaft with linear internal or “rotating” damping and nonlinear external damping. The shaft is excited by external random forces. Analysis of resulting random vibrations is based on stochastic averaging method which yields separated (in the linear approximation) equations for complex amplitudes of forward and backward whirling motions. The former of these motions is shown to be dominant at rotation speeds in the vicinity of the instability threshold. Using this approximation an analytical solution is obtained for probability density of squared radius of the shaft's whirl. This solution can be used to detect on-line shaft's instability from its observed response. Solution is also obtained for expected time for reaching given level by the squared whirl radius of the shaft.