Stochastic stability of a rotating shaft

The stochastic stability problem of an elastic, balanced rotating shaft subjected to action of axial forces at the ends is studied. The shaft is of circular cross-section, it rotates at a constant rate about its longitudinal axis of symmetry. The effect of rotatory inertia of the shaft cross-section is included in the present formulation. Each force consists of a constant part and a time-dependent stochastic function. Closed form analytical solutions are obtained for simply supported boundary conditions. By using the direct Liapunov method almost sure asymptotic stability conditions are obtained as the function of stochastic process variance, damping coefficient, damping ratio, angular velocity, mode number and geometric and physical parameters of the shaft. Numerical calculations are performed for the Gaussian process with a zero mean and as well as an harmonic process with random phase.