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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.Keywords Random loading · Liapunov functional · Almost sure stability · Rotatory inertia · Gaussian and harmonic process IntroductionRotating shafts, as elements of construction, often can take position to lose stability. The stability problem of rotating shafts arises when shafts are required to run smoothly at high speed. Destabilizing factors can be compressive force, the normal inertia force as well as certain types of damping. Suppose, that viscous damping forces act on the rotating shafts, these being external damping in absolute motion relatively to fixed internal axes, internal damping in motion relatively to rotating axes having the same angular velocity as the shaft.The dynamic stability of rotating shafts, with omission of the compressive force, was first analyzed by Bishop [1] using a modal approach. The same problem using the direct Liapunov method was examined by Parks and Pritchard [2]. Shaw and Shaw [3] considered instabilities and bifurcations in non-linear rotating shaft made of viscoelastic Voigt-Kelvin material without compressive force. Uniform stochastic stability of the rotating shafts, when the axial force is a wide-band Gaussian process with zero mean was studied by Tylikowski [4]. Stability problem of a closed cylindrical shell subjected to a time-dependent temperature field and thermally activated shape memory alloy hybrid rotating cylindrical shell was also analyzed by Tylikowski [5]. Tylikowski and Hetnarski [6] examined the influence of the activation through the change of the temperature on dynamic stability of the shape memory alloy hybrid rotating shaft. Young and Gau [7,8] investigated dynamic stability of a pretwisted cantilever beam with constant and non-constant spin rates, subjected to axial random forces. By using stochastic averaging method, they determined mean-square stability R. Pavlović (B) · P. Kozić · S. Mitić · I. Pavlović

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a b s t r a c tThe Lyapunov exponent and moment Lyapunov exponents of two degrees-of-freedom linear systems subjected to white noise parametric excitation are investigated. The method of regular perturbation is used to determine the explicit asymptotic expressions for these exponents in the presence of small intensity noises. The Lyapunov exponent and moment Lyapunov exponents are important characteristics for determining the almost-sure and moment stability of a stochastic dynamical system. As an example, we study the almost-sure and moment stability of the double-beam system under stochastic compressive axial loading. The validity of the approximate results for moment Lyapunov exponents is checked by the numerical Monte Carlo simulation method for this stochastic system.

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