The use of Markov chain theory in the problem of the dynamic stability of an elastic rod

The stability of the vertical position of an elastic rods, hinged at the ends and loaded at the upper end with a time-dependent longitudinal force containing a stochastic component, is considered. A homogeneous Markov chain is used to model the stochastic component. An inequality is obtained that depends on the characteristics of the stochastic force and the damping factor of the bar and, depending on whether this inequality is satisfied, the above equilibrium position will either be attracting or unstable.