Stochastic stability of the inverted pendulum subject to support motion

This paper is concerned with the stochastic stability of an inverted pendulum with a point mass at the top and a spring at the base; the bar is massless. The base is subjected to a vertical acceleration A(t) that is supposed to be a Gaussian stochastic process. A line-like structure excited by a vertical ground motion can be idealized in this way. Without simplifying assumptions the study of the stochastic stability gives rise to a non-trivial problem as the equation of motion belongs to the class of damped Mathieu equations. Thus, it is assumed that during the motion the angle of rotation theta remains small so that sin(theta) = theta . In this way, the motion equation assumes the classical form of the second order oscillator, but the excitation is parametric so that there is a possibility of stochastic instability. Among the different definitions of stochastic stability, the almost sure (sample) stability and the stability in the second moments are considered herein. They are compared in the numerical analyses: it is found that they lead to notable differences in the stability boundaries and the almost sure stability is not conservative.