Stochastic stability of the inverted pendulum subjected to delta- correlated base excitation. (June 2018) - Floris, Claudio
Highlights: This paper is aimed at analyzing the stochastic stability of an inverted pendulum subjected to a parametric excitation having a white spectrum. It may be either a Gaussian white noise or a Poisson white noise. Under the hypotheses of small rotations the motion equation is lead to the standard form of the motion equation of a second order oscillator having a random perturbation in its stiffness. Both the almost sure (sample) stochastic stability and the stability in the response statistical moments are considered. The almost sure stability is analyzed by computing the top Lyapunov exponents of the system, while the criterion of stability in the moments is obtained by studying the eigenvalues of the matrix of the system of differential equations ruling the time evolution of the moments. The principal finding of the paper are: (1) the almost sure stability criterion foresees a stability regions markedly larger than those yielded by the stability in the second moments; (2) in the case of Poisson white noise, the mean square stability is determined only by the arrival rate of underlying Poisson counting process and by the mean square amplitude of the pulses: the cumulants beyond the second order do not affect it. Abstract: 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 at the base to a vertical acceleration A ( t ) that is supposed to be a white noise (delta-correlated) stochastic process. Both Gaussian and Poissonian white noises are considered. A line-like structure excited by a vertical ground motion can be idealized in this way. It is assumed that during the motion the angle of rotation ϑ remains small so that sin ϑ≅ϑ. 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. The almost sure (sample) stability and the stability in the second moments are considered herein. It is found that the two stability criteria lead to notable differences in the stability boundaries and the almost sure stability is not conservative. The mean square stability under the Poisson white noise is determined only by the arrival rate of underlying Poisson counting process and by the mean square amplitude of the pulses: the cumulants beyond the second order do not affect it.