The Melnikov criterion of instability for random rocking of a rigid block with a fractional derivative element

This paper discusses the problem of the cumulative influence of fractional dissipation and external noise on escape of a rigid block from a prescribed set of rocking oscillations. For simplicity, the dynamics of a weakly dissipative slender block affected by low energy noise is considered. Under these assumptions, the generic motion is depicted by the linearized equation of an undamped block with the well-known periodic vibro-impact solution. The stochastic Melnikov criterion based on the generic solution allows calculating the upper bound of escape probability of the structure in the presence of fractional dissipation of the Caputo type. The analytical results demonstrate an explicit dependence of escape probability on the fractional dissipation parameters.