Stability Analysis of the Rocking Block: Numerical Investigations on Analytical Stability Boundaries

The present paper illustrates some recently developed techniques to evaluate stability features, such as characteristic multipliers and Lyapunov’s exponents, for a problem with strong discontinuities. The problem analyzed concerns the plane dynamics of a rigid block simply supported on a harmonically moving rigid ground. In previous papers [1–3] many aspects have been investigated and the following results have been carried out: 1. the general procedure to approach numerically the problem has been outlined; 2. a rigorous method has been established to calculate characteristic multipliers and Lyapunov’s exponents at the instants of discontinuities; 3. the stability boundaries of symmetric sub-harmonic responses have been drafted by means of closed-form analytical methods based on various hypotheses of linearization. In this paper the new capacities allowed by the techniques and methods pointed out at the previous points 1., 2. and 3. are exploited with the aims: • to investigate numerically in the occurrence of dissipative impacts the features of dynamic responses across the upper boundary of a stability range (i.e. that of the (1,3) sub-harmonic response) included in a larger one (i.e. that of the (1,1) response); • to analyze the responses attainable with a value of the restitution coefficient equal to 1 to describe the impulsive phases (namely for non-dissipative impacts); in previous works, these responses have been classified as quasi-periodic or chaotic [4]. By means of the new techniques proposed and implemented, it has been possible to classify and analyze more deeply such presumed quasi-periodic or chaotic responses and at the same time to clarify the role played by the initial conditions.