Moving mass over a viscoelastic system: asymptotic behaviours and insights into nonlinear dynamics

Moving masses are of interest in many applications of structural dynamics, soliciting in the last decades a vast debate in the scientific literature. However, despite the attention devoted to the subject, to the best of the authors’ knowledge, there is a lack of analysis about the fate of a movable mass when it rolls or slips with friction on a structure. With the aim of elucidating the dynamics of the simplest paradigm of this system and to investigate its asymptotic response, we make reference to a two-degree-of-freedom model made of an elastically vibrating carriage surmounted by a spherical mass, facing the problem both theoretically and experimentally. In case of linear systems, the analytical solutions and the laboratory tests performed on ad hoc constructed prototypes highlighted a counterintuitive asymptotic dynamics, here called binary: in the absence of friction at the interface of the bodies’ system, the mass holds its initial position or, if nonzero damping acts, at the end of the motion it is in a position that exactly recovers the initial relative distance carriage–sphere. While the first result might be somewhat obvious, the second appears rather surprising. Such a binary behaviour is also confirmed for a Duffing-like system, equipped with cubic springs, while it can be lost when non-smooth friction phenomena occur, as well as in the case of elastic springs restraining the motion of the sphere. The obtained analytical results and the numerical findings, also confirmed by experimental evidences, contribute to the basic understanding of the role played by the damping parameters governing the systems’ dynamics with respect to its asymptotic behaviour and could pave the way for designing active or passive vibration controllers of interest in engineering.