Soft impact dynamics of a cantilever beam: equivalent SDOF model versus infinite-dimensional system

Abstract

Non-smooth dynamics of a cantilever beam subjected to a transverse harmonic force and impacting onto a soft obstacle is studied. Upon formulating the equations of motion of the beam, proper attention is paid to identifying the mechanical properties of an equivalent single-degree-of-freedom (SDOF) piecewise linear impacting model. A multi-degree-of-freedom (MDOF) model of the impacting beam is also derived via standard finite elements. An ‘optimal’ identification curve of the obstacle spring rigidities in the two models is obtained by comparing the relevant pseudo-resonance frequencies. The identification is then exploited in the non-linear dynamic regime to get hints on some main, mostly regular, features of non-linear dynamic response of the impacting beam by the actual investigation of the behaviour of the sole equivalent SDOF model, with a definitely lower computational effort. Sample regular and non-regular responses of the MDOF model are also presented where the identification does not work. Overall, useful points are made as regards the possibility and the limitations of referring to an SDOF impacting model to investigate the non-linear response of the underlying infinite-dimensional system.