A new impact model for the flexible rocking oscillator

Motivated by the rocking motion of non-rigid structures during earthquakes, this paper investigates the rocking behaviour of laterally flexible oscillators. Previous studies in this area utilised ad hoc assumptions to consider transitions between different motion phases at impact. Some models explored direct transitions between two rocking phases assuming that elastic translational velocities remain the same, while others assumed the dissipation of all vertical momentum after a rocking phase, leading to a full contact phase upon impact. In this paper, an improved impact model is proposed, where consistent mechanical principles are used to determine the phase transition criteria. The model departs from the principle that the spring and damper elements of the oscillator cannot transfer horizontal impulses over infinitesimally small durations. Alongside other mechanical constraints, this principle yields a series of momentum equations, which are solved to determine the post-impact velocities when transiting from one rocking phase to another. Through these equations, it is shown that a transition to full contact implies a specific locus of the vertical impulse from the support medium. Analytical and numerical investigations are then conducted to comparatively evaluate the new impact model. It is demonstrated that the new impact model yields equivalent results to established rigid rocking impact models for effectively rigid structures and resolves long-standing issues associated with excessive energy dissipation observed in previous models. Time histories of free rocking motion and stability analyses under pulse excitations are used to illustrate and generalise the findings for a range of geometries.