Beneficial effects of TMDs with embedded vibro-impacting mass on fully isolated rigid block-like structures

This study investigates the nonlinear seismic response of rigid block-like structures supported by a dual horizontal-vertical base-isolation system and equipped with a Tuned Mass Damper (TMD) incorporating an embedded Vibro-Impacting Mass (VIM). The proposed configuration combines three strongly interacting mechanisms—base isolation, rocking dynamics, and vibro-impact interactions—and explores a protection strategy that has not yet been systematically investigated for isolated rigid block-like structures. A unified Lagrangian formulation is developed to consistently describe both full-contact and rocking phases, together with impact events occurring at the block base and within the TMD–VIM device. The nonlinear hysteretic behavior of the horizontal isolation system is represented through a Bouc–Wen model, whereas the vertical isolation system and the TMD restoring forces are described by Kelvin–Voigt viscoelastic formulations. The resulting framework enables the analysis of the coupled isolation–rocking–impact dynamics under strong horizontal and vertical seismic excitations. A comprehensive parametric investigation is performed by varying the horizontal and vertical isolation periods, the TMD dynamic properties, and the mass ratio and clearance of the embedded vibro-impacting mass. Rocking and displacement maps are constructed to identify the boundaries separating full-contact, rocking, and overturning regimes. The results demonstrate that the integration of a vibro-impacting TMD into a dual horizontal–vertical seismic isolation system significantly enhances the seismic performance of the protected structure by modifying the stability boundaries that govern its response. In particular, the embedded VIM enlarges the full-contact region, converts rocking responses into full-contact motion over significant portions of the parameter space, and prevents overturning in configurations that would otherwise be unstable. The effectiveness of the proposed device is shown to depend strongly on the distribution of mass between the TMD and the VIM, with the best performance achieved when the primary TMD mass remains dominant. The findings provide new insight into the nonlinear mechanisms governing the interaction among seismic isolation, rocking motion, and vibro-impact phenomena, and demonstrate the potential of TMD–VIM systems as an effective strategy for enhancing the seismic stability of isolated rigid block-like structures.