Mitigation of vertical vibrations in coupled meta-rods through internal interactions and boundary constraint

Environmental vibrations caused by seismic waves and traffic loads pose increasing risks in urban areas due to their low attenuation and structural impact. To mitigate the in-plane bulk waves and vertical Raleigh wave component generated by ambient vibrations, we proposed a coupled meta-rod assembled from a local resonance (LR) bar and a negative-stiffness (NS) system. This meta-rod is designed in two forms to redistribute the wave energy, where the NS system is introduced as a boundary constraint. We develop analytical models based on the Bloch conditions and the transfer matrix method to solve and investigate dispersion relationship and transmission abilities. Complementarily, finite elements (FE) models are also established to validate the analytical results and further visualise the interaction between the LR bar and NS system during longitudinal wave propagation. The results indicate that most of long-wavelength waves propagate through the low-stiffness medium within the coupled system. As wave energy is redistributed across frequency, an additional attenuation region emerges beyond the LR bandgap due to the maximum in-phase motion between the LR bar and NS system. This internal interaction gradually dominates the attenuation mechanism as the LR bandgap shifts toward lower frequencies under boundary constraints, providing more effective suppression before the resonance frequency. This study offers a promising strategy for the design of efficient vertical vibration mitigation solutions.