Structural control strategies have attracted great attention in reducing unwanted structural responses. To mitigate the vibrations of seismically excited buildings, this paper proposes a novel vibration control device: an inerter-enhanced bistable nonlinear energy sink (BNESI). A three-story building structure is selected as an example, where the BNESI is attached to the top of the structure and linked to the top second floor of the structure, to explore the control effect of the device under earthquakes. The restoring force of the BNESI and equations of motion for the BNESI-controlled system are derived first. Subsequently, a multi-objective optimization design method of the BNESI is proposed and applied in building structures. The input energy robustness of the BNESI is then accessed by various initial velocities subjected to the structures. Moreover, the seismic vibration mitigation capacity of the BNESI is systematically investigated under a group of near-field and far-field excitations. Analysis results show that the proposed design algorithm for BNESI can be an attractive alternative to effectively obtain optimal parameters of the control device. The BNESI is less sensitive to the input energy levels, and can stably capture and dissipate energy from the primary structure. Also, the BNESI system provides good control effectiveness with less sensitivity to the uncertainty of seismic excitations. In addition, with less standard deviation, the damper stroke of the BNESI is only about a quarter of that of the TMD under the same excitation. Moreover, the damage to the primary structure has a negligible effect on the control effect of the BNESI system. Meanwhile, the BNESI can decrease the energy dissipation demand on the structure, and the energy distribution mode of the BNESI system shows less sensitivity to the frequency decrease of the structure.
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