Parameters optimization of series–parallel inerter system with negative stiffness in controlling a single–degree–of–freedom system under base excitation

This study proposes a series–parallel inerter system with negative stiffness for the passive vibration control of an undamped single–degree–of–freedom system under base excitation. The necessary and sufficient conditions for stability of series-parallel inerter system with negative stiffness are established by Routh–Hurwitz criterion, and the stability boundary is obtained. The tuning parameters of the series-parallel inerter system with negative stiffness are determined through fixed point theory, and a comparison between the vibration mitigation performance of series-parallel inerter system with negative stiffness, series–parallel inerter system (without negative stiffness), and tuned mass damper is presented considering both harmonic excitation, transient excitation, and random (white noise) excitation. The results of this study demonstrate that under base harmonic excitation, series-parallel inerter system with negative stiffness outperforms the series–parallel inerter system and tuned mass damper in terms of suppression bandwidth and reducing the peak vibration amplitude of the primary mass. In the case of base acceleration–excited primary structure, more than 49.84% and 67.53% improvement can be obtained from series-parallel inerter system with negative stiffness as compared with tuned mass damper in terms of suppression bandwidth and reducing the peak vibration amplitude, respectively. Whereas in the case of base displacement–excited primary structure, more than 78% and 80% improvement can be obtained from series-parallel inerter system with negative stiffness, respectively, following the same criteria. A slightly lower improvement has been obtained from series-parallel inerter system with negative stiffness as compared with series–parallel inerter system, which justified the superiority of series–parallel inerter system compared to tuned mass damper. The transient response investigation showed that series-parallel inerter system with negative stiffness outperforms the series–parallel inerter system and tuned mass damper in terms of much shorter stabilization times and lower peak amplitude of the primary mass. Finally, the further comparison among these devices (series-parallel inerter system with negative stiffness, series–parallel inerter system, and tuned mass damper) under white noise excitation also shows that series-parallel inerter system with negative stiffness is superior to series–parallel inerter system and tuned mass damper for a small inertance mass ratio. This result could provide a theoretical basis for the design of inerter-based isolators with negative stiffness.