H₂ Optimization of Tuned Liquid Column Damper With Inerter and Negative Stiffness According to Different Seismic Response Objectives

A nonlinear tuned liquid column damper with inerter and negative stiffness (NS-TLCDI) is proposed for structural vibration control with frequency tunability. The close-form transfer functions of different responses are presented by introducing an equivalent linearization method. The Routh–Hurwitz stability criterion is applied to confirm the stability of system with negative stiffness. Subsequently, analytical solutions for the H₂ optimized frequency ratio and damping ratio of NS-TLCDI under a white noise excitation are derived based on different engineering requirements that prioritize either absolute acceleration or relative displacement, the validity of which is verified by comparison with corresponding numerical solutions. Furthermore, the vibration suppression of NS-TLCDI in the frequency domain is assessed compared with a tuned liquid column damper inerter (TLCDI). The results indicate that NS-TLCDI offers a significant advantage over TLCDI in reducing the mean square values of the dynamic responses of the main structure, which becomes more pronounced as the absolute value of negative stiffness increases within the stability requirements. Moreover, the robustness of the system equipped with NS-TLCDI has also been considerably enhanced under this trend. Finally, a detailed time-history response analysis is conducted to evaluate the vibration control performance of NS-TLCDI under 44 real earthquake excitations. Both in terms of peak values and overall, the time-history responses of the structure equipped with optimal NS-TLCDI are consistently lower than that of optimal TLCDI. The energy dissipation of systems is further examined, indicating that the dashpot of NS-TLCDI remains the dominant dissipation component.