H ∞ closed-form solution of tuned mass damper enhanced with negative stiffness element (TMD-NS) for structural vibration control

Negative stiffness element has been applied to improve the control performance of tuned mass dampers (TMDs) recently, and two tuned mass dampers enhanced with negative stiffness (TMDNS) element, namely KDamper (Without loss of generality, it is referred to as TMD-NS I in the present study) and Extended KDamper (EKD, TMD-NS II), have been developed. Previous studies have demonstrated the control effectiveness of TMD-NS I and II. However, there still exist some issues to be addressed: (1) previous studies normally optimize TMD-NS via the intricate and timeconsuming numerical methods, the analytical solutions for the optimal design parameters of TMD-NS II remain unknown; (2) a comprehensive and exhaustive evaluation that compares the control effectiveness of TMD-NS I and II is absent from existing literature. To fill these research gaps, this study derives closed-form optimum solutions for TMD-NS II using the H infinity approach. The control effectiveness of TMD-NS I and II in suppressing the seismic responses of structures is investigated systematically. Specifically, the analytical model of an undamped SDOF system equipped with TMD-NS I or II is first developed within a unified framework, and corresponding dynamic equations of motion are formulated. Subsequently, the optimal parameters of TMD-NS I and II are derived based on the classical "fixed-point" theory. Based on the derived optimal parameters, the control effectiveness of TMD-NS I and II are examined by using a damped SDOF system subjected to harmonic excitations and real earthquake ground motions. Finally, a 5-storey isolated benchmark building model is adopted to further investigate the effectiveness of TMD-NS in the seismic protection of engineering structures. The results reveal that the derived closed-form solutions are accurate in capturing the optimal parameters of TMD-NS. In addition, both the optimized TMD-NS I and II outperform the conventional TMD in reducing the seismic responses of structures. Furthermore, TMD-NS I proves more effective in reducing the absolute acceleration of the isolated building, whereas TMD-NS II demonstrates better performance in mitigating the isolating deformation. In a nutshell, both the TMD-NS I and II are highly effective alternatives to conventional TMDs, showcasing superior performance in vibration reduction and robustness.