Optimal design of negative stiffness amplifying damper for seismic response control of damped civil structures

The negative stiffness amplifying damper (NSAD), as one of the passive control devices, has been shown to effectively mitigate seismic vibrations in engineering structures. However, existing studies have predominantly concentrated on the simplified closed-form solution for the optimal design parameters of NSAD derived under the assumption that the primary structure is undamped. Therefore, one of the contributions of this study lies in proposing a closed-form solution for determining the optimal design parameters of NSAD while considering the damping ratio of the primary structure. To achieve this objective, the analytical formulation for the effective damping ratio (EDR) of the NSAD is established to assess its control performance. The optimal closed-form solution for NSAD parameters (referred to as the EDR solution) is derived by maximizing the EDR to achieve optimal control performance. The applicable ranges of the negative stiffness ratio and the positive stiffness ratio for the EDR solution are specified with respect to the stability condition and the effective damping ratio enhancement (EDRE) effect, respectively. The applicability of the EDR solution is rigorously validated through comparative analysis against the fixed-point solution, employing both frequency-domain and time-history analyses. The results indicate that the EDR solution can be regarded as more accurate and reasonable compared to the simplified closed-form solution derived from the undamped primary structure hypothesis, particularly for positive stiffness ratios exceeding 0.1. Moreover, the enhanced effectiveness of the EDR solution in mitigating displacement responses of the primary structure, as compared to the fixed-point solution, is attributable to its superior ability to control system input energy.