Optimal Design of Tuned Mass and Negative Stiffness Amplifier Dampers with Inerter by H₂ Optimal Control Under Bidirectional Seismic Load

The combination of a negative stiffness damper and an inerter is a novel system that acts as an energy dissipation device for the structures under seismic loading. In the present study, a damped system with a single degree of freedom (SDOF) and supplementary dampers, including negative stiffness and inerter-based damper, was considered to control the response based on H₂ optimum control strategies. Five different configurations were investigated as supplementary dampers that included a tuned mass damper inerter (TMDI), clutching inerter damper (CID), and three configurations of negative stiffness amplifier damper inerter (NSADI). Structural responses under stochastic or random excitations were controlled by using the H₂ optimal control strategy based on minimizing the root mean square. The random white noise process was modeled as ground acceleration excitations. Optimum parameters were obtained from closed-form of expressions, and the corresponding equations of motion of the SDOF system with supplementary dampers were expressed in a state space form. Closed-forms of expressions were obtained for TMDI, NSADI, and CID from two-stage processes that consisted of firstly using techniques to search for the minimum of the H₂ form and secondly using a numerical search technique of curve fitting at a sequence scheme for arriving at the closed-form of expressions. Two different ground motions (horizontal and vertical ground motion excitations) and two sets of ground motions (near-fault and far-field) for input excitations for base-isolated structures were considered. A parametric study was carried out to optimize TMDI, CID and NSADI parameters, including mass ratio, negative stiffness ratio, positive stiffness ratio, natural frequency and damping ratio, according to the maximum reduction of the response maxima. The H₂ optimum technique plays a vital role in the response mitigation of base isolated structures under real seismic excitations. In addition, the base-isolated structure with NSADI and CID performs better compared to other supplementary dampers, considering the response reduction. Consequently, the response of both the negative stiffness damper and the tuned mass damper is boosted by adding an inerter mechanism for controlling the response of structures under seismic load.