H 2 and H ∞ optimal designs of tuned inerter dampers for base motion excited structures with inherent damping

Tuned inerter damper (TID) has recently gained increasing attention as a new structural control mechanism for seismic protection of structures. Currently, theoretical investigations are undertaken by researchers to reveal its fundamentals and to understand its underline principles in altering the structural performances of structures against dynamic loadings. However, the comprehensive study of optimization design of TID for undamped structures is lacking and the majority of the research focuses on the optimization of TID for structures without any damping. This research evaluates the H 2 and H ∞ optimal designs of TIDs on the structures with damping. Using SDOF structure as an example, the frequency response function of the system equipped with TID underground motion excitation is obtained. The H 2 and H ∞ designs of TID for structures without damping are derived considering various response parameters using analytical method. A numerical search method is utilized for the H 2 and H ∞ designs of TID for structures with damping; meanwhile, a set of explicit formulae are obtained by curve-fitting for convenience in the application. Finally, the relative motion response of the inerter is explored, and an optimal design formula of TID which can reduce the displacements of primary mass and inerter simultaneously, is proposed.