Equal modal damping-based optimal design of a grounded tuned mass-damper-inerter for flexible structures

For design of tuned mass-damper-inerter (TMDI) in flexible multi-degrees-of-freedom (MDOF) structures, the classic 2-DOF model is incapable of accounting for the background flexibility that comes from non-resonant modes, leading to an unbalance in the frequency response of the flexible structure-TMDI system. This becomes more critical for the grounded TMDI attached to a lower floor (a practical installation location). This paper proposes a set of closed-form formulas for optimal design of the grounded TMDI based on equal modal damping principle. An analogue 2-DOF model accounting for the background flexibility of the structure is first constructed. The root locus analysis is then performed, and three optimal frequency-tuning formulas are derived by the equal modal damping ratio criterion corresponding to three different damping levels of TMDI. A straightforward approach for determining the optimal TMDI damping ratio is proposed based on the bifurcation point present in the root locus. All the derived optimal design formulas turn out to be dependent on the structural inherent property. The effectiveness of the proposed closed-form formulas are assessed via both frequency-domain and time-history (with earthquake ground motions) analyses of a 10-story building. Results indicate that the proposed design formulas leads to superior performance of the grounded TMDI compared to the conventional formula ignoring background flexibility. When equal modal damping of the system is guaranteed, an excellent level of vibration reduction of structural seismic responses can be achieved.