Closed-form optimization of tuned mass-damper-inerter (TMDI) in flexible structures

Conventional optimal calibration of TMDI is based on the classic 2-degree-of-freedom (2-DOF) model with one structural DOF in terms of the resonant mode. For flexible MDOF structures, this implies that background flexibility from other non-resonant modes is omitted, resulting in an unbalanced frequency response of the flexible structure-TMDI coupled system. To address this issue while avoiding computationally-demanding numerical optimization, this paper develops closed-form optimal calibration formulas for the two-terminal TMDI with arbitrary installation topologies in flexible structures. The TMDI is characterized by the mechanical property that the two terminal forces are different in magnitude, and all previous calibration methods fail in such a case. In this paper, a novel analogue SDOF-TMDI model is proposed with two equivalent background stiffness coefficients representing the non-resonance response (the background flexibility) of the flexible structure at the two damper installation locations. These two stiffness coefficients are derived by modal analysis of the original system. Based on the seventh order characteristic equation of this analogue model, analytical formula is derived for the optimal TMDI frequency ratio using the pole-placement approach, where the background flexibility contribution is explicitly represented. For comparison, optimal TMDI frequency ratio is also derived from the classic 2-DOF model ignoring background flexibility, and optimal TMDI damping ratio is developed from this model as well. All the developed formulas are capable of accommodating arbitrary installation topologies of the TMDI. The analytical design formulas are evaluated via both frequency-domain and time-history (with earthquake ground motions) analyses of a 10-storey shear frame. The optimal frequency-tuning formula accounting for background flexibility lead to a balanced frequency response curve of the primary structure with any TMDI installation configuration, as well as superior vibration control performance of the TMDI in flexible structures than the classic formula.