Closed-form optimal solution of two-degree-of-freedom system with Inerter based on equal modal damping with potential application in non-structural elevator for seismic control

Targeting the great demand for adding non-structural elevators to old residential buildings, this article proposes an updated configuration of tuned mass damper inerter (U-TMDI) applied in external non-structural elevators. An analog 2-DOF system is established to describe the residential building controlled by an inerter elevator. Then, three closed-form optimal solutions for designing the U-TMDI are derived via the fixed-point method, equal modal damping criterion, and the infinity damping assumption. Subsequently, these optimal solutions are compared and discussed involving the expressions of tuning frequency ratio and the optimal damping parameters, root locus diagram and supplemental damping ratios, transfer function, and robustness to frequency variation, respectively. Finally, a residential building example is adopted to validate the feasibility of the proposed retrofitting strategy and the closed-form optimal design solutions. It is demonstrated that the optimal tuning frequency ratios derived by the fixed-point method and equal modal damping criterion are different for U-TMDI due to the influence of elevator stiffness ratio eta$\eta $, while its degradation forms for tuned mass damper (TMD) are identical, recognizing the importance of the elevator stiffness. Moreover, the proposed retrofitting strategy of using an inerter elevator can significantly mitigate the main structure displacement by about 18%similar to 23% for both far-field and near-fault earthquakes.