Closed-form solutions for the optimal design of lever-arm tuned mass damper inerter (LTMDI)

Tuned mass damper inerter (TMDI) has become a popular choice for vibration control in flexible structures, including long-span bridges, due to its superior performance compared to traditional tuned mass dampers (TMDs), such as improved control effect and smaller static displacement. However, the control effect of TMDI is heavily reliant on the spanning distance of inerter, which typically generates a significant internal additional moment due to the force imbalance, limiting the practical application of TMDI on flexible structures. For this reason, a novel lever-arm TMDI (LTMDI) is proposed in this study, to suppress the vibrations of long-span bridges. The nondimensional governing equations of the bridge-LTMDI system are established in modal coordinates, and the closed-form solutions of the optimum frequency and damping ratios of LTMDI are derived based on the classical fixed-point theory. Parametric studies are conducted to investigate the influences of the fulcrum position, modal shape ratio, mass ratio, inertance ratio and asymmetric mode on the control effectiveness. Moreover, for comparison, the control effectiveness of LTMDI is further compared to those of the TMD and TMDI. Finally, a long-span bridge is selected as an example to demonstrate the effectiveness of LTMDI in reducing vortexinduced vibration (VIV). The results show that the LTMDI has superior control effect and space feasibility as compared to the TMD and TMDI in both frequency and time domains. The derived analytical formulas can accurately determine the optimal parameters, providing a simple yet efficient guideline for the application of LTMDI in flexible structures.