Parameter optimization of tuned inerter damper for vibration suppression in structures with damping

This study investigates the parameter optimization of tuned inerter damper (TID) in a single degree of freedom (SDOF) system, with a focus on vibration mitigation. As a type of inerter based dynamic vibration absorber (IDVA), TID achieves vibration suppression of the primary structure by replacing the mass block in traditional dynamic vibration absorber (DVA) with an inerter, thereby minimizing the increase in physical mass. With the displacement response of the primary structure as the objective function, this study first employs the classical fixed point theory (FPT) to derive the analytical formulations for the optimal parameters of TID following the $H_{\infty }$ optimization approach and neglecting the inherent damping of primary structure. The influence of optimal parameter deviations on the vibration mitigation effect of TID is also analyzed, revealing that the deviation of the optimal natural frequency ratio has a significant impact on the vibration mitigation performance. By flattening the amplitude curve of the displacement transfer function of primary structure between the fixed points, this study derives the analytical solution for the optimal damping ratio of TID using the extended fixed point theory (EFPT). When the inherent damping of primary structure is considered, this study employs the approximate extended fixed point theory (AEFPT) to derive the approximate optimal parameters of TID. A comparative study of the optimal natural frequency ratios obtained using FPT, AEFPT and numerical searches reveals that the discrepancies among the three methods are minimal for structures with low inherent damping. However, as the inherent damping ratio of the primary structure increases, the optimal natural frequency ratio obtained using FPT deviates significantly from the exact value, whereas the approximate optimal natural frequency ratio derived using AEFPT can significantly reduce this deviation. Combining the conclusion that the vibration mitigation effect of TID is significantly influenced by the natural frequency ratio, this study suggests that for highly damped primary structures, the use of AEFPT can yield more optimal TID parameters, thereby enhancing the robustness of vibration mitigation performance.