H2 and H∞ Norms-Based Optimization of TID Parameters in Damped Structures for Enhanced Vibration Reduction

This study investigates the optimal parameter design of the tuned inerter damper (TID) in single-degree-of-freedom (SDOF) systems. First, based on the [Formula: see text] and [Formula: see text] optimization criteria, the analytical formulas for the optimal natural frequency ratio and optimal damping ratio of the TID subsystem are derived for undamped main structures. Subsequently, considering the significant impact of the optimal natural frequency ratio on the TID’s vibration reduction effect, the analytical formulas for the optimal natural frequency ratio and optimal damping ratio are derived based on the mixed [Formula: see text] optimization criterion. Taking into account the inherent damping of the main structure, the analytical formulas for the optimal natural frequency ratio of the TID are derived using the approximate extended fixed-point theory (AEFPT) with the mixed [Formula: see text] optimization criterion. Meanwhile, the analytical formula for the optimal damping ratio of the TID is derived based on the [Formula: see text] optimization criterion. To validate the vibration reduction effect of the TID under different types of excitations, harmonic, white noise, and seismic excitations are applied to the system, and the effects on the peak displacement response and displacement response variance are analyzed under different optimal TID design schemes. Numerical results indicate that for SDOF-damped main structures subjected to harmonic and white noise excitations, the C4 scheme, which considers the main structure’s damping using the mixed [Formula: see text] optimization criterion achieves a better balance between peak response control and response variance control compared to the C1–C3 schemes based on single [Formula: see text] or [Formula: see text] optimization criteria. Furthermore, under seismic excitation, the C4 scheme exhibits significant advantages in displacement response control for main structures with higher inherent damping.