Parametric Optimization and Vibration Attenuation Analysis of Hierarchical Tuned Mass Damper Inerter

While the classic ungrounded tuned mass damper inerter (TMDI) benefits from structural simplicity and engineering adaptability, its performance is diminished due to the negative stiffness effect resulting from direct inerter-primary structure coupling. This study introduces a novel hierarchical tuned mass damper inerter (H-TMDI), which incorporates a TMD subsystem between the conventional ungrounded TMDI and the primary structure. Hierarchical configuration mitigates the negative stiffness effects inherent in traditional ungrounded TMDI systems while achieving enhanced control performance. We establish the mechanical model of the inerter element and derive the governing equations for the H-TMDI dynamic system. Through H∞ norm and H 2 norm optimization, we systematically evaluate the control parameters and performance characteristics. The results demonstrate that H-TMDI achieves superior control efficacy under harmonic excitation. A comprehensive parametric study examines the influence of critical factors, including mass distribution ratio of two-stage systems, normalized apparent mass, and total mass ratio on vibration attenuation performance. It is revealed by robustness analysis that the H-TMDI outperforms the conventional TMD in three key aspects: enhanced peak attenuation in the main structure's transfer function, broader vibration reduction bandwidth, and reduced sensitivity to parameter variations. A closed-form expression for the H-TMDI control force is formulated, revealing the fundamental mechanism responsible for eliminating the negative stiffness effects. From a control system structural standpoint, the hierarchical design effectively decouples the inerter from the main structure, thereby eliminating the inerter's negative stiffness effect while preserving its mass amplification capability, ultimately leading to enhanced system performance.