Wave attenuation mechanisms in inerter-integrated locally resonant metamaterials

To address the trade-off between bandgap (BG) tunability and energy dissipation in low-frequency vibration suppression, this study proposes a one-dimensional locally resonant acoustic metamaterial (1D LRAM) with inter-unit-cell inerter-based mechanical networks (MNs). A unified framework combining the damped Floquet–Bloch theorem, power flow analysis (PFA), and structural impedance analysis (SIA) is developed to investigate wave propagation and energy transfer. A controlled comparison with a T1 reference employing the same coupling damping coefficient shows that the inerter-coupled T6 configuration increases the total intrinsic BG width from 1.67 to 2.67 Hz and the BG-integrated attenuation from 2.58 to 3.95 Hz m −1 . Complex band analysis shows that damping merges the separated propagating and evanescent branches into continuous complex-wave branches with coexisting phase progression and spatial attenuation. SIA reveals a frequency-dependent redistribution of input energy: under low damping, reactive-power dominance, high input impedance, and low transmitted power indicate impedance-controlled attenuation near the BGs, whereas at higher damping, active-power transfer intensifies over selected frequency intervals and inerter-induced relative motion promotes energy dissipation. Independent time-domain simulations reproduce the frequency-domain wavefields, power transmission, and spatial attenuation of the finite T6 lattice. A reduced-order non-ideal inerter analysis further shows that the intrinsic BG-width and BG-integrated attenuation advantages remain robust within the investigated ranges of parasitic stiffness and equivalent viscous loss, although fixed-band gains remain sensitive to the prescribed evaluation interval. Linear hysteretic damping (LHD) provides stronger low-frequency attenuation, whereas viscous damping (VD) becomes more effective at higher frequencies. Overall, the proposed framework links complex-wave propagation, impedance-controlled attenuation, and damping-dependent dissipation, providing a physically transparent basis for the dynamic design of inerter-based metamaterials.