Merging Acoustic Black Holes and Local Resonators to Enhance Vibration Attenuation in Periodic Metamaterial Beams

This paper studies the bandgap properties and wave attenuation mechanisms of periodic beams embedded with a combination of acoustic black holes (ABHs) and local resonators (LRs). ABH refers to a retarding structure with a decreasing, power-lawed thickness profile, which gradually reduces the local phase velocity of incoming bending waves and thus traps the structural vibration energy within a confined area. Combining LR with ABH provides a practical approach to enhance structure vibration attenuation. To characterize the combined effects of ABH and LR, an energy-based formulation that uses B-splines as admissible functions is proposed. The B-spline basis functions can be allocated in a unique way such that the power-lawed variation of the beam profile can be accurately described despite the sharp thickness reductions and strong wave fluctuations in the ABH part. The vibration characteristics of the periodic beam are investigated under two scenarios: the resonance frequency of the LRs is tuned to coincide with the passband of the beam or the stopband of the beam. Improved vibration attenuations are observed in both scenarios, but the coupling behaviors and the underlying mechanisms are drastically different. To seek a clear explanation, an equivalent model of three degrees of freedom is established. By correlating the dynamics of the equivalent model with those of the beam model, it is found that the ratio between the stiffness of the resonator and that of the host beam plays an important role in forming new bandgaps. When the resonance of the LRs occurs in the passband of the ABH beam, the new bandgaps are a super-positioned effect of the original ABH bandgap and the LR bandgap. When the resonance of the LRs occurs outside the ABH bandgap, interactions between the LRs and the host beam are greatly enhanced, leading to an interesting frequency-splitting effect that dominates the formation of new bandgaps. Finally, the vibration responses of the proposed beam are investigated through experiments.