We investigate the dynamic behavior and topology of quasiperiodic resonant metastructures. We show that the quasiperiodic arrangement of resonators introduces frequency band gaps in addition to the locally resonant band gap defined by the natural frequency of the resonators. The concept is illustrated on a beam with an array of mechanical resonators. Numerical evaluation of the spectrum as a function of the quasiperiodic arrangement of resonators reveals a structure reminiscent of a Hofstadter butterfly and allows the study of key topological properties. The results illustrate the occurrence of additional band gaps that are topologically nontrivial and that host edge-localized modes in finite structures. The occurrence of these gaps and of the associated edge states is demonstrated experimentally by measuring the frequency response of the beam and by evaluating the spatial distribution of selected operational deflection shapes. The results unveil the potential of deterministic quasiperiodic structural designs to induce wave localization and attenuation over multiple frequency bands, which may find applications in vibration isolation and energy harvesting, among others.

中文翻译：

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准周期局部共振元结构中的拓扑边缘状态

我们研究准周期共振亚结构的动力学行为和拓扑。我们表明，谐振器的准周期性布置除了由谐振器的固有频率定义的局部谐振带隙之外，还引入了频带隙。该概念在具有机械谐振器阵列的横梁上进行了说明。作为谐振器的准周期布置函数的频谱的数值评估揭示了让人联想起霍夫施塔特蝴蝶的结构，并允许研究关键的拓扑特性。结果说明了在拓扑学上微不足道的其他带隙的发生，这些带隙在有限的结构中具有边缘定位模式。通过测量光束的频率响应并评估所选工作偏转形状的空间分布，可以通过实验证明这些间隙和相关边缘状态的出现。结果揭示了确定性准周期结构设计在多个频带上引起波局部化和衰减的潜力，这可能会在振动隔离和能量收集等方面得到应用。