Abstract This study presents a strategy of vibration suppression of a beam when multiple resonant frequencies of the structure are excited. The methodology operates on the ability of local resonators attached to the beam to create frequency ranges corresponding to locally resonant bandgaps in which the vibration of the beam is attenuated. This study discusses the mechanism of multiple bandgap formation by first deriving the equations of motion of the metastructure based on Hamilton’s principle and then utilizing a modal analysis approach to obtain analytical expressions for the edge frequencies of the created bandgaps. As a consequence of this assumption, the lattice constant of the resulting metastructure is much smaller than the operating flexural wavelength of the host beam. These edge frequencies are derived by assuming an infinite number of resonators tuned to different resonant frequencies of the beam at which a bandgap is desired to be centered at. Parametric studies on the steady state dynamic response of the beam, however, reveal that only a finite number of resonators is required to create these bandgaps and that their bandwidths largely depend on the ratio of the mass of the resonators to that of the beam. The proposed analytical approach is used to demonstrate bandgap formation at the first and second resonant frequencies of a cantilever beam both numerically using a commercial finite element solver as well as experimentally. Numerical modal analysis results of the metastructure compare well with experimentally measured modal analysis results and the steady-state response of the tip displacement of the structure clearly demonstrates the creation of two bandgaps both numerically as well as experimentally.

中文翻译：

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局部谐振线性超材料光束中的多带隙形成：理论与实验

摘要 本研究提出了当结构的多个共振频率被激发时梁的振动抑制策略。该方法利用连接到梁的局部谐振器的能力来创建对应于梁的振动衰减的局部谐振带隙的频率范围。本研究首先根据哈密尔顿原理推导出元结构的运动方程，然后利用模态分析方法获得所产生带隙边缘频率的解析表达式，从而讨论多带隙形成的机制。作为这一假设的结果，所得超微结构的晶格常数远小于主光束的工作弯曲波长。这些边缘频率是通过假设无限数量的谐振器被调谐到光束的不同谐振频率而得出的，带隙希望以该频率为中心。然而，对梁的稳态动态响应的参数研究表明，只需要有限数量的谐振器来产生这些带隙，并且它们的带宽在很大程度上取决于谐振器的质量与梁的质量之比。所提出的分析方法用于证明悬臂梁的第一和第二共振频率下的带隙形成，这两种方法都使用商业有限元求解器以数值方式和实验方式进行。