A theoretical framework is presented to compute the dispersive properties of a locally-resonant Euler–Bernoulli beam hosting a periodic array of beam-like resonators, the bending-torsion response of which may be either uncoupled or coupled depending on the cross-section geometry. It is shown that, in the frequency domain, the reaction of every beam-like resonator can be expressed in an exact analytical form depending only on deflection and rotation of the host beam at the connection points. In this way, the beam-like resonators can be substituted by equivalent fictitious external supports with frequency dependent stiffness. On this basis: (a) band gaps of the infinite locally-resonant beam are easily computed employing a transfer matrix approach; (b) the frequency response, as well as the transmittance of the finite locally-resonant beam, are computed in closed analytical forms extending an efficient formulation recently proposed by the authors, based on generalized functions. In addition to presenting the novel theoretical framework, the main purpose of the paper is promoting coupled bending-torsion beam-like resonators as a novel concept for locally-resonant structures, which may prove even more efficient than conventional uncoupled bending-torsion beam-like resonators. Indeed, it is shown the former offer more and better opportunities to open band gaps, taking advantage of the inherent coupling between bending and torsional vibrations.

提出了一个理论框架来计算局部谐振欧拉-伯努利梁的色散特性，该梁承载周期性的梁状谐振器阵列，其弯曲扭转响应可能是非耦合的，也可能是耦合的，具体取决于横截面几何形状。结果表明，在频域中，每个梁状谐振器的反应都可以用精确的解析形式表示，这仅取决于连接点处主梁的偏转和旋转。通过这种方式，梁状谐振器可以由具有频率相关刚度的等效虚拟外部支撑代替。在此基础上： (a) 使用传递矩阵方法可以轻松计算无限局部谐振光束的带隙；(b) 频率响应，以及有限局部共振光束的透射率，以封闭分析形式计算，扩展了作者最近提出的基于广义函数的有效公式。除了提出新的理论框架外，本文的主要目的是将耦合弯扭梁状谐振器作为局部谐振结构的新概念进行推广，这可能比传统的非耦合弯扭梁状谐振器更有效谐振器。事实上，事实表明，前者利用弯曲和扭转振动之间的固有耦合提供了更多更好的打开带隙的机会。该论文的主要目的是将耦合弯扭梁状谐振器作为局部谐振结构的新概念进行推广，这可能比传统的非耦合弯扭梁状谐振器更有效。事实上，事实表明，前者利用弯曲和扭转振动之间的固有耦合提供了更多更好的打开带隙的机会。该论文的主要目的是将耦合弯扭梁状谐振器作为局部谐振结构的新概念进行推广，这可能比传统的非耦合弯扭梁状谐振器更有效。事实上，事实表明，前者利用弯曲和扭转振动之间的固有耦合提供了更多更好的打开带隙的机会。