In this study, a Maxwell–Rayleigh-type model is investigated, describing a unidimensional lattice with a finite length, where the unit cell includes hosting and resonant masses mutually connected by elastic springs. The configuration selected is inspired by the specific engineering design to be discussed: however, the theoretical approach pursued is rather general and can be easily generalized to different scenarios. By the heuristic homogenization based on a Piola’s Ansatz, an equivalent continuum is specified: through a variational approach, resting on the minimization of the Hamilton’s action functional, a dispersion relation is deduced, revealing the existence of a band gap. Such a continuous interval of frequencies, inside which the propagation of waves is inhibited, can be tuned controlling the features of the original system. Thereafter, exploiting the same equations, the stationary elasto-dynamic response for a chain with a finite length is also deduced, corresponding to standing waves. On the basis of such results, the preliminary design of a linear chain with a finite number of cells is carried out. By the present strategy proper boundary conditions are deduced to be prescribed at the ends of the 1D lattice sample, and not over the unit cell (as it occurs in Bloch–Floquet approach with periodicity conditions). To realize the mutual elastic connections between unequal masses fitting the problem constraints, tensegrity prisms are selected, including compressed bars and tensioned cables, for which it is possible to govern the tangent axial stiffness through the cable pre-tensioning. We analyze the scenario in which the band gap coincides with the interval , indicating an appropriate geometry and suitable engineering materials for the tensegrity elements.

中文翻译：

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用于带隙可调张拉整体链初步设计的麦克斯韦-瑞利型有限长度模型的变分研究

在这项研究中，研究了麦克斯韦-瑞利型模型，描述了具有有限长度的一维晶格，其中晶胞包括通过弹性弹簧相互连接的宿主质量和谐振质量。所选配置的灵感来自于要讨论的具体工程设计：然而，所追求的理论方法相当通用，可以轻松推广到不同的场景。通过基于 Piola 的 Ansatz 的启发式均质化，指定了等效连续体：通过变分方法，基于 Hamilton 作用泛函的最小化，推导了色散关系，揭示了带隙的存在。这种连续的频率间隔（波的传播在该频率间隔内被抑制）可以通过调整来控制原始系统的特征。此后，利用相同的方程，还推导了有限长度链的稳态弹动力响应，对应于驻波。在此结果的基础上，进行了有限单元线性链的初步设计。通过本策略，适当的边界条件被推论为在一维晶格样本的末端规定，而不是在晶胞上（如在具有周期性条件的 Bloch-Floquet 方法中发生的那样）。为了实现满足问题约束的不等质量之间的相互弹性连接，选择了张拉整体棱柱，包括压缩杆和张拉索，可以通过索预张拉来控制切向轴向刚度。我们分析了带隙与间隔 重合的情况，表明张拉整体单元具有适当的几何形状和合适的工程材料。