The study of mono-coupled periodic structures has gained renewed interest by the scientific community due to the new applications of metamaterials and meta-structures. Much research has focused on the wave propagation properties of infinite structures. However, this paper focuses on finite periodic structures, in particular the parameters that govern the behaviour of a low frequency stop-band of such a structure. From an engineering perspective, these are the lower and upper cut-off frequencies, i.e., the bandwidth, and the minimum transmission of vibration within the band. Using the Caley-Hamilton theorem, analytical expressions are derived for the receptance, dynamic stiffness and transmissibility of a finite mono-coupled structure. It is shown that the properties of the whole structure can be determined from the transmissibility of a single element. If the element is symmetric, then the expressions describing the stop-band are particularly simple. An approximate analytical expression has been derived that allows the number of elements needed for a given maximum attenuation in a low frequency stop-band to be determined. To illustrate the approach, lumped parameter systems are considered, in which the stop-band behaviour is governed by the addition of mass, stiffness and a vibration absorber. Expressions are derived for the maximum vibration attenuation within the first stop-band, for each case, enabling clear physical insight into the controlling parameters. Expressions are provided for the lower and upper cut-off frequencies of the stop-band. Some experimental results are also presented to support the theoretical analysis.

由于超材料和超结构的新应用，单耦合周期结构的研究引起了科学界的重新关注。许多研究集中在无限结构的波传播特性上。然而，本文着重于有限周期结构，尤其是控制这种结构的低频阻带行为的参数。从工程角度来看，这些是下限和上限频率，即带宽，以及频带内振动的最小传递。利用Caley-Hamilton定理，得出了有限单联结构的接受度，动态刚度和透射率的解析表达式。结果表明，整个结构的特性可以由单个元素的透射率来确定。如果元素是对称的，则描述阻带的表达式特别简单。得出了一个近似分析表达式，该表达式允许确定低频阻带中给定最大衰减所需的元素数量。为了说明该方法，考虑了集总参数系统，其中阻带的行为由质量，刚度和减振器的增加决定。在每种情况下，都可以得出第一阻带内最大振动衰减的表达式，从而可以清楚地了解控制参数。提供了针对阻带的下限和上限截止频率的表达式。