Mechanical metamaterials composed of a non dissipative periodic microstructure of flexible ligaments and stiff rings, viscoelastically coupled with local resonators, are considered. By following a variational approach, the linear damped dynamics are described according to a beam lattice formulation, valid for a generic coordination number. A consistent theoretical description of the viscoelastic ring-resonator coupling is introduced, leading to an integral–differential form of the governing equations. The enlargement of the model dimension by addition of viscoelastic states allows to conveniently describe the metamaterial dynamics through a system of ordinary differential equations. Therefore, the application of the $\mathcal{Z}$-transform and a suitable mapping in the wavevector space allows to obtain the system of ordinary differential equations governing the free and forced propagation of damped waves. Subsequently, the polynomial eigenproblem ruling the harmonic free wave propagation in the complex frequency domain is stated by applying the bilateral Laplace transform, whereas an analytical procedure is outlined to solve the forced wave propagation problem in the time domain. The triangular beam lattice metamaterial is considered as significant benchmark for application of the theoretical and methodological framework. Considering first the free wave propagation, the properties of the dynamic stiffness matrix are discussed and the dispersion frequency spectrum is parametrically investigated, with focus on the spectral effects of the viscoelastic coupling. The dependence of the stop bandwidth amplitude between the low and high-frequency spectral branches on the relaxation time characterizing the viscous function is also highlighted and discussed. Moreover, the forced response to a harmonic external point source is numerically investigated in the time domain. The results are qualitatively compared and quantitatively discussed by distinguishing the fundamental cases of non-resonant, resonant and quasi-resonant excitation frequency.

考虑由柔性韧带和刚性环的非耗散周期性微结构组成的机械超材料，与局部谐振器粘弹性耦合。通过遵循变分方法，线性阻尼动力学根据梁格公式描述，适用于通用配位数。引入了粘弹性环形谐振器耦合的一致理论描述，导致控制方程的积分 - 微分形式。通过添加粘弹性状态来扩大模型尺寸允许通过常微分方程系统方便地描述超材料动力学。因此，应用 $\mathcal{Z}$-变换和波向量空间中的适当映射允许获得控制阻尼波的自由和强制传播的常微分方程组。随后，应用双边拉普拉斯变换说明了支配复频域中谐波自由波传播的多项式特征问题，并概述了解决时域中的强迫波传播问题的解析过程。三角梁晶格超材料被认为是应用理论和方法框架的重要基准。首先考虑自由波传播，动态特性讨论了刚度矩阵，并对色散频谱进行了参数化研究，重点关注粘弹性耦合的光谱效应。还强调并讨论了低和高频谱分支之间的停止带宽幅度对表征粘性函数的弛豫时间的依赖性。此外，在时域中对谐波外点源的强制响应进行了数值研究。通过区分非谐振、谐振和准谐振激励频率的基本情况，对结果进行定性比较和定量讨论。