The optimal absorption of flexural energy by the critical coupling of a locally resonant grating embedded in a thin plate is reported in this work for the reflection and transmission problems. The grating is made of a 1D-periodic array of local resonators. A viscoelastic coating is also placed on top of each resonator to control the intrinsic losses of the system. The scattering matrices for the propagative waves of both problems are obtained by means of the Layered Multiple Scattering Theory and validated by the Finite Element Method. In this work, we find that the perfect absorption can be obtained in the reflection problem and the maximal absorption in the transmission problem is limited to 50% by tuning the losses only. These results agree with the theoretical predictions since the eigenvalues reduce to the reflection coefficient in the reflection problem and only one of the two eigenvalues of the scattering matrix is critically coupled in the transmission problem. These results highlight the adaptability of the critical coupling method to optimize the absorption of locally resonant materials for flexural waves in 2D transmission and reflection problems, and pave the way to the design of resonators for efficient flexural wave absorption.

对于反射和透射问题，在这项工作中报道了通过嵌入薄板中的局部共振光栅的临界耦合来最佳吸收弯曲能。光栅由局部共振器的一维周期阵列组成。粘弹性涂层也放置在每个谐振器的顶部，以控制系统的固有损耗。通过分层多重散射理论获得了两个问题的传播波的散射矩阵，并通过有限元方法对其进行了验证。在这项工作中，我们发现在反射问题中可以获得理想的吸收，而仅通过调整损耗，可以在透射问题中将最大吸收限制为50％。这些结果与理论预测相符，因为在反射问题中特征值减小到反射系数，并且散射矩阵的两个特征值中只有一个在透射问题中被严格耦合。这些结果凸显了临界耦合方法的适应性，以优化二维透射和反射问题中挠曲波的局部共振材料的吸收，并为有效吸收挠曲波的共振器设计铺平了道路。