A new perspective of physical understanding is presented in this paper for the propagation and attenuation behaviours of the beam coupled with periodic resonators. Wavenumber spectral relation of the beam-resonators coupling system is established based on the Timoshenko beam theory and its wave form solution, the formulation of the method of reverberation-ray matrix (MRRM) and the Bloch theorem of periodic structures. The complex wavenumbers of the coupling system are calculated by numerical techniques. The validity and accuracy of the MRRM in analysing the propagation and attenuation characteristics of the beam-resonators coupling system are verified by the analytical solution of the homogeneous beam and the numerical results of the beam coupled with periodic resonators reported in the published literature. Numerical examples are analysed to demonstrate the general wavenumber spectral characteristics of the beam-resonators coupling system, together with those of the homogeneous beam and the beam on elastic foundation. The effects of the parameters of the beam and the resonators on the wavenumber spectral characteristics are respectively evaluated. Numerical results show that the coupling between the beam and the resonators results in a local resonant attenuation band and multiple non-local resonant attenuation bands. In all these attenuation bands, the real wavenumber of the flexural wave is an integer multiple of $\pi$, and it does not vary with frequency within each of the attenuation bands. There are two resonant frequencies respectively corresponding to the translational and rotational coupling between the resonators and the beam. The effect of the parameters of the beam on the wavenumber spectral characteristics of the coupling system is mainly reflected in shifting all the attenuation bands to the higher or lower frequency ranges. The parameters of the resonators have a more significant effect on the local resonant attenuation band with respect to the non-local resonant ones. With the increase of the spacing of two adjacent resonators, the bandwidth and the attenuation factor of all the attenuation bands decrease, the number of the non-local resonant attenuation bands increases linearly, and the real wavenumber spectrum curve of the beam coupled with periodic resonators gradually converges to the real wavenumber spectrum curve of the homogeneous beam. Mergence of the local and non-local resonant attenuation bands enlarges the bandwidth of the local resonant attenuation band significantly. The occurrence of mergence of the local and non-local resonant attenuation bands is tunable by selecting appropriate structural parameters such as the mass-related parameters of the beam, the translational coupling stiffness and the spacing of two adjacent resonators. The innovative findings and practical suggestions could provide potential references for the researchers and engineers in vibration reduction design of engineering structures.

本文提出了一种物理理解的新观点，用于耦合周期谐振器的光束的传播和衰减行为。基于Timoshenko束理论及其波形解，混响射线矩阵法（MRRM）和周期结构布洛赫定理，建立了束谐振耦合系统的波数谱关系。耦合系统的复波数通过数值技术来计算。MRRM在分析束-谐振器耦合系统的传播和衰减特性中的有效性和准确性通过已发表文献中报道的均质束的解析解以及带有周期性谐振器的束的数值结果得到验证。通过数值算例分析，证明了梁－谐振器耦合系统的一般波数频谱特性，以及均质梁和弹性地基上的梁的频谱特征。分别评估了光束和谐振器的参数对波数频谱特性的影响。数值结果表明，梁与谐振器之间的耦合导致一个局部谐振衰减带和多个非局部谐振衰减带。在所有这些衰减带中，弯曲波的实际波数是的整数倍。分别评估了光束和谐振器的参数对波数频谱特性的影响。数值结果表明，梁与谐振器之间的耦合导致一个局部谐振衰减带和多个非局部谐振衰减带。在所有这些衰减带中，弯曲波的实际波数是的整数倍。分别评估了光束和谐振器的参数对波数频谱特性的影响。数值结果表明，梁与谐振器之间的耦合导致一个局部谐振衰减带和多个非局部谐振衰减带。在所有这些衰减带中，弯曲波的实际波数是的整数倍。$\pi$，并且在每个衰减频带内不会随频率变化。存在两个谐振频率，分别对应于谐振器和光束之间的平移和旋转耦合。光束参数对耦合系统波数频谱特性的影响主要体现在将所有衰减带移到较高或较低的频率范围。相对于非局部谐振衰减器，谐振器的参数对局部谐振衰减带具有更大的影响。随着两个相邻谐振器间距的增加，所有衰减带的带宽和衰减因子减小，非局部谐振衰减带的数量线性增加，耦合周期谐振器的光束的实波数频谱曲线逐渐收敛到均匀光束的实波数频谱曲线。局部和非局部共振衰减带的合并显着增大了局部共振衰减带的带宽。通过选择适当的结构参数，例如梁的质量相关参数，平移耦合刚度和两个相邻谐振器的间距，可以调节局部和非局部谐振衰减带合并的发生。这些创新的发现和实践建议可为工程结构减振设计中的研究人员和工程师提供参考。局部和非局部共振衰减带的合并显着增大了局部共振衰减带的带宽。通过选择适当的结构参数，例如梁的质量相关参数，平移耦合刚度和两个相邻谐振器的间距，可以调节局部和非局部谐振衰减带合并的发生。这些创新的发现和实践建议可为工程结构减振设计中的研究人员和工程师提供参考。局部和非局部共振衰减带的合并显着增大了局部共振衰减带的带宽。通过选择适当的结构参数，例如梁的质量相关参数，平移耦合刚度和两个相邻谐振器的间距，可以调节局部和非局部谐振衰减带合并的发生。这些创新的发现和实践建议可为工程结构减振设计中的研究人员和工程师提供参考。通过选择适当的结构参数，例如梁的质量相关参数，平移耦合刚度和两个相邻谐振器的间距，可以调节局部和非局部谐振衰减带合并的发生。这些创新的发现和实践建议可为工程结构减振设计中的研究人员和工程师提供参考。通过选择适当的结构参数，例如梁的质量相关参数，平移耦合刚度和两个相邻谐振器的间距，可以调节局部和非局部谐振衰减带合并的发生。这些创新的发现和实践建议可为工程结构减振设计中的研究人员和工程师提供参考。