## 1 Introduction
[58] designed periodic binary beams for supporting the ocean platform to reduce the propagation of the ocean noise to the platform, analyzed with a method based on EBT and TMM the band structure and the frequency response of the infinite and finite periodic binary beams, respectively, and verified the results by comparing them with an experiment of the scaled model of the ocean platform. Fourth, bending waves in periodic beams of the above-classified fourth kind have been investigated since the 1980s. All the unit cells in the numerical examples considered up to now were binary. As the attachments are lumped, [PERSON] and [PERSON] [20] analyzed the band structure of infinite periodic beams supported by translational springs and discussed the free vibration properties based on the EBT and the TMM. As the attachments are distributed, [PERSON] [59] analyzed the band structures of periodic beams with spring and viscously-damped supports based on EBT and Floquet theory. [PERSON] et al. [60] studied the band structures of prestressed periodic perfect and defected beams on an elastic foundation based on EBT, with special attention on the effects of prestress on the position/width of the band-gaps. The band-gap localized modes and effective negative mass effects of periodic perfect beams on an elastic foundation with an additional mass were also discussed using a Green's function formulation. [PERSON] and [PERSON] [61] analyzed the band gaps and the dynamic responses of periodic and quasi-periodic beams on a Pasternak foundation based on TBT and using the DQM. [PERSON] et al. [43] studied the complex band structures and FRF of periodic beams on elastic Winkler foundations based on EBT. Also based on EBT, [PERSON] et al. [62; 63] analyzed the vibration band gaps of periodic beams on Winkler and two-parameter foundations by the proposed MTM and by the TMM, respectively. [PERSON] et al. [64] developed the EDQEM to study the effect of initial stress on the attenuation zones and attenuation coefficients of periodic Timoshenko beams resting on a two-parameter elastic foundation. Fifth, the fifth periodic beams classified above have been devised since 2006. In all the unit cells considered, actually the beam itself was homogenous along the whole structure. In the case of attached two DOF resonators, [PERSON] et al. [65] analyzed the band structures of infinite periodic beams using the TMM based on EBT. The FRF and the characteristic modes of a finite periodic beam were calculated by the FEM. Theoretical results were validated by experimentally measured FRF on a finite periodic beam specimen, with special attention on the material damping in the rubber. [PERSON] et al. [66] investigated the band gaps by TMM with the results validated by FEM, and observed the effects of band-gap coupling, the intervals, and oscillator parameters. In the case of attached resonators with more finite DOF, [PERSON] et al. [67] extended the PWE method to study the complex band structures of locally resonant beams attached with multiple periodic arrays of spring-mass resonators based on EBT, which were validated by the FRF obtained with commercial FEM software. [PERSON] et al. [68] suggested a chiral lattice-based elastic metamaterial beam with multiple embedded local resonators to achieve broadband vibration suppression without sacrificing its load-bearing capacity. In the case of attached resonators with infinite DOF, [PERSON] and [PERSON] [69] investigated the frequency band structure by TMM based on EBT, and showed richer dispersion properties in their proposed periodic beam with distributed DOF than those with the conventional force-only resonators. Reviewing the above literature, we can summarize the current status of research on bending wave propagation in bi-coupled periodic beams:1. There is plenty of literature on bi-coupled periodic mono-component beams with the unit cell containing only one beam segment.