## 1 Introduction
We hope that the proposed method and interpretation will push forward better design and optimization of bi-coupled periodic multi-component beams for controlling bending waves and vibration isolation in engineering practice, such as for aircraft/automobile engines [57] and ocean platforms [58]. The outline of the paper is as follows. In Section 2 we derive the formulation of MRRM. The theoretical derivation and the computer program are verified by numerical examples in Section 3, where the advantages of the proposed MRRM as compared to TMM, the general dispersion characteristics, and the influence of various construction factors of the unit cell on the band structures, are discussed. Some conclusions are drawn in Section 4. The Method of Reverberation-Ray Matrix (MRRM) for the Analysis of Bending Waves in Bi-Coupled Periodic Beams  
Consider the bending waves in an infinite periodic structure whose unit cell of length \(l\) containing \(m\) beams and \(n\) (\(=\ m\ +\ 1\)) joints are depicted in Figure 1. Each constituent beam in the unit cell is rigidly connected to its adjacent beam and is a carrier member of bending waves including the coupled flexural and thickness-shear modes. Thus, each joint in the unit cell has two coupling coordinates, at which the bending waves scatter. Therefore, the structure considered is a bi-coupled periodic beam [1]. Because the infinite periodic structures satisfy the Floquet-Bloch principle [1; 2; 3], the unit cell model in Figure 1 is adequate for determining the dispersion characteristics of bending waves in the infinite bi-coupled periodic beam as considered. In the unit cell, any constituent beam is assumed to have a straight axis, homogenous material and uniform cross-section, and is deemed as one structural member. If it does not satisfy this assumption, the beam can always be further re-divided into more members to meet this requirement. In this case, the model in Figure 1 also applies but with enlarged \(m\), in which the geometrical parameters \(A\), \(I\) and \(\kappa\) are the area, moment of inertia and shear coefficient of the cross-section, respectively, and the material parameters \(\rho\), \(E\) and \(G\) are the mass density, Young's modulus and shear modulus, respectively. These geometrical and material parameters are all constant for each member. In addition, the cross-sectional dimensions of each constituent beam are comparatively smaller than the length, so that beam theories such as the Timoshenko or Euler-Bernoulli theory can be adopted to model the bending waves in the member. Thus, the deflection \(v\), rotation \(\varphi\), shear force \(Q\) and bending moment \(M\) should be the variables to describe the dynamic state of the member. The amplitudes \(a_{r}\) and \(d_{r}\) of the respective backward (arriving) and forward (departing) waves of flexural (\(r\ =\ 2\)) or thickness-shear (\(r\ =\ 3\)) mode are used to represent the bending wave motion in the member. All members from left to right are denoted in turn by numbers (1) to \((m)\) with their lengths signified by \(l_{(1)}\) to \(l_{(m)}\). In the unit cell, the end joints connecting the current unit cell to its neighbors and the intermediate joints linking the contiguous members are referred to as the exterior joints and the interior joints, respectively. All joints from left to right are signified in sequence as 1 to \(N\ (=n)\), which include exterior joints 1, \(N\) and interior joints 2 through \(N\ -\ 1\). For an arbitrary member \((j)\) with left joint \(J\) and right joint \(K\), a pair of dual local coordinate systems \((x,y,z)^{JK}\) and \((x,y,z)^{KJ}\) of right-handed type are introduced, as shown in Figure 2a, to describe the cross-section and physical variables of this member within the framework of MRRM, and the variables of member \((j)\) in respective coordinates are described by superscripts \(JK\) and \(KJ\), as shown in Figure 2b-d. By this means, the axial coordinate of any one end cross-section in the local coordinate system with its origin on this cross-section is always zero.