## 1 Introduction
There already exist extensive investigations on the bending waves in mono-coupled periodic beams, and without exception periodic multi-span beams [1; 4; 12; 13; 14; 15; 16; 17; 18; 19; 20], have led to the very wide applications of these structures in engineering practice, such as periodically supported bridges for instance. Thus, here we focus on bi-coupled periodic beams in this paper. Note that there are five kinds of bi-coupled periodic beams that have been presented so far: (1) a periodic beam composed of a uniform beam periodically attached with elastic supports with/without lumped masses; (2) a periodic beam composed of a uniform beam periodically attached with mass-spring systems (together with lumped masses and/or elastic supports); (3) a periodic beam made of periodically arranged two or more beam segments of different material/geometrical properties; (4) a periodic beam formed by combining the third and the first concepts; (5) a periodic beam formed by combining the third and the second concepts. It should be pointed out that periodic beams formed by combining the first and the second concepts belong to the second kind itself. It should also be emphasized that the second and the fifth kinds of bi-coupled periodic beams introduce a locally resonant mechanism [2] other than the Bragg scattering mechanism [2; 3] into the formation of band gaps, so that low-frequency band gaps may possibly provide a new way in controlling low-frequency waves/vibrations without using unrealistically large structures. Note further that the first two kinds of periodic beams all have unit cells containing only one beam segment with attachments, and these structures are referred here to as bi-coupled periodic mono-component beams. In contrast, the last three kinds of periodic beams all have unit cells containing more than one beam segments without or with attachments, which are referred to here as bi-coupled periodic multi-component beams. Up until now, the bending waves in all the five types of bi-coupled periodic beams have been of concern. First, the bending waves in the first kind of periodic beams have gained attention since the 1960s. Some literature considered the bending wave/vibration properties of infinite/finite periodic beams. For the periodic beams supported by translational/rotational springs: [PERSON] [21] pioneered the analysis of the natural frequencies and modes of finite beams; then, [PERSON] [16] derived the dispersion equation of characteristic bending waves in the infinite beams and discussed the interaction between characteristic bending waves and the alternating pressure field using the receptance method (RM); [PERSON] and [PERSON] [22] constructed the mutual variational principle of bending wave propagation in periodic structures, which provided the theoretical basis for the corresponding h-type finite element method (FEM). With the h-p hybrid FEM, [PERSON] et al. [17] analyzed the phase constant curves of the bending waves in infinite periodic Euler-Bernoulli beams; [PERSON] and [PERSON] [23] presented the design of periodic undamped and damped beams aimed at reducing the transmitted vibrations by referring the bi-dimensional analytical maps of the single unit free-wave propagation domains (stop, pass and complex domains) as the optimal choice of the cell properties and ordering. For periodic beams supported by translational/rotational springs with lumped masses: [PERSON] and [PERSON] [18; 19] introduced the extended Rayleigh-Ritz method to analyze the pass band properties and the natural properties of infinite and finite periodic structures, respectively; [PERSON] [24] discussed the propagation properties of bending waves. For periodic beams with only added masses, [PERSON] et al. [25] investigated numerically and experimentally the dispersion curves and the transmission loss coefficient of infinite and semi-infinite beams, respectively.