## 1 Introduction  
A periodic structure consists of repeated unit cells which are joined together end-to-end/side-by-side [1]. The most important property determined by this constructional feature is that the frequency domain of the waves falls into pass-bands and stop-bands that correspond respectively to those frequencies at which waves propagate without attenuation and to others at which waves attenuate with no propagation [1; 2; 3]. Consequently, periodic structures may have wide applications in various fields requiring wave filtering or vibration isolation, which drives the investigations of wave propagation in periodic structures in various scientific fields such as solid-state physics, electrical engineering and mechanical engineering, that are concerned with electron waves, electromagnetic waves, and electric current and elastic waves [2; 3]. As one of the most important periodic structures, a periodic beam consists of beam segments possessing frequency bands of bending waves. Since the beam structure is a basic component in engineering and the bending wave is one of the main vibration forms and noise origins in engineering structures [4], the study of bending waves in periodic beams hence provides a method for vibration control and noise reduction in engineering structures, e.g., railway trains and tracks. Using the elastically point-supported, flexibly interconnected periodic beam and Timoshenko beam theory (TBT) or Euler-Bernoulli beam theory (EBT) to model railway train, [PERSON] et al. [5] analyzed the harmonic and time-history responses of high-speed trains and train-like articulated systems travelling in confined fluid. Adopting the elastically point-supported periodic beam and based on the TBT, EBT or extended beam theory to model the railway track, [PERSON] and [PERSON] [6] and [PERSON] and [PERSON] [7] studied bending-wave propagation together with the forced responses of railway tracks and floating ladder tracks, respectively. [PERSON] et al. [8] derived the propagation constant equations and used them for ballasted track to study the harmonic responses to steadily moving or static loads [8] and to discuss the propagation and resonance properties [9]. Using the Green's function approach, [PERSON] [10] predicted the interaction between a simple moving vehicle and the infinite periodic Timoshenko tract beam. Employing the elastically point-supported periodic Euler-Bernoulli beam to model the direct fixation track and the periodic beam with a unit cell consisting of a double beam on an elastic foundation to model the floating slab track, [PERSON] and [PERSON] [11] computed the transfer functions of the coupled track-tunnel-soil system and the dynamic forces accounting for the interaction between the moving vehicle and the periodic track. Although in recent years active periodic beams with smart components have been proposed on the basis of initially presented passive periodic beams made of only elastic materials in order for the bending-wave bands to be adjusted intelligently for controlling vibration and reducing noise in a more effective way, in this paper we still exclusively consider passive periodic beams in which only the construction of unit cell affects the band properties and the dispersion characteristics remain after the unit cell is determined. This is because there are still pending problems on this research objective, as will be summarized after an extensive literature review. Passive periodic elastic beams may fall into two categories according to the minimum number of coupling coordinates between the structural beams, i.e., mono-coupled and bi-coupled periodic beams that have one and two minimum coupling coordinates, respectively [1; 12]. For example, as the flexible interconnection of the articulated train system in Reference [5] transfers one or two generalized forces, the analysis models are mono-coupled and bi-coupled periodic beams, respectively. Mono-coupled periodic beams are mainly introduced in the form of periodic multi-span beams.