## 1 Introduction
Others consider the responses of infinite/finite periodic beams under external excitation: [PERSON] [26] discussed the propagation constants of bending waves in infinite periodic beams supported by translational springs and analyzed the steady-state response at the supported joint under harmonic loads by considering the effect of damping and fluid loads; using Fourier transform, [PERSON] [27] analyzed the transient responses of infinite periodic Euler-Bernoulli beams with translational springs under a concentrated load; [PERSON] et al. [28; 29] presented the wavenumber-harmonic method to analyze the frequency responses and sound radiations of a translational point-spring supported periodic Timoshenko beam subjected to a convected or travelling harmonic loading by considering the fluid-loading effect and discussing the influences of the spring support and the loading speed; [PERSON] and [PERSON] [30] and [PERSON] and [PERSON] [31] analyzed the steady-state response of infinite periodic Euler-Bernoulli beams with translational/rotational springs under alternating random loads by the space harmonic method and complex modal method, respectively; [PERSON] [32] proposed a method to analyze the propagation constants of bending waves in infinite-period Timoshenko beams attached with translational/rotational springs and lumped-mass and their responses to an alternating harmonic pressure field; [PERSON] and [PERSON] [33] analyzed the steady-state responses of infinite periodic beams with translational/rotational spring supports or lumped-mass under harmonic loads based on propagation constants. Second, the bending waves in the second periodic beams classified above have been studied ever since 2005. The main concerns have been the band gaps and frequency responses of bending waves caused by Bragg scattering and locally resonant in infinite/finite periodic beams. For periodic beams with single DOF (degree-of-freedom) resonators: [PERSON] et al. [34] and [PERSON] et al. [35] analyzed the complex band structures using the transfer matrix method (TMM) based on TBT and EBT, respectively. [PERSON] et al. [34] further analyzed the frequency response function (FRF, i.e., transmission) using the FEM and validated the results by experiments. [PERSON] et al. [35] further analyzed the FRF using TMM,particularly concerning the influence of different local resonators, and summarized some design guidelines for this kind of structure. [PERSON] et al. [36] analyzed the complex band structures by the spectral element method (SEM) and examined the effects of various system parameters on band-gap behavior. The band formation mechanisms and formulas determining the transition and near-coupling conditions between local resonance and Bragg scattering were obtained. [PERSON] and [PERSON] [37] studied the attenuation behavior of low-frequency waves in sandwich beams by closed form expression obtained from a phased array approach and verified the results by FEM. For periodic beams with two DOF resonators: [PERSON] et al. [38] analyzed the band structures by TMM and the FRF by FEM, respectively, with a simplified model proposed to determine the edge frequencies of the band gaps and to enhance the understanding of the locally resonant mechanism. [PERSON] et al. [39] calculated the dynamic behaviors by FEM and explained how the spring-mass-damper subsystem created two stopbands and negative effective mass and stiffness. For periodic beams with three DOF resonators: [PERSON] et al. [40] studied the suppression of flexural vibration theoretically and numerically by the dispersion relation obtained using [PERSON]'s principle and [PERSON]'s theorem. For periodic beams with more than three DOF resonators: [PERSON] et al. [41] analyzed the band structures and characteristic modes of infinite periodic beams using the lumped mass method (LMM) and calculated the FRF of finite periodic beams using the FEM, based on three-node triangular plane stress elements. The theoretical FRF was validated by experimental measurements on a finite periodic beam specimen.