Free and steady forced vibration characteristics of elastic metamaterial beam

The propagation of elastic waves in infinite elastic metamaterials (EMs) is studied by using the effective medium theory (EMT). However, when EMs are applied in engineering practice, finite EM structures should obviously be the most cases, in which the propagation of elastic waves corresponds to the problem of vibration. Therefore, it is necessary to establish a reasonable vibration analysis method for typical EM waveguides and investigate their unique modal characteristics and related vibration characteristics. At present, the relevant exploration is relatively limited, especially analytical analysis. As one typical EM finite structure, an EM beam was chosen to be investigated in this study. Based on the EMT, the Euler beam theory is extended to the analysis of EM finite beams, and the analysis process of free and steady forced vibration is established, the particular characteristics are revealed, including the gathering of natural frequencies in the vicinity of band edges, the absence of natural frequencies within bandgap (BG), and the particular modal correspondence before and after BG. Then, the formation mechanisms of the characteristics are explained from the perspective of standing waves. The obtained results may give insights into the vibration analysis of other finite EM structures, such as EM rods, shafts, plates, and shells.