Flexural Wave Attenuation in a Multi-Frequency Locally Resonant Phononic Crystals Beam Resting on Elastic Foundations

Phononic crystals (PCs) beam has band gaps' characterization and the potential to inhibit flexural wave vibrations, and it can be used for vibration isolation in beam-foundation systems. To realize the broadband gaps, a multi-frequency locally resonant (LR) PCs model, in which two types of spring-oscillator are periodically attached to an Euler beam, is introduced into the beam-foundation systems. Based on the motion equations of flexural wave in beam, the complex band structures of beam-foundation systems are derived by transfer matrix method (TMM) and Bloch's theorem. The derivation is verified by the frequency response function (FRF) of the system, which is conducted with finite element method (FEM). Then, the band gap characteristics in the beam-foundation systems are investigated theoretically. Furthermore, the effects of oscillator and foundation parameters on the band gaps are analyzed in detail. Results indicate that multi-band gaps exist in the novel LR PCs beam-foundation system and the multi-frequency LR PCs beam can effectively enlarge the range of flexural band gaps.