Flexural Vibration Band Gaps Characteristics in Phononic Crystal Euler Beams on Two-Parameter Foundation

The vibration band gaps in the phononic crystal (PC) Euler beam on two-parameter foundation are investigated theoretically by the combination of the transfer matrix (TM) method and Bloch theorem. Compared with the Winkler foundation with only the foundation reaction modulus kappa, the shear modulus G(p) is taken into account simultaneously to consider the continuity of deformation of the foundation. The band structures of a lead-copper PC Euler beam either on two-parameter foundation or on Winkler (one-parameter) foundation show that the introduction of the shear modulus G(p) has significant impact on the distribution of band gaps. The first band gap has not any evolution with the increase of G(p), while the other band gaps trend to higher frequency and their ranges become narrower. The analysis shows that the parameters of foundation and the length ratio of segments can be conditionally selected to obtain adjustable band gaps.