This work presents an effort to understand the evolution of Bragg scattering band gaps in the context of the transfer functions of finite Phononic Crystals (PCs). Following the dispersion analysis of an infinite PC based on a single unit cell, an analytical derivation of the natural frequencies of a finite PC with a given number of cells is presented. Next, the transfer function between the tip displacement of a finite PC and a force exerted at the other end is derived in closed-form, and used to establish an understanding of the band gap formation in the finite setting. The analysis reveals that the phenomenon can be attributed to the split of poles around the center of the band gap and the absence of any poles within it. The formation mechanism is then discussed in light of several numerical examples with different combinations of system parameters and number of cells.
H. Al Ba'ba'a, T. Singh, and M. Nouh, "Interpreting phononic Bragg band gaps through finite system dynamics and transfer functions," Proc. SPIE 10600, Health Monitoring of Structural and Biological Systems XII, 106001T (Presented at SPIE Smart Structures and Materials + Nondestructive Evaluation and Health Monitoring: March 08, 2018; Published: 27 March 2018); https://doi.org/10.1117/12.2296658.
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