关键词: phononic crystals, bandgaps, Sierpinski carpet, fractal

Abstract: The effects of rotation, size, and geometric shape of rods within a quasi-Sierpinski carpet fractal structure on bandgap behavior were investigated. In the first stage of the Sierpinski carpet ($L=1$), rotating the central rod from 0$^{\circ}$ to 45$^{\circ}$ led to a slight increase in the bandgap width at frequencies between 0 kHz and 5 kHz. In the second stage ($L=2$), rotating the side rods gradually expanded the bandgap width, increasing it from approximately 2.7 kHz to over 3.3 kHz. Although the bandgaps in the third stage ($L=3$) were narrower compared to those in the previous two stages, new bandgaps appeared at frequencies between 16 kHz and 20 kHz. Varying the size of the central rod from $a/27$ to $a/2$ led to an overall increase in the bandgap size from 1 kHz to 3.65 kHz. Geometric modifications also played a significant role in bandgap size; for instance, the use of a dodecagonal central rod generated the widest bandgap of 3.42 kHz, while the combination of a diamond-shaped central rod with circular side rods resulted in a bandgap size of 2.19 kHz. The results indicate that such modifications can significantly optimize the bandgaps in terms of both frequency position and width. These findings can contribute to the design of advanced materials for applications such as acoustic insulation, vibration reduction, and mechanical wave control.

Key words: phononic crystals, bandgaps, Sierpinski carpet, fractal