Energy Distribution and Exchange Between Spatial Harmonics in Bending Wave Phononic Crystals

Abstract

Resonant metamaterials and phononic crystals are widely studied, with the aim of controlling and altering elastic wave propagation in solids. Beams with varying stiffness along the length are an archetypal example of a phononic crystal for bending waves, since the resulting nontrivial dispersion relation can be modeled analytically. In this work, the complex dispersion curve over the complete audible frequency range is measured using the inhomogeneous wave-correlation (IWC) technique, and compared to a unitcell model and an explicit model of the finite beam. Multiple band gaps, each with a different attenuation coefficient, are predicted and validated. An iterative implementation of IWC allows the identification of a spatial spectrum of wave numbers, responsible for the intricate beam displacement shape. The energy carried by the fundamental wave number and its Bloch (sub)harmonics are determined, showing that the main dispersion branch is monotonically increasing. It is shown that only a small fraction of the wave energy travels along branches with a negative slope in the first Brillouin zone. Additionally, in band gaps affecting only bending waves, energy is converted to other wave types, such as longitudinal waves. This approach offers some important insights into the physics of the wave propagation, which cannot be gathered from unit-cell modeling alone.

Bending-wave localization and interaction band gaps in quasiperiodic beams

“…The measured values for the periodic beam 1 are much affected by mode conversion, and they do not reach the predicted high values from the transfer-matrix method. This was already described in [31]. However, the interaction band gaps are narrow enough not to suffer much from mode conversion, and the typical band-gap pattern for phononic crystals can be measured for frequencies up to 15 kHz.…”

An INverse COnvolution MEthod for wavenumber extraction (INCOME): Formulations and applications

“…The corresponding assumptions are that n w = 1 and that 2D exponential functions with different directions of propagation are orthogonal. Nonetheless, an extension of the IWC dealing with n w > 1 was proposed in [13] for 1D wave propagation. By contrast, the IWD achieves recovery of the k-space by fitting the displacement field with a discretized version of Eq.…”

Efficient sound radiation using a bandgap structure

“…[2][3][4][5] Two common uses of bandgaps are attenuation of the transmission of the wave energy and guiding the propagation of waves. Based on these two elements, many interesting devices such as passive filters, [6][7][8][9][10] wave guides, [11][12][13][14][15] and acoustic lenses [16][17][18][19] have been developed.…”

“…For simulations concerning the resonances of studs, one can consider the stiffness of the studs with specific parameters. The hexagonal shape is chosen for the stud to match the shape with the practical fabrication, but a circular 27 or rectangular 11 shape can be considered with very similar results. The defect is created by removing four studs in the center part.…”