Abhigna Bhatt

Abhigna Bhatt

she/her

ANRF National Postdoctoral Fellow at IISc Bengaluru, working on elastic waves, metamaterials and vibration isolation.

When Bragg Meets Resonance

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A uniform bar carries a mass-spring resonator in every cell. The resonator mass stays fixed while its stiffness runs from zero to infinity. Choose a stiffness and a frequency, and see how the local-resonance gap moves through the Bragg gaps and merges with each one in turn.

The stiffness slider runs from k = 0 (left end) to k → ∞ (right end), with the resonator mass fixed at a quarter of the bar mass per cell. Drag the frequency line on the plots, or use the slider. In the propagation plot both axes use the same scale, so the slope of the line from the origin is the phase velocity and the slope of the tangent is the group velocity, in units of the bar wave speed c. The dashed diagonal is the bare bar. At each Bragg gap, the node mode has the resonators standing still, and the sign of meff sets which side of nωB the gap opens. The bar's longitudinal motion is drawn vertically, and resonator motion is capped for display.

Two kinds of gap in one bar

A periodic array of resonators can stop waves in two ways. Near the resonance frequency \(\omega_r=\sqrt{k/m}\), each resonator moves against the bar and acts as a large or negative mass, which opens a local-resonance gap. Because the resonators repeat every cell, they also scatter waves coherently and open Bragg gaps near \(\omega=n\,\omega_B\), where \(\omega_B=\pi c/a\) is the frequency at which half a wavelength fits in one cell and \(c=\sqrt{E/\rho}\) is the wave speed in the bare bar. With \(k=0\) the resonators are detached and the bar has no gaps. As \(k\) grows, the local-resonance gap rises from zero frequency and the Bragg gaps widen. The frequency \(n\,\omega_B\) is normally a band edge whose standing wave has a node at every resonator, but when \(\omega_r=n\,\omega_B\) it falls inside a gap, and the local-resonance gap and Bragg gap \(n\) merge into one wide gap (references 1 and 2). As \(k\to\infty\) the resonators become rigid point masses: only Bragg gaps remain, and the local-resonance gap has moved to infinite frequency.

References

  1. Y. Xiao, B. R. Mace, J. Wen and X. Wen, Formation and coupling of band gaps in a locally resonant elastic system comprising a string with attached resonators, Physics Letters A, 375, 1485–1491, 2011.
  2. L. Liu and M. I. Hussein, Wave motion in periodic flexural beams and characterization of the transition between Bragg scattering and local resonance, Journal of Applied Mechanics, 79, 011003, 2012.
  3. Y. Xiao et al., Longitudinal wave band gaps in metamaterial-based elastic rods containing multi-degree-of-freedom resonators, New Journal of Physics, 14, 033042, 2012.
  4. L. Brillouin, Wave Propagation in Periodic Structures, McGraw-Hill, 1946.
  5. M. I. Hussein, M. J. Leamy and M. Ruzzene, Dynamics of phononic materials and structures: Historical origins, recent progress, and future outlook, Applied Mechanics Reviews, 66, 040802, 2014.