Periodic media exhibit band gaps, frequency bands where waves are not allowed to propagate. This occurs due to Bragg scattering effects at their periodically repeating interfaces, which lead to destructive interferences of waves within the band-gap region. However, a finite phononic crystal may exhibit resonance frequencies within the repeating unit-cell Bragg band gap. This is due to truncating the infinite medium into a finite number of unit cells with structural end boundaries that interact the dispersion characteristics of the parent infinite medium. This work focuses on the case of phononic crystals modeled as discrete periodic chains. This model offers a good representation for a case in which particles are seen to vibrate as rigid bodies around their equilibrium positions, for instance linear granular media. Most literature studies in this area focus on band-gap resonance frequencies in finite periodic chains with free-free boundary conditions. In this work, a method is presented to formulate the band-gap mode existence conditions in a discrete finite diatomic chain with general boundary conditions. The developed method is based on the notion that the characteristic equation of this periodic chain can be regarded as that of a 2-Teoplitz matrix, which can be obtained in closed form. It can also be generalized to chains made of unit cells with different compositions.

 

The band-gap modes existence conditions are found to depend on both the diatomic unit cell features, e.g., the mass ratio or the springs’ stiffness ratio, and the finite-chain features, e.g., the number of unit cells and the type of boundary conditions. Finite diatomic chains with common boundary conditions, e.g., fixed, or free boundaries, are studied as examples. The derived band-gap modes existence conditions match those previously reported in theoretical and experimental works in the literature for linear granular-type free-free chains.  Furthermore, due to the generality of the proposed approach, existence conditions for chains with fixed-free and fixed-fixed boundary conditions are derived for the first time.  Moreover, another example is shown in which an even-numbered free-free chain with an arbitrary right-end mass is considered. The variation of the band-gap resonance frequencies existence conditions with the right-end mass value demonstrates a gradual transition between the distinct existence conditions found under various chain boundary conditions and parity (even or odd chains). Therefore, this method and the resulting findings pave the way towards manipulating band-gap resonance frequencies in linear discrete granular-type periodic structures, which is useful for sensing, energy harvesting, flow control applications, among others.


Band-Gap Resonances in a Linear Diatomic Chain With General Boundary Conditions: Closed-Form Existence Conditions

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Paper Proceedings: https://asmedigitalcollection.asme.org/IMECE/IMECE2020/volume/84478

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