Pole distribution in finite phononic crystals: Understanding Bragg-effects through closed-form system dynamics

Abstract

Bragg band gaps associated with infinite phononic crystals are predicted using wave dispersion models. This paper departs from the Bloch-wave solution and presents a comprehensive dynamic systems analysis of finite phononic systems. Closed form transfer functions are derived for two systems where phononic effects are achieved by periodic variation of material property and boundary conditions. Using band structures, differences in dispersion characteristics are highlighted and followed by an analytical derivation of the eigenvalues. The latter is used to derive the end-to-end transfer function of a finite phononic crystal as a function of any given parameters. The analysis reveals intriguing features that explain the evolution of Bragg band gaps in the frequency response. It quantifies how the split of eigenvalues into sub- and super-band-gap natural frequencies contribute to band gap formation. The unique distribution of poles allows the closely packed sub-band gap natural frequencies to achieve maximum attenuation in the Bode response. At that point, the impact of the super-band-gap frequencies on the opposing side becomes significant causing the attenuation to fade and the band gap to come to an end. Finally, the effect of splitting the poles further apart is presented in both phononic systems, with material and boundary condition periodicities.

Uncertainty quantification of tunable elastic metamaterials using polynomial chaos

“…In the finite regime, the band gaps are confined by poles (i.e. natural frequencies) at both ends which also agrees the theoretical limits listed in Table II with sufficiently large n [21]. Unlike the LRM case, the FRF slightly changes in the MV case and remains unchanged in the EF one as ϑ flips its sign, owing to the symmetrical response of the band gap limits about ϑ = 0 ( Figure 3).…”

“…In all three cases, a band gap of width ∆Ω = Ω u − Ω l splits the acoustical and optical branches of the dispersion relation, where Ω u and Ω l denote the upper and lower limits, respectively, normalized by ω 0 . An additional band gap opens up in the EF and LRM cases as a result of the grounded elastic supports [21], which starts at Ω = 0 and ends at Ω z .…”

“…The poles of the system can then be found from the determinant of D(s) and are indifferent to the excitation and measurement (sensing) locations. Analytical expressions for the aforementioned transfer functions for all three EM types considered here and free-free boundary conditions have been recently reported and the derivations are therefore omitted here for brevity [20,21]. However, final expressions for Z(s) and P (s) for the MV, EF, and LRM cases are summarized in Table III. The frequency response functions (FRFs) of the three EMs are graphically presented in Figure 4 for ϑ = ±0.5ϑ max and n = 40 cells.…”

“…It can be clearly observed that all the MC runs are sandwiched within the analytical Bernstein bounds. In finite metamaterials, it has been shown that the band gap limits coincide with the two natural frequencies flanking the band gap, the latter being a natural-frequency-free range [21]. As such, the global set of natural frequencies of the finite metamaterials can be effectively split into two groups representing poles which lie on the acoustical and optical branches, respectively.…”

“…extended frequency ranges of forbidden wave propagation; a hallmark feature of phononic, acoustic and elastic metamaterials [15][16][17][18][19]. Such gaps are theoretically predicted in infinitely long periodic structures and are derived from unit-cell-based models which often fail to capture truncation and boundary effects in finite realizations of such metamaterials; the ramifications of which have been extensively reviewed in literature [20][21][22][23]. As such, the * Corresponding author: mnouh@buffalo.edu analysis conducted here spans both infinite and finite configurations of three distinct types of elastic metamaterials, namely (1) Periodic structures with periodic variations -reminiscent of traditional phononic crystals, (2) A lattice with a periodic elastic foundation, and (3) A locally resonant metamaterial.…”