Free vibrations of finite periodic structures in pass- and stop-bands of the counterpart infinite waveguides

Hvatov, Alexander; Sorokin, Sergey

doi:10.1016/j.jsv.2015.03.003

Cited by 59 publications

(43 citation statements)

References 16 publications

(40 reference statements)

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“…The occurrence of band gaps in these infinite structures has been explained in light of gaps in the unit cell's dispersion curve (band diagram) and/or the negative effective mass density concept [29,30]. Discrepancies in the response of actual metamaterials motivated several efforts to understand band gap realizations in finite structures and the effect of imposed boundary conditions [31][32][33]. Significant among those is the investigation of the relationship between the borders of Bragg-effect band gaps in phononic (periodic) structures and the corresponding eigenfrequencies, explained using the phase-closure principle [32,33].…”

Section: Introductionmentioning

confidence: 99%

“…Discrepancies in the response of actual metamaterials motivated several efforts to understand band gap realizations in finite structures and the effect of imposed boundary conditions [31][32][33]. Significant among those is the investigation of the relationship between the borders of Bragg-effect band gaps in phononic (periodic) structures and the corresponding eigenfrequencies, explained using the phase-closure principle [32,33]. Modal analysis has also been utilized to develop a mathematical formulation to estimate locally resonant band gaps and provide design guidelines and insights into the choice of resonators and their optimal locations [34].…”

Section: Introductionmentioning

confidence: 99%

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Formation of local resonance band gaps in finite acoustic metamaterials: A closed-form transfer function model

Ba’ba’a

Nouh

Singh

2017

Journal of Sound and Vibration

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The objective of this paper is to use transfer functions to comprehend the formation of band gaps in locally resonant acoustic metamaterials. Identifying a recursive approach for any number of serially arranged locally resonant mass in mass cells, a closed form expression for the transfer function is derived. Analysis of the end-to-end transfer function helps identify the fundamental mechanism for the band gap formation in a finite metamaterial. This mechanism includes (a) repeated complex conjugate zeros located at the natural frequency of the individual local resonators, (b) the presence of two poles which flank the band gap, and (c) the absence of poles in the band-gap. Analysis of the finite cell dynamics are compared to the Bloch-wave analysis of infinitely long metamaterials to confirm the theoretical limits of the band gap estimated by the transfer function modeling. The analysis also explains how the band gap evolves as the number of cells in the metamaterial chain increases and highlights how the response varies depending on the chosen sensing location along the length of the metamaterial. The proposed transfer function approach to compute and evaluate band gaps in locally resonant structures provides a framework for the exploitation of control techniques to modify and tune band gaps in finite metamaterial realizations.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Formation of local resonance band gaps in finite acoustic metamaterials: A closed-form transfer function model

Ba’ba’a

Nouh

Singh

2017

Journal of Sound and Vibration

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“…Since the energy flow evaluation is computationally expensive it is required to build the approximation that allows one to assess attenuation zones without computing insertion losses. One of the approximations are the eigenfrequencies of single periodicity cells [6][7]. The process of eigenfrequency obtaining is described below.…”

Section: =+mentioning

confidence: 99%

“…The second step in the analysis in consider finite symmetrical counterpart, most interesting is to consider structures, that have a one-period length as shown in Finite structure, shown in Fig.6 is called symmetrical periodicity cell. When symmetrical periodicity cell is supported with the symmetrical boundary conditions, it has eigenfrequencies that are located on the stop-band boundaries [6]. Despite the structure is non-linear, one can still find eigenfrequencies of the linearized with the harmonic balance method system.…”

Section: The Eigenfrequency Analysismentioning

confidence: 99%

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Floquet Theory Analysis of a Weakly Non-Linear Periodic Structure

Hvatov¹,

Sorokin²

2019

Proceedings of the 7th International Conference on Computational Methods in Structural Dynamics and Earthquake Engineering (COM

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In the paper, the question of applicability of Floquet theory in a weakly non-linear structure is considered. Usually, references contain numerical approaches for the simple non-linear spring masses case. Nevertheless, there is no readily available analytical solution for that problem. Moreover, the theory expansion to a continuous to a non-linear case is still questionable. In the paper, the structure consisting of a waveguide connected with a non-linear spring is considered. One can use the eigenfrequency of a symmetrical periodicity cell approach, which has no restrictions on a periodic structure form. Two approaches for the stop-band predictions in the infinite periodic weakly nonlinear structure on a simple example of rods connected with the non-linear spring are considered. The approaches considered in the paper such as force balance and symmetrical cell eigenfrequencies could be used for more complicated structures.

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Theory of Truncation Resonances in Continuum Rod‐Based Phononic Crystals with Generally Asymmetric Unit Cells

Ba’ba’a

Willey

Chen

et al. 2023

Advcd Theory and Sims

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Phononic crystals exhibit Bragg bandgaps, frequency regions within which wave propagation is forbidden. In solid continua, bandgaps are the outcome of destructive interferences resulting from periodically alternating material layers. Under certain conditions, natural frequencies emerge within these bandgaps in the form of high‐amplitude localized vibrations near a structural boundary, referred to as truncation resonances. In this paper, the vibrational spectrum of finite phononic crystals which take the form of a one‐dimensional rod is investigated and the factors that contribute to the origination of truncation resonances are explained. By identifying a unit cell symmetry parameter, a family of finite phononic rods, which share the same dispersion relation, yet distinct truncated forms, is defined. A transfer matrix method is utilized to derive closed‐form expressions of the characteristic equations governing the natural frequencies of the finite system and decipher the truncation resonances emerging across different boundary conditions. The analysis establishes concrete connections between the localized vibrations associated with a truncation resonance, boundary conditions, and the overall configuration of the truncated chain as dictated by unit cell choice. The study provides tools to predict, tune, and selectively design truncation resonances, to meet the demands of various applications that require and uniquely benefit from such truncation resonances.

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Free vibrations of finite periodic structures in pass- and stop-bands of the counterpart infinite waveguides

Cited by 59 publications

References 16 publications

Formation of local resonance band gaps in finite acoustic metamaterials: A closed-form transfer function model

Formation of local resonance band gaps in finite acoustic metamaterials: A closed-form transfer function model

Floquet Theory Analysis of a Weakly Non-Linear Periodic Structure

Theory of Truncation Resonances in Continuum Rod‐Based Phononic Crystals with Generally Asymmetric Unit Cells

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