Closed-form existence conditions for bandgap resonances in a finite periodic chain under general boundary conditions

Bastawrous, Mary V.; Hussein, Mahmoud I.

doi:10.1121/10.0009229

Title: Closed-form existence conditions for bandgap resonances in a finite periodic chain under general boundary conditions

Journal Article · 18 January 2022 · Journal of the Acoustical Society of America

DOI: https://doi.org/10.1121/10.0009229 · OSTI ID:1841039

Bastawrous, Mary V.; Hussein, Mahmoud I.

Bragg scattering in periodic media generates bandgaps, frequency bands where waves attenuate rather than propagate. Yet, a finite periodic structure may exhibit resonance frequencies within these bandgaps. This is caused by boundary effects introduced by the truncation of the nominal infinite medium. Previous studies of discrete systems determined existence conditions for bandgap resonances, although the focus has been limited to mainly periodic chains with free–free boundaries. In this paper, we present closed-form existence conditions for bandgap resonances in discrete diatomic chains with general boundary conditions (free–free, free–fixed, fixed–free, or fixed–fixed), odd or even chain parity (contrasting or identical masses at the ends), and the possibility of attaching a unique component (mass and/or spring) at one or both ends. The derived conditions are consistent with those theoretically presented or experimentally observed in prior studies of structures that can be modeled as linear discrete diatomic chains with free–free boundary conditions. An intriguing case is a free–free chain with even parity and an arbitrary additional mass at one end of the chain. Introducing such an arbitrary mass underscores a transition among a set of distinct existence conditions, depending on the type of chain boundaries and parity. The proposed analysis is applicable to linear periodic chains in the form of lumped-parameter models, examined across the frequency spectrum, as well as continuous granular media models, or similar configurations, examined in the low-frequency regime.

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Sponsoring Organization:: USDOE Advanced Research Projects Agency - Energy (ARPA-E)

Grant/Contract Number:: AR0001056

OSTI ID:: 1841039

Journal Information:: Journal of the Acoustical Society of America, Journal Name: Journal of the Acoustical Society of America Journal Issue: 1 Vol. 151; ISSN 0001-4966

Publisher:: Acoustical Society of America (ASA)Copyright Statement

Country of Publication:: United States

Language:: English

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