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. 2018 Feb 2;8:2262. doi: 10.1038/s41598-018-19963-1

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© The Author(s) 2018

Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/.

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Simplified models for bandgap limiting modes. (a) Model for closing mode: schematic view of the undeformed and deformed configurations of the U-shaped elastic suspensions. (b) Frequency tuning of opening and closing modes in function of the u displacement induced by external tensile loads. The frequency in the undeformed and deformed configurations is indicated by f₀ and f respectively. (c) Modal shape in point Y of the bottom limit of the first bandgap in the undeformed configuration. (d) Cross-sectional view of the modal shape of the bottom limit of the first bandgap and its simplified model.