{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,28]],"date-time":"2026-01-28T19:59:09Z","timestamp":1769630349148,"version":"3.49.0"},"reference-count":36,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2022,5,18]],"date-time":"2022-05-18T00:00:00Z","timestamp":1652832000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Education and Scientific Research Foundation for Young Teachers in Fujian Province","award":["JAT190009"],"award-info":[{"award-number":["JAT190009"]}]},{"name":"Education and Scientific Research Foundation for Young Teachers in Fujian Province","award":["VSN202103"],"award-info":[{"award-number":["VSN202103"]}]},{"name":"Education and Scientific Research Foundation for Young Teachers in Fujian Province","award":["52105091"],"award-info":[{"award-number":["52105091"]}]},{"name":"Education and Scientific Research Foundation for Young Teachers in Fujian Province","award":["GXRC-20014"],"award-info":[{"award-number":["GXRC-20014"]}]},{"name":"Education and Scientific Research Foundation for Young Teachers in Fujian Province","award":["GXRC-20018"],"award-info":[{"award-number":["GXRC-20018"]}]},{"name":"Education and Scientific Research Foundation for Young Teachers in Fujian Province","award":["2021J05115"],"award-info":[{"award-number":["2021J05115"]}]},{"name":"State Key Discipline Laboratory of Noise and Vibration Control of Ship Equipment in Shanghai Jiaotong University","award":["JAT190009"],"award-info":[{"award-number":["JAT190009"]}]},{"name":"State Key Discipline Laboratory of Noise and Vibration Control of Ship Equipment in Shanghai Jiaotong University","award":["VSN202103"],"award-info":[{"award-number":["VSN202103"]}]},{"name":"State Key Discipline Laboratory of Noise and Vibration Control of Ship Equipment in Shanghai Jiaotong University","award":["52105091"],"award-info":[{"award-number":["52105091"]}]},{"name":"State Key Discipline Laboratory of Noise and Vibration Control of Ship Equipment in Shanghai Jiaotong University","award":["GXRC-20014"],"award-info":[{"award-number":["GXRC-20014"]}]},{"name":"State Key Discipline Laboratory of Noise and Vibration Control of Ship Equipment in Shanghai Jiaotong University","award":["GXRC-20018"],"award-info":[{"award-number":["GXRC-20018"]}]},{"name":"State Key Discipline Laboratory of Noise and Vibration Control of Ship Equipment in Shanghai Jiaotong University","award":["2021J05115"],"award-info":[{"award-number":["2021J05115"]}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["JAT190009"],"award-info":[{"award-number":["JAT190009"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["VSN202103"],"award-info":[{"award-number":["VSN202103"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["52105091"],"award-info":[{"award-number":["52105091"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["GXRC-20014"],"award-info":[{"award-number":["GXRC-20014"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["GXRC-20018"],"award-info":[{"award-number":["GXRC-20018"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["2021J05115"],"award-info":[{"award-number":["2021J05115"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100008859","name":"Starting Grants of Fuzhou University","doi-asserted-by":"publisher","award":["JAT190009"],"award-info":[{"award-number":["JAT190009"]}],"id":[{"id":"10.13039\/501100008859","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100008859","name":"Starting Grants of Fuzhou University","doi-asserted-by":"publisher","award":["VSN202103"],"award-info":[{"award-number":["VSN202103"]}],"id":[{"id":"10.13039\/501100008859","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100008859","name":"Starting Grants of Fuzhou University","doi-asserted-by":"publisher","award":["52105091"],"award-info":[{"award-number":["52105091"]}],"id":[{"id":"10.13039\/501100008859","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100008859","name":"Starting Grants of Fuzhou University","doi-asserted-by":"publisher","award":["GXRC-20014"],"award-info":[{"award-number":["GXRC-20014"]}],"id":[{"id":"10.13039\/501100008859","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100008859","name":"Starting Grants of Fuzhou University","doi-asserted-by":"publisher","award":["GXRC-20018"],"award-info":[{"award-number":["GXRC-20018"]}],"id":[{"id":"10.13039\/501100008859","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100008859","name":"Starting Grants of Fuzhou University","doi-asserted-by":"publisher","award":["2021J05115"],"award-info":[{"award-number":["2021J05115"]}],"id":[{"id":"10.13039\/501100008859","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100008859","name":"Starting Grants of Fuzhou University","doi-asserted-by":"publisher","award":["JAT190009"],"award-info":[{"award-number":["JAT190009"]}],"id":[{"id":"10.13039\/501100008859","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100008859","name":"Starting Grants of Fuzhou University","doi-asserted-by":"publisher","award":["VSN202103"],"award-info":[{"award-number":["VSN202103"]}],"id":[{"id":"10.13039\/501100008859","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100008859","name":"Starting Grants of Fuzhou University","doi-asserted-by":"publisher","award":["52105091"],"award-info":[{"award-number":["52105091"]}],"id":[{"id":"10.13039\/501100008859","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100008859","name":"Starting Grants of Fuzhou University","doi-asserted-by":"publisher","award":["GXRC-20014"],"award-info":[{"award-number":["GXRC-20014"]}],"id":[{"id":"10.13039\/501100008859","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100008859","name":"Starting Grants of Fuzhou University","doi-asserted-by":"publisher","award":["GXRC-20018"],"award-info":[{"award-number":["GXRC-20018"]}],"id":[{"id":"10.13039\/501100008859","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100008859","name":"Starting Grants of Fuzhou University","doi-asserted-by":"publisher","award":["2021J05115"],"award-info":[{"award-number":["2021J05115"]}],"id":[{"id":"10.13039\/501100008859","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100003392","name":"Fujian Provincial Natural Science Foundation","doi-asserted-by":"publisher","award":["JAT190009"],"award-info":[{"award-number":["JAT190009"]}],"id":[{"id":"10.13039\/501100003392","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100003392","name":"Fujian Provincial Natural Science Foundation","doi-asserted-by":"publisher","award":["VSN202103"],"award-info":[{"award-number":["VSN202103"]}],"id":[{"id":"10.13039\/501100003392","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100003392","name":"Fujian Provincial Natural Science Foundation","doi-asserted-by":"publisher","award":["52105091"],"award-info":[{"award-number":["52105091"]}],"id":[{"id":"10.13039\/501100003392","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100003392","name":"Fujian Provincial Natural Science Foundation","doi-asserted-by":"publisher","award":["GXRC-20014"],"award-info":[{"award-number":["GXRC-20014"]}],"id":[{"id":"10.13039\/501100003392","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100003392","name":"Fujian Provincial Natural Science Foundation","doi-asserted-by":"publisher","award":["GXRC-20018"],"award-info":[{"award-number":["GXRC-20018"]}],"id":[{"id":"10.13039\/501100003392","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100003392","name":"Fujian Provincial Natural Science Foundation","doi-asserted-by":"publisher","award":["2021J05115"],"award-info":[{"award-number":["2021J05115"]}],"id":[{"id":"10.13039\/501100003392","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>The wave propagation and vibration transmission in metamaterial-based elastic rods with periodically attached multi-degree-of-freedom (MDOF) dynamic anti-resonant vibration isolator (DAVI) resonators are investigated. A methodology based on a combination of the transfer matrix (TM) method and the Bloch theorem is developed, yielding an explicit formulation for the complex band structure calculation. The bandgap behavior of the periodic structure arrayed with single-degree-of-freedom (SDOF) DAVI resonators and two-degree-of-freedom (2DOF) DAVI resonators are investigated, respectively. A comparative study indicates that the structure consisting of SDOF DAVI resonators periodically jointed on the metamaterial-based elastic rod can obtain an initial locally resonant band gap 500 Hz smaller than the one given in the published literature. The periodic structure containing 2DOF DAVI resonators has an advantage over the periodic structure with SDOF DAVI resonators in achieving two resonance band gaps. By analyzing the equivalent dynamic mass of a DAVI resonator, the underlying mechanism of achieving a lower initial locally resonant band gap by this periodic structure is revealed. The parameters of the 2DOF DAVI resonator are optimized to obtain the lowest band gap of the periodic structure. The numerical results show that, with the optimal 2DOF DAVI parameters, the periodic structure can generate a much lower initial locally resonant band gap compared with the case before the optimization.<\/jats:p>","DOI":"10.3390\/sym14051030","type":"journal-article","created":{"date-parts":[[2022,5,18]],"date-time":"2022-05-18T03:20:43Z","timestamp":1652844043000},"page":"1030","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["Longitudinal Wave Locally Resonant Band Gaps in Metamaterial-Based Elastic Rods Comprising Multi-Degree-of-Freedom DAVI Resonators"],"prefix":"10.3390","volume":"14","author":[{"given":"Niuniu","family":"Liu","sequence":"first","affiliation":[{"name":"School of Mechanical Engineering and Automation, Fuzhou University, Fuzhou 350116, China"}]},{"given":"Xianliang","family":"Lei","sequence":"additional","affiliation":[{"name":"School of Mechanical Engineering and Automation, Fuzhou University, Fuzhou 350116, China"}]},{"given":"Fuqiang","family":"Lai","sequence":"additional","affiliation":[{"name":"School of Mechanical Engineering and Automation, Fuzhou University, Fuzhou 350116, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8021-6685","authenticated-orcid":false,"given":"Xin","family":"Xue","sequence":"additional","affiliation":[{"name":"School of Mechanical Engineering and Automation, Fuzhou University, Fuzhou 350116, China"}]}],"member":"1968","published-online":{"date-parts":[[2022,5,18]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"765","DOI":"10.1524\/zkri.2005.220.9-10.765","article-title":"Classical vibrational modes in phononic lattices: Theory and experiment","volume":"220","author":"Sigalas","year":"2005","journal-title":"Zeitschrift f\u00fcr Kristallographie-Cryst. Mater."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"1485","DOI":"10.1016\/j.physleta.2011.02.044","article-title":"Formation and coupling of band gaps in a locally resonant elastic system comprising a string with attached resonators","volume":"375","author":"Xiao","year":"2011","journal-title":"Phys. Lett. A"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"2022","DOI":"10.1103\/PhysRevLett.71.2022","article-title":"Acoustic band structure of periodic elastic composites","volume":"71","author":"Kushwaha","year":"1993","journal-title":"Phys. Rev. Lett."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"101072","DOI":"10.1016\/j.nantod.2020.101072","article-title":"Controlled self-assembly of plasmon-based photonic nanocrystals for high performance photonic technologies","volume":"37","author":"Yadav","year":"2021","journal-title":"Nano Today"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"410","DOI":"10.1016\/j.physleta.2005.08.067","article-title":"Low frequency torsional vibration gaps in the shaft with locally resonant structures","volume":"348","author":"Yu","year":"2006","journal-title":"Phys. Lett. A"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"1734","DOI":"10.1126\/science.289.5485.1734","article-title":"Locally resonant sonic materials","volume":"289","author":"Liu","year":"2000","journal-title":"Science"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"083049","DOI":"10.1088\/1367-2630\/12\/8\/083049","article-title":"A sonic band gap based on the locally resonant phononic plates with stubs","volume":"12","author":"Oudich","year":"2010","journal-title":"New J. Phys."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"225502","DOI":"10.1103\/PhysRevLett.88.225502","article-title":"Evidence of fano-like interference phenomena in locally resonant materials","volume":"88","author":"Goffaux","year":"2002","journal-title":"Phys. Rev. Lett."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"165116","DOI":"10.1103\/PhysRevB.65.165116","article-title":"Three-component elastic wave band-gap material","volume":"65","author":"Liu","year":"2002","journal-title":"Phys. Rev. B"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"154302","DOI":"10.1103\/PhysRevLett.93.154302","article-title":"Two-dimensional locally resonant phononic crystals with binary structures","volume":"93","author":"Wang","year":"2004","journal-title":"Phys. Rev. Lett."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"104302","DOI":"10.1103\/PhysRevB.71.104302","article-title":"Quasi-one-dimensional phononic crystals studied using the improved lumped-mass method: Application to locally resonant beams with flexural wave band gap","volume":"71","author":"Wang","year":"2005","journal-title":"Phys. Rev. B"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"201904","DOI":"10.1063\/1.2739369","article-title":"Lamb waves in binary locally resonant phononic plates with two-dimensional lattices","volume":"90","author":"Hsu","year":"2007","journal-title":"Appl. Phys. Lett."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"041906","DOI":"10.1063\/1.3299007","article-title":"Acoustic metamaterial panels for sound attenuation in the 50\u20131000 Hz regime","volume":"96","author":"Yang","year":"2010","journal-title":"Appl. Phys. Lett."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"055602","DOI":"10.1103\/PhysRevE.70.055602","article-title":"Double-negative acoustic metamaterial","volume":"70","author":"Li","year":"2004","journal-title":"Phys. Rev. E"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"452","DOI":"10.1038\/nmat1644","article-title":"Ultrasonic metamaterials with negative modulus","volume":"5","author":"Fang","year":"2006","journal-title":"Nat. Mater."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"51","DOI":"10.1121\/10.0005479","article-title":"Modeling acoustic metamaterials based on reused buttons using data fitting with neural network","volume":"150","author":"Ciaburro","year":"2021","journal-title":"J. Acoust. Soc. Am."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"108605","DOI":"10.1016\/j.apacoust.2021.108605","article-title":"Membrane-type acoustic metamaterial using cork sheets and attached masses based on reused materials","volume":"189","author":"Ciaburro","year":"2022","journal-title":"Appl. Acoust."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"248","DOI":"10.1088\/1367-2630\/8\/10\/248","article-title":"On cloaking for elasticity and physical equations with a transformation invariant form","volume":"8","author":"Milton","year":"2006","journal-title":"New J. Phys."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"014301","DOI":"10.1103\/PhysRevB.77.014301","article-title":"Negative refraction and focusing of ultrasound in two-dimensional phononic crystals","volume":"77","author":"Sukhovich","year":"2008","journal-title":"Phys. Rev. B"},{"key":"ref_20","first-page":"2411","article-title":"Acoustic cloaking theory","volume":"464","author":"Norris","year":"2008","journal-title":"Proc. R. Soc. Lond. A Math. Phys. Eng. Sci."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"063015","DOI":"10.1088\/1367-2630\/10\/6\/063015","article-title":"Acoustic cloaking in two dimensions: A feasible approach","volume":"10","author":"Torrent","year":"2008","journal-title":"New J. Phys."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"024301","DOI":"10.1103\/PhysRevLett.106.024301","article-title":"Broadband acoustic cloak for ultrasound waves","volume":"106","author":"Zhang","year":"2011","journal-title":"Phys. Rev. Lett."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"167","DOI":"10.1115\/1.2061947","article-title":"Quasi-one-dimensional periodic structure with locally resonant band gap","volume":"73","author":"Wang","year":"2006","journal-title":"J. Appl. Mech."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"033042","DOI":"10.1088\/1367-2630\/14\/3\/033042","article-title":"Longitudinal wave band gaps in metamaterial-based elastic rods containing multi-degree-of-freedom resonators","volume":"14","author":"Xiao","year":"2012","journal-title":"New J. Phys."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"3031","DOI":"10.1016\/j.jsv.2014.02.002","article-title":"Reduction of vibration and noise radiation of an underwater vehicle due to propeller forces using periodically layered isolators","volume":"333","author":"Song","year":"2014","journal-title":"J. Sound Vib."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"3141","DOI":"10.1016\/j.physleta.2017.08.020","article-title":"Multi-low-frequency flexural wave attenuation in Euler\u2013Bernoulli beams using local resonators containing negative-stiffness mechanisms","volume":"381","author":"Zhou","year":"2017","journal-title":"Phys. Lett. A"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"647","DOI":"10.1007\/s11071-019-04812-1","article-title":"A nonlinear resonator with inertial amplification for very low-frequency flexural wave attenuations in beams","volume":"96","author":"Zhou","year":"2019","journal-title":"Nonlinear Dyn."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"664","DOI":"10.1016\/j.ymssp.2019.02.008","article-title":"Lower band gaps of longitudinal wave in a one-dimensional periodic rod by exploiting geometrical nonlinearity","volume":"124","author":"Wang","year":"2019","journal-title":"Mech. Syst. Signal Process."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"111862","DOI":"10.1016\/j.compstruct.2020.111862","article-title":"Design and numerical validation of quasi-zero-stiffness metamaterials for very low-frequency band gaps","volume":"236","author":"Cai","year":"2020","journal-title":"Compos. Struct."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"105548","DOI":"10.1016\/j.ijmecsci.2020.105548","article-title":"A semi-active metamaterial beam with electromagnetic quasi-zero-stiffness resonators for ultralow-frequency band gap tuning","volume":"176","author":"Wang","year":"2020","journal-title":"Int. J. Mech. Sci."},{"key":"ref_31","unstructured":"Flannelly, W.G. (1967). Dynamic Antiresonant Vibration Isolator. (3,322,379), U.S. Patent."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"3819","DOI":"10.1177\/1077546317711538","article-title":"Application of a dynamic antiresonant vibration isolator to minimize the vibration transmission in underwater vehicles","volume":"24","author":"Liu","year":"2018","journal-title":"J. Vib. Control"},{"key":"ref_33","unstructured":"Wells, D.A. (1967). Theory and Problems of Lagrangian Dynamics, Schaum-Publishing."},{"key":"ref_34","doi-asserted-by":"crossref","unstructured":"Rao, S.S. (2019). Vibration of Continuous Systems, John Wiley & Sons.","DOI":"10.1002\/9781119424284"},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"235","DOI":"10.1016\/0022-460X(73)90064-3","article-title":"A general theory of harmonic wave propagation in linear periodic systems with multiple coupling","volume":"27","author":"Mead","year":"1973","journal-title":"J. Sound Vib."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"254","DOI":"10.1038\/s42005-021-00757-3","article-title":"Exceptional points in oligomer chains","volume":"4","author":"Downing","year":"2021","journal-title":"Commun. Phys."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/5\/1030\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T23:13:52Z","timestamp":1760138032000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/5\/1030"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,5,18]]},"references-count":36,"journal-issue":{"issue":"5","published-online":{"date-parts":[[2022,5]]}},"alternative-id":["sym14051030"],"URL":"https:\/\/doi.org\/10.3390\/sym14051030","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,5,18]]}}}