{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,17]],"date-time":"2026-04-17T14:28:21Z","timestamp":1776436101838,"version":"3.51.2"},"reference-count":57,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2021,9,22]],"date-time":"2021-09-22T00:00:00Z","timestamp":1632268800000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"},{"start":{"date-parts":[[2021,9,22]],"date-time":"2021-09-22T00:00:00Z","timestamp":1632268800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"}],"funder":[{"DOI":"10.13039\/501100001871","name":"Funda\u00e7\u00e3o para a Ci\u00eancia e a Tecnologia","doi-asserted-by":"publisher","award":["POCI-01-0145-FEDER-030348"],"award-info":[{"award-number":["POCI-01-0145-FEDER-030348"]}],"id":[{"id":"10.13039\/501100001871","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["J. Vib. Eng. Technol."],"published-print":{"date-parts":[[2022,1]]},"DOI":"10.1007\/s42417-021-00381-z","type":"journal-article","created":{"date-parts":[[2021,9,22]],"date-time":"2021-09-22T16:03:02Z","timestamp":1632326582000},"page":"375-393","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":6,"title":["Modes of Vibration of Single- and Double-Walled CNTs with an Attached Mass by a Non-local Shell Model"],"prefix":"10.1007","volume":"10","author":[{"given":"Eduardo Henrique","family":"Gon\u00e7alves","sequence":"first","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7718-6910","authenticated-orcid":false,"given":"Pedro","family":"Ribeiro","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2021,9,22]]},"reference":[{"issue":"6348","key":"381_CR1","doi-asserted-by":"publisher","first-page":"56","DOI":"10.1038\/354056a0","volume":"354","author":"Iijima Sumio","year":"1991","unstructured":"Sumio Iijima (1991) Helical microtubules of graphitic carbon. Nature 354(6348):56\u201358. https:\/\/doi.org\/10.1038\/354056a0 (issn: 1476-4687)","journal-title":"Nature"},{"key":"381_CR2","doi-asserted-by":"publisher","unstructured":"Gibson Ronald F, Ayorinde Emmanuel O, Wen Y-F (2007) Vibrations of carbon nanotubes and their composites: a review. Compos Sci Technol 67(1):1\u201328. https:\/\/doi.org\/10.1016\/j.compscitech.2006.03.031. http:\/\/www.sciencedirect.com\/science\/article\/pii\/S026635380600131X (issn: 0266-3538)","DOI":"10.1016\/j.compscitech.2006.03.031"},{"key":"381_CR3","doi-asserted-by":"publisher","unstructured":"Han Z, Fina A (2011) Thermal conductivity of carbon nanotubes and their polymer nanocomposites: a review. Progress Polym Sci 36(7):914\u2013944. https:\/\/doi.org\/10.1016\/j.progpolymsci.2010.11.004. http:\/\/www.sciencedirect.com\/science\/article\/pii\/S0079670010001243 (issn: 0079- 6700)","DOI":"10.1016\/j.progpolymsci.2010.11.004"},{"key":"381_CR4","doi-asserted-by":"publisher","unstructured":"Naik AK, Hanay MS, Hiebert WK, Feng XL, Roukes ML (2009) Towards single-molecule nanomechanical mass spectrometry. Nat Nanotechnol 4(7):445\u2013450 https:\/\/doi.org\/10.1038\/nnano.2009.152 (issn: 1748-3395)","DOI":"10.1038\/nnano.2009.152"},{"key":"381_CR5","doi-asserted-by":"publisher","unstructured":"Kong J, Franklin NR, Zhou C, Chapline MG, Peng S, Cho K, Dai H (2000) Nanotube molecular wires as chemical sensors. Science 287(5453):622\u2013625. https:\/\/doi.org\/10.1126\/science.287.5453.622","DOI":"10.1126\/science.287.5453.622"},{"issue":"10","key":"381_CR6","doi-asserted-by":"publisher","first-page":"1238","DOI":"10.1039\/B707401H","volume":"7","author":"S Waggoner Philip","year":"2007","unstructured":"Waggoner Philip S, Craighead Harold G (2007) Micro- and nanomechanical sensors for environmental, chemical, and biological detection. Lab Chip 7(10):1238\u20131255. https:\/\/doi.org\/10.1039\/B707401H (issn: 1473-0197)","journal-title":"Lab Chip"},{"key":"381_CR7","doi-asserted-by":"publisher","unstructured":"Ilic B, Czaplewski D, Zalalutdinov M, Craighead HG (2001) Single cell detection with micromechanical oscillators. J Vacuum Sci Technol 19(6):2825\u20132828. https:\/\/doi.org\/10.1116\/1.1421572. https:\/\/avs.scitation.org\/doi\/abs\/10.1116\/1.1421572","DOI":"10.1116\/1.1421572"},{"issue":"11","key":"381_CR8","doi-asserted-by":"publisher","first-page":"1976","DOI":"10.1063\/1.1667011","volume":"84","author":"A Gupta","year":"2004","unstructured":"Gupta A, Akin D, Bashir R (2004) Single virus particle mass detection using microresonators with nanoscale thickness. Appl Phys Lett 84(11):1976\u20131978. https:\/\/doi.org\/10.1063\/1.1667011 (issn:0003-6951)","journal-title":"Appl Phys Lett"},{"issue":"2","key":"381_CR9","doi-asserted-by":"publisher","first-page":"522","DOI":"10.1021\/cr0681041","volume":"108","author":"M Goeders Karen","year":"2008","unstructured":"Goeders Karen M, Colton Jonathan S, Bottomley Lawrence A (2008) Microcantilevers: sensing chemical interactions via mechanical motion. Chem Rev 108(2):522\u2013542. https:\/\/doi.org\/10.1021\/cr0681041 (issn:0009- 2665)","journal-title":"Chem Rev"},{"key":"381_CR10","doi-asserted-by":"publisher","unstructured":"Stachowiak Jeanne C, Yue M, Castelino K, Chakraborty A, Majumdar A (2006) Chemomechanics of surface stresses induced by DNA hybridization. Langmuir 22(1):263\u2013268. https:\/\/doi.org\/10.1021\/la0521645 (issn: 0743- 7463)","DOI":"10.1021\/la0521645"},{"key":"381_CR11","doi-asserted-by":"publisher","unstructured":"Kwon T, Park J, Yang J, Yoon DS, Na S, Kim C-W, Suh J-S, Huh Y-M, Haam S, Eom K (2009) Nanomechanical in situ monitoring of proteolysis of peptide by Cathepsin B. PloS One 4(7):e6248\u2013e6248. https:\/\/doi.org\/10.1371\/journal.pone.0006248. https:\/\/pubmed.ncbi.nlm.nih.gov\/19606222%20, https:\/\/www.ncbi.nlm.nih.gov\/pmc\/articles\/PMC2707113\/ (issn: 1932- 6203)","DOI":"10.1371\/journal.pone.0006248"},{"issue":"25","key":"381_CR12","doi-asserted-by":"publisher","first-page":"5246","DOI":"10.1063\/1.1764933","volume":"84","author":"Li Chunyu","year":"2004","unstructured":"Chunyu Li, Tsu-Wei Chou (2004) Mass detection using carbon nanotube-based nanomechanical resonators. Appl Phys Lett 84(25):5246\u20135248. https:\/\/doi.org\/10.1063\/1.1764933 (issn: 0003-6951)","journal-title":"Appl Phys Lett"},{"key":"381_CR13","doi-asserted-by":"publisher","unstructured":"Lassagne B, Garcia-Sanchez D, Aguasca A, Bachtold A (2008) Ultrasensitive mass sensing with a nanotube electromechanical resonator. Nano Lett 8(11):3735\u20133738. https:\/\/doi.org\/10.1021\/nl801982v (PMID: 18939809)","DOI":"10.1021\/nl801982v"},{"key":"381_CR14","doi-asserted-by":"publisher","first-page":"533","DOI":"10.1038\/nnano.2008.200","volume":"3","author":"K Jensen","year":"2008","unstructured":"Jensen K, Kim K, Zettl A (2008) An atomicresolution nanomechanical mass sensor. Nat Nanotechnol 3:533\u2013537. https:\/\/doi.org\/10.1038\/nnano.2008.200 (issn: 1748-3395)","journal-title":"Nat Nanotechnol"},{"key":"381_CR15","doi-asserted-by":"publisher","unstructured":"Chiu H-Y, Hung P, Postma HWCh, Bockrath M (2008) Atomic-scale mass sensing using carbon nanotube resonators. Nano Lett 8(12):4342\u20134346. https:\/\/doi.org\/10.1021\/nl802181c (PMID: 19053791)","DOI":"10.1021\/nl802181c"},{"key":"381_CR16","doi-asserted-by":"publisher","unstructured":"Lassagne B, Bachtold A (2010) Carbon nanotube electromechanical resonator for ultrasensitive mass\/force sensing. Comptes Rendus Physique 11(5):355\u2013361. https:\/\/doi.org\/10.1016\/j.crhy.2010.06.006. https:\/\/www.sciencedirect.com\/science\/article\/pii\/S163107051000054X. (Optical properties of nanotubes, issn: 1631-0705)","DOI":"10.1016\/j.crhy.2010.06.006"},{"key":"381_CR17","doi-asserted-by":"publisher","unstructured":"Farokhi H, Pa\u00efdoussis MP, Misra AK (2018) Nonlinear behaviour and mass detection sensitivity of geometrically imperfect cantilevered carbon nanotube resonators. In: Communications in Nonlinear Science and Numerical Simulation vol 65, pp 272\u2013298. https:\/\/doi.org\/10.1016\/j.cnsns.2018.05.013. https:\/\/www.sciencedirect.com\/science\/article\/pii\/S1007570418301588 (issn: 1007- 5704)","DOI":"10.1016\/j.cnsns.2018.05.013"},{"key":"381_CR18","doi-asserted-by":"publisher","unstructured":"Shaat M, Abdelkefi A (2017) Reporting the sensitivities and resolutions of CNT-based resonators for mass sensing. Mater Des 114:591\u2013598. https:\/\/doi.org\/10.1016\/j.matdes.2016.11.104. https:\/\/www.sciencedirect.com\/science\/article\/pii\/S0264127516315064 (issn: 0264-1275)","DOI":"10.1016\/j.matdes.2016.11.104"},{"issue":"30","key":"381_CR19","doi-asserted-by":"publisher","first-page":"6714","DOI":"10.1021\/acs.jpcc.0c04998","volume":"124","author":"A Besley Nicholas","year":"2020","unstructured":"Besley Nicholas A (2020) Vibrational analysis of carbon nanotube-based nanomechanical resonators. J Phys Chem C 124(30):6714\u201316721. https:\/\/doi.org\/10.1021\/acs.jpcc.0c04998","journal-title":"J Phys Chem C"},{"key":"381_CR20","doi-asserted-by":"publisher","unstructured":"Arash B, Wang Q (2012) A review on the application of nonlocal elastic models in modeling of carbon nanotubes and graphenes. Comput Mater Sci 51(1):303\u2013313. https:\/\/doi.org\/10.1016\/j.commatsci.2011.07.040. http:\/\/www.sciencedirect.com\/science\/article\/pii\/S0927025611004320 (issn: 0927-0256)","DOI":"10.1016\/j.commatsci.2011.07.040"},{"key":"381_CR21","doi-asserted-by":"publisher","unstructured":"Khademolhosseini F, Phani AS, Nojeh A, Rajapakse N (2012) Nonlocal continuum modeling and molecular dynamics simulation of torsional vibration of carbon nanotubes. IEEE Trans Nanotechnol 11(1):34\u201343. https:\/\/doi.org\/10.1109\/TNANO.2011.2111380 (issn: 1941-0085)","DOI":"10.1109\/TNANO.2011.2111380"},{"key":"381_CR22","doi-asserted-by":"publisher","unstructured":"Strozzi M, Smirnov VV, Manevitch LI, Milania M, Pellicano F (2016) Nonlinear vibrations and energy exchange of singlewalled carbon nanotubes. Circumferential flexural modes. J Sound Vib. https:\/\/doi.org\/10.1016\/j.jsv.2016.06.013","DOI":"10.1016\/j.jsv.2016.06.013"},{"issue":"2","key":"381_CR23","doi-asserted-by":"publisher","first-page":"024305","DOI":"10.1063\/1.2423140","volume":"101","author":"WH Duan","year":"2007","unstructured":"Duan WH, Wang CM, Zhang YY (2007) Calibration of nonlocal scaling effect parameter for free vibration of carbon nanotubes by molecular dynamics. J Appl Phys 101(2):024305. https:\/\/doi.org\/10.1063\/1.2423140 (issn: 0021-8979)","journal-title":"J Appl Phys"},{"key":"381_CR24","doi-asserted-by":"publisher","unstructured":"Ansari R, Rouhi H, Sahmani S (2011) Calibration of the analytical nonlocal shell model for vibrations of double-walled carbon nanotubes with arbitrary boundary conditions using molecular dynamics. Int J Mech Sci 53(9):786\u2013792. https:\/\/doi.org\/10.1016\/j.ijmecsci.2011.06.010. http:\/\/www.sciencedirect.com\/science\/article\/pii\/S0020740311001299 (issn: 0020- 7403)","DOI":"10.1016\/j.ijmecsci.2011.06.010"},{"issue":"04","key":"381_CR25","doi-asserted-by":"publisher","first-page":"1540017","DOI":"10.1142\/S0219876215400174","volume":"12","author":"R Liu","year":"2015","unstructured":"Liu R, Wang L (2015) Vibration of cantilevered double-walled carbon nanotubes predicted by timoshenko beam model and molecular dynamics. Int J Comput Methods 12(04):1540017. https:\/\/doi.org\/10.1142\/S0219876215400174 (issn: 0219-8762)","journal-title":"Int J Comput Methods"},{"key":"381_CR26","doi-asserted-by":"publisher","first-page":"381","DOI":"10.1504\/IJMPT.2003.002498","volume":"18","author":"B Bodily","year":"2003","unstructured":"Bodily B, Sun C (2003) Structural and equivalent continuum properties of single-walled carbon nanotubes. Int J Mater Prod Technol 18:381\u2013397. https:\/\/doi.org\/10.1504\/IJMPT.2003.002498","journal-title":"Int J Mater Prod Technol"},{"key":"381_CR27","doi-asserted-by":"publisher","unstructured":"Sakhaee-Pour A, Ahmadian MT, Vafai A (2009) Vibrational analysis of single-walled carbon nanotubes using beam element. Thin-Walled Struct 47(6):646\u2013652. https:\/\/doi.org\/10.1016\/j.tws.2008.11.002. http:\/\/www.sciencedirect.com\/science\/article\/pii\/S0263823108002413 (issn: 0263-8231)","DOI":"10.1016\/j.tws.2008.11.002"},{"key":"381_CR28","doi-asserted-by":"publisher","first-page":"205","DOI":"10.1007\/s42417-019-00113-4","volume":"7","author":"JW Yan","year":"2019","unstructured":"Yan JW, He JB, Tong LH (2019) Longitudinal and torsional vibration characteristics of boron nitride nanotubes. J Vib Eng Technol 7:205\u2013215. https:\/\/doi.org\/10.1007\/s42417-019-00113-4","journal-title":"J Vib Eng Technol"},{"key":"381_CR29","doi-asserted-by":"publisher","unstructured":"Strozzi M, Smirnov VV, Manevitch LI, Pellicano F (2018) Nonlinear vibrations and energy exchange of single-walled carbon nanotubes. Radial breathing modes. Compos Struct 184:613\u2013632. https:\/\/doi.org\/10.1016\/j.compstruct.2017.09.108. http:\/\/www.sciencedirect.com\/science\/article\/pii\/S026382231732055X (issn: 0263-8223)","DOI":"10.1016\/j.compstruct.2017.09.108"},{"key":"381_CR30","doi-asserted-by":"publisher","unstructured":"Avramov KV (2018) Nonlinear vibrations characteristics of single-walled carbon nanotubes by nonlocal elastic shell model. Int J Non-Linear Mech 107:149\u2013160. https:\/\/doi.org\/10.1016\/j.ijnonlinmec.2018.08.017. http:\/\/www.sciencedirect.com\/science\/article\/pii\/S0020746218303238 (issn: 0020-7462)","DOI":"10.1016\/j.ijnonlinmec.2018.08.017"},{"key":"381_CR31","doi-asserted-by":"publisher","unstructured":"Strozzi M, Manevitch LI, Pellicano F, Smirnov VV, Shepelev DS (2014) Low-frequency linear vibrations of single-walled carbon nanotubes: analytical and numerical models. J Sound Vib 333(13):2936\u20132957. https:\/\/doi.org\/10.1016\/j.jsv.2014.01.016. http:\/\/www.sciencedirect.com\/science\/article\/pii\/S0022460X14000601 (issn: 0022-460X)","DOI":"10.1016\/j.jsv.2014.01.016"},{"issue":"5","key":"381_CR32","doi-asserted-by":"publisher","first-page":"670","DOI":"10.1177\/1077546312463750","volume":"20","author":"R Ansari","year":"2012","unstructured":"Ansari R, Rouhi H, Sahmani S (2012) Free vibration analysis of single- and double- vibrations of CNTs with attached masses 21 walled carbon nanotubes based on nonlocal elastic shell models. J Vib Control 20(5):670\u2013678. https:\/\/doi.org\/10.1177\/1077546312463750 (issn: 1077-5463)","journal-title":"J Vib Control"},{"issue":"11","key":"381_CR33","doi-asserted-by":"publisher","first-page":"114323","DOI":"10.1063\/1.1898445","volume":"97","author":"CY Wang","year":"2005","unstructured":"Wang CY, Ru CQ, Mioduchowski A (2005) Free vibration of multiwall carbon nanotubes. J Appl Phys 97(11):114323. https:\/\/doi.org\/10.1063\/1.1898445 (issn: 0021-8979)","journal-title":"J Appl Phys"},{"key":"381_CR34","doi-asserted-by":"publisher","first-page":"1960","DOI":"10.1007\/978-3-662-01028-0","volume-title":"Stresses in shells","author":"W Fl\u00fcgge","year":"1960","unstructured":"Fl\u00fcgge W (1960) Stresses in shells. Springer, Berlin, p 1960"},{"key":"381_CR35","doi-asserted-by":"publisher","unstructured":"Silvestre N, Wang CM, Zhang YY, Xiang Y (2011) Sanders shell model for buckling of single-walled carbon nanotubes with small aspect ratio. Compos Struct 93(7):1683\u20131691. https:\/\/doi.org\/10.1016\/j.compstruct.2011.01.004. http:\/\/www.sciencedirect.com\/science\/article\/pii\/S0263822311000183 (issn: 0263-8223)","DOI":"10.1016\/j.compstruct.2011.01.004"},{"key":"381_CR36","doi-asserted-by":"publisher","unstructured":"Silvestre N (2012) On the accuracy of shell models for torsional buckling of carbon nanotubes. Eur J Mech 32:103\u2013108. https:\/\/doi.org\/10.1016\/j.euromechsol.2011.09.005. http:\/\/www.sciencedirect.com\/science\/article\/pii\/S0997753811001379 (issn: 0997-7538)","DOI":"10.1016\/j.euromechsol.2011.09.005"},{"issue":"1","key":"381_CR37","doi-asserted-by":"publisher","first-page":"178","DOI":"10.1088\/0964-1726\/16\/1\/022","volume":"16","author":"Q Wang","year":"2007","unstructured":"Wang Q, Varadan VK (2007) Application of nonlocal elastic shell theory in wave propagation analysis of carbon nanotubes. Smart Mater Struct 16(1):178\u2013190. https:\/\/doi.org\/10.1088\/0964-1726\/16\/1\/022 (issn: 0964-1726 1361-665X)","journal-title":"Smart Mater Struct"},{"issue":"9","key":"381_CR38","doi-asserted-by":"publisher","first-page":"4703","DOI":"10.1063\/1.332803","volume":"54","author":"A Cemal Eringen","year":"1983","unstructured":"Cemal Eringen A (1983) On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves. J Appl Phys 54(9):4703\u20134710. https:\/\/doi.org\/10.1063\/1.332803 (issn: 0021- 8979)","journal-title":"J Appl Phys"},{"key":"381_CR39","doi-asserted-by":"publisher","unstructured":"Hu Y-G, Liew KM, Wang Q, Hea XQ, Yakobson BI (2008) Nonlocal shell model for elastic wave propagation in single- and double-walled carbon nanotubes. J Mech Phys Solids 56(12):3475\u20133485. https:\/\/doi.org\/10.1016\/j.jmps.2008.08.010. http:\/\/www.sciencedirect.com\/science\/article\/pii\/S0022509608001452 (issn: 0022-5096)","DOI":"10.1016\/j.jmps.2008.08.010"},{"key":"381_CR40","doi-asserted-by":"publisher","first-page":"773","DOI":"10.1007\/s42417-019-00194-1","volume":"8","author":"Y Yang","year":"2020","unstructured":"Yang Y, Lin Q, Guo R (2020) Axisymmetric wave propagation behavior in fluid conveying carbon nanotubes based on nonlocal fluid dynamics and nonlocal strain gradient theory. J Vib Eng Technol 8:773\u2013780. https:\/\/doi.org\/10.1007\/s42417-019-00194-1","journal-title":"J Vib Eng Technol"},{"key":"381_CR41","doi-asserted-by":"publisher","unstructured":"Yan JW, Tong LH, Li C, Zhua Y, Wang ZW (2015) Exact solutions of bending deflections for nano-beams and nano-plates based on nonlocal elasticity theory. Compos Struct 125:304\u2013313. https:\/\/doi.org\/10.1016\/j.compstruct.2015.02.017. https:\/\/www.sciencedirect.com\/science\/article\/pii\/S0263822315000938 (issn: 0263-8223)","DOI":"10.1016\/j.compstruct.2015.02.017"},{"key":"381_CR42","volume-title":"Nonlinear vibrations and stability of shells and plates","author":"A Marco","year":"2008","unstructured":"Amabili M (2008) Nonlinear vibrations and stability of shells and plates. Cambrigde University Press, New York"},{"key":"381_CR43","doi-asserted-by":"publisher","unstructured":"Chang T (2010) A molecular based anisotropic shell model for single-walled carbon nanotubes. J Mech Phys Solids 58(9):1422\u20131433. https:\/\/doi.org\/10.1016\/j.jmps.2010.05.004. http:\/\/www.sciencedirect.com\/science\/article\/pii\/S0022509610000943 (issn: 0022-5096)","DOI":"10.1016\/j.jmps.2010.05.004"},{"key":"381_CR44","unstructured":"Chuaqui TRC (2016) Linear and nonlinear vibrations of single-layer graphene sheets. DEMec. MA thesis. Porto: Faculdade de Engenharia da Universidade do Porto"},{"key":"381_CR45","doi-asserted-by":"publisher","unstructured":"He XQ, Kitipornchai S, Liew KM (2005) Buckling analysis of multi-walled carbon nanotubes: a continuum model accounting for van der Waals interaction. J Mech Phys Solids 53(2):303\u2013326. https:\/\/doi.org\/10.1016\/j.jmps.2004.08.003. http:\/\/www.sciencedirect.com\/science\/article\/pii\/S0022509604001413 (issn: 0022-5096)","DOI":"10.1016\/j.jmps.2004.08.003"},{"key":"381_CR46","doi-asserted-by":"publisher","unstructured":"Bardell NS (1989) The application of symbolic computing to the hierarchical finite element method. Int J Numer Methods Eng 28(5):1181\u20131204. https:\/\/doi.org\/10.1002\/nme.1620280513. https:\/\/onlinelibrary.wiley.com\/doi\/abs\/10.1002\/nme.1620280513 (issn: 0029-5981)","DOI":"10.1002\/nme.1620280513"},{"key":"381_CR47","doi-asserted-by":"publisher","unstructured":"Han W, Petyt M, Hsiao K-M (1994) An investigation into geometrically nonlinear analysis of rectangular laminated plates using the hierarchical finite element method. Finite Elements Anal Des 18(1):273\u2013288. https:\/\/doi.org\/10.1016\/0168-874X(94)90107-4. http:\/\/www.sciencedirect.com\/science\/article\/pii\/0168874X94901074 (issn:0168-874X)","DOI":"10.1016\/0168-874X(94)90107-4"},{"key":"381_CR48","doi-asserted-by":"publisher","unstructured":"Yakobson BI, Brabec CJ, Bernholc J (1996) Nanomechanics of carbon tubes: instabilities beyond linear response. Phys Rev Lett 76(14):2511\u20132514. https:\/\/doi.org\/10.1103\/PhysRevLett.76.2511. https:\/\/link.aps.org\/doi\/10.1103\/PhysRevLett.76.2511. (22 Eduardo Henrique Gon\u00e7alves, Pedro Ribeiro)","DOI":"10.1103\/PhysRevLett.76.2511"},{"key":"381_CR49","doi-asserted-by":"publisher","unstructured":"Huang Y, Wu J, Hwang KC (2006) Thickness of graphene and single-wall carbon nanotubes. Phys Rev B 74:245413. https:\/\/doi.org\/10.1103\/PhysRevB.74.245413. https:\/\/link.aps.org\/doi\/10.1103\/PhysRevB.74.245413","DOI":"10.1103\/PhysRevB.74.245413"},{"key":"381_CR50","doi-asserted-by":"publisher","unstructured":"Dresselhaus MS, Dresselhaus G, Saito R (1995) Physics of carbon nanotubes. Carbon 33(7):883\u2013891. https:\/\/doi.org\/10.1016\/0008-6223(95)00017-8. https:\/\/www.sciencedirect.com\/science\/article\/pii\/0008622395000178 (Nanotubes; issn: 0008-6223)","DOI":"10.1016\/0008-6223(95)00017-8"},{"issue":"2","key":"381_CR51","doi-asserted-by":"publisher","first-page":"281","DOI":"10.1002\/nme.1620190209","volume":"19","author":"L Meirovitch","year":"1983","unstructured":"Meirovitch L, Baruh H (1983) On the inclusion principle for the hierarchical finite element method. Int J Numer Methods Eng 19(2):281\u2013291. https:\/\/doi.org\/10.1002\/nme.1620190209 (issn: 0029-5981)","journal-title":"Int J Numer Methods Eng"},{"key":"381_CR52","unstructured":"Eduardo Henrique Lopes Marinho Gon\u00e7alves (2020) Vibrations of carbon nanotubes by a nonlocal shell model. DEMec. MA thesis. Porto: Faculdade de Engenharia da Universidade do Porto"},{"key":"381_CR53","doi-asserted-by":"publisher","unstructured":"Gupta SS, Bosco FG, Batra RC (2010) Wall thickness and elastic moduli of single-walled carbon nanotubes from frequencies of axial, torsional and inextensional modes of vibration. Comput Mater Sci 47(4):1049\u20131059. https:\/\/doi.org\/10.1016\/j.commatsci.2009.12.007. http:\/\/www.sciencedirect.com\/science\/article\/pii\/S092702560900456X (issn: 0927-0256)","DOI":"10.1016\/j.commatsci.2009.12.007"},{"key":"381_CR54","doi-asserted-by":"publisher","unstructured":"Ribeiro P, Thomas O (2017) Nonlinear modes of vibration and internal resonances in nonlocal beams. J Comput Nonlinear Dyn 12(3):031017. https:\/\/asmedigitalcollection.asme.org\/computationalnonlinear\/article-pdf\/12\/3\/031017\/6109267\/cnd_012_03_031017.pdf. https:\/\/doi.org\/10.1115\/1.4035060 (issn: 1555-1415)","DOI":"10.1115\/1.4035060"},{"key":"381_CR55","doi-asserted-by":"publisher","unstructured":"Alves M, Ribeiro P (2017) Non-linear modes of vibration of Timoshenko nanobeams under electrostatic actuation. Int J Mech Sci 130:88\u2013202. https:\/\/doi.org\/10.1016\/j.ijmecsci.2017.06.003. https:\/\/www.sciencedirect.com\/science\/article\/pii\/S0020740317300383 (issn: 0020-7403)","DOI":"10.1016\/j.ijmecsci.2017.06.003"},{"key":"381_CR56","doi-asserted-by":"publisher","unstructured":"Pellicano F (2007) Vibrations of circular cylindrical shells: theory and experiments. J Sound Vib 303(1):154\u2013170. https:\/\/doi.org\/10.1016\/j.jsv.2007.01.022. http:\/\/www.sciencedirect.com\/science\/article\/pii\/S0022460X07000247 (issn: 0022-460X)","DOI":"10.1016\/j.jsv.2007.01.022"},{"key":"381_CR57","doi-asserted-by":"publisher","unstructured":"Ansari R, Ajori S, Arash B (2012) Vibrations of single- and double-walled carbon nanotubes with layerwise boundary conditions: a molecular dynamics study. Curr Appl Phys 12(3):707\u2013711. https:\/\/doi.org\/10.1016\/j.cap.2011.10.007. http:\/\/www.sciencedirect.com\/science\/article\/pii\/S1567173911005384 (issn: 1567-1739)","DOI":"10.1016\/j.cap.2011.10.007"}],"container-title":["Journal of Vibration Engineering &amp; Technologies"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s42417-021-00381-z.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s42417-021-00381-z\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s42417-021-00381-z.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,3,29]],"date-time":"2022-03-29T17:17:28Z","timestamp":1648574248000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s42417-021-00381-z"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,9,22]]},"references-count":57,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2022,1]]}},"alternative-id":["381"],"URL":"https:\/\/doi.org\/10.1007\/s42417-021-00381-z","relation":{},"ISSN":["2523-3920","2523-3939"],"issn-type":[{"value":"2523-3920","type":"print"},{"value":"2523-3939","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,9,22]]},"assertion":[{"value":"8 March 2021","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"27 July 2021","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"16 August 2021","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"22 September 2021","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"On behalf of all authors, the corresponding author states that there is no conflict of interest.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}]}}