{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,25]],"date-time":"2025-10-25T12:08:01Z","timestamp":1761394081243,"version":"build-2065373602"},"reference-count":53,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2011,8,25]],"date-time":"2011-08-25T00:00:00Z","timestamp":1314230400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Sensors"],"abstract":"<jats:p>Multilayered microresonators commonly use sensitive coating or piezoelectric layers for detection of mass and gas. Most of these microresonators have a variable cross-section that complicates the prediction of their fundamental resonant frequency (generally of the bending mode) through conventional analytical models. In this paper, we present an analytical model to estimate the first resonant frequency and deflection curve of single-clamped multilayered microresonators with variable cross-section. The analytical model is obtained using the Rayleigh and Macaulay methods, as well as the Euler-Bernoulli beam theory. Our model is applied to two multilayered microresonators with piezoelectric excitation reported in the literature. Both microresonators are composed by layers of seven different materials. The results of our analytical model agree very well with those obtained from finite element models (FEMs) and experimental data. Our analytical model can be used to determine the suitable dimensions of the microresonator\u2019s layers in order to obtain a microresonator that operates at a resonant frequency necessary for a particular application.<\/jats:p>","DOI":"10.3390\/s110908203","type":"journal-article","created":{"date-parts":[[2011,8,30]],"date-time":"2011-08-30T06:04:23Z","timestamp":1314684263000},"page":"8203-8226","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":15,"title":["Analytical Modeling for the Bending Resonant Frequency of Multilayered Microresonators with Variable Cross-Section"],"prefix":"10.3390","volume":"11","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-7373-9258","authenticated-orcid":false,"given":"Agust\u00edn L.","family":"Herrera-May","sequence":"first","affiliation":[{"name":"Centro de Investigaci\u00f3n en Micro y Nanotecnolog\u00eda, Universidad Veracruzana, Calzada Ruiz Cortines 455, 94292, Boca del R\u00edo, Veracruz, Mexico"},{"name":"Depto. Ingenier\u00eda Mec\u00e1nica, DICIS, Universidad de Guanajuato, Carr. Salamanca-Valle 3.5+1.8 km, Palo Blanco, 36885, Salamanca, Guanajuato, Mexico"}]},{"given":"Luz A.","family":"Aguilera-Cort\u00e9s","sequence":"additional","affiliation":[{"name":"Depto. Ingenier\u00eda Mec\u00e1nica, DICIS, Universidad de Guanajuato, Carr. Salamanca-Valle 3.5+1.8 km, Palo Blanco, 36885, Salamanca, Guanajuato, Mexico"}]},{"given":"Hector","family":"Plascencia-Mora","sequence":"additional","affiliation":[{"name":"Depto. Ingenier\u00eda Mec\u00e1nica, DICIS, Universidad de Guanajuato, Carr. Salamanca-Valle 3.5+1.8 km, Palo Blanco, 36885, Salamanca, Guanajuato, Mexico"}]},{"given":"\u00c1ngel L.","family":"Rodr\u00edguez-Morales","sequence":"additional","affiliation":[{"name":"Depto. Ingenier\u00eda Molecular de Materiales, Centro de F\u00edsica Aplicada y Tecnolog\u00eda Aplicada, UNAM, Boulevard Juriquilla 3001, 76230, Juriquilla, Queretaro, Mexico"}]},{"given":"Jian","family":"Lu","sequence":"additional","affiliation":[{"name":"Research Center for Ubiquitous MEMS and Micro Engineering (UMEMSME), National Institute of Advanced Industrial Science and Technology (AIST), Namiki 1-2-1, 305-8564, Tsukuba, Ibaraki, Japan"}]}],"member":"1968","published-online":{"date-parts":[[2011,8,25]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"62","DOI":"10.1049\/el.2010.3051","article-title":"Silicon carbide electrothermal mixer-filters","volume":"46","author":"Mastropaolo","year":"2010","journal-title":"Electron. Lett"},{"key":"ref_2","first-page":"87","article-title":"Silicon carbide electromechanical resonators","volume":"223","author":"Mastropaolo","year":"2010","journal-title":"Proc. IMechE. Part N: J. Nanoeng. 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