{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,12]],"date-time":"2026-05-12T00:21:14Z","timestamp":1778545274308,"version":"3.51.4"},"reference-count":26,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2021,4,5]],"date-time":"2021-04-05T00:00:00Z","timestamp":1617580800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["61973308; 61703270"],"award-info":[{"award-number":["61973308; 61703270"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100012226","name":"Fundamental Research Funds for the Central Universities of China","doi-asserted-by":"publisher","award":["800015Z1160"],"award-info":[{"award-number":["800015Z1160"]}],"id":[{"id":"10.13039\/501100012226","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Sensors"],"abstract":"<jats:p>A compensation method for nonlinear vibration of a silicon micro resonant sensor is proposed and evaluated to be effective through simulation and experimental analysis. Firstly, the parameter characterization model of the silicon micro resonant sensor is established, which presents significant nonlinearity because of the nonlinear vibration of the resonant beam. A verification circuit is devised to imitate the nonlinear behavior of the model by matching the simulation measurement error of the frequency offset produced by the circuit block with the theoretical counterparts obtained from the model. Secondly, the principle of measurement error compensation is studied, and the compensation method dealing with nonlinear characteristics of the resonant beam is proposed by introducing a compensation beam and corresponding differential operations. The measurement error, compensation rate, and measurement residual between the two scenarios that use single beam and double beams, respectively, are derived and are compared with their simulation and experimental counterparts. The results coincide with the predicted trend, which verifies the effectiveness of the compensation method.<\/jats:p>","DOI":"10.3390\/s21072545","type":"journal-article","created":{"date-parts":[[2021,4,5]],"date-time":"2021-04-05T11:48:29Z","timestamp":1617623309000},"page":"2545","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":9,"title":["A Compensation Method for Nonlinear Vibration of Silicon-Micro Resonant Sensor"],"prefix":"10.3390","volume":"21","author":[{"given":"Yan","family":"Li","sequence":"first","affiliation":[{"name":"School of Mechanical Electronic &amp; Information Engineering, China University of Mining and Technology-Beijing, Beijing 100083, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Hao","family":"Li","sequence":"additional","affiliation":[{"name":"School of Mechanical Electronic &amp; Information Engineering, China University of Mining and Technology-Beijing, Beijing 100083, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yifeng","family":"Xiao","sequence":"additional","affiliation":[{"name":"School of Mechanical Electronic &amp; Information Engineering, China University of Mining and Technology-Beijing, Beijing 100083, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Le","family":"Cao","sequence":"additional","affiliation":[{"name":"School of Electrical and Electronic Engineering, Shanghai University of Engineering Science, Shanghai 201620, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Zhan-She","family":"Guo","sequence":"additional","affiliation":[{"name":"School of Instrumentation and Optoelectronic Engineering, Beihang University, Beijing 100191, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2021,4,5]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"757","DOI":"10.1007\/s00542-014-2109-8","article-title":"Design and optimization based on uncertainty analysis in electro-thermal excited MEMS resonant sensor","volume":"21","author":"Shi","year":"2014","journal-title":"Microsyst. Technol."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"279","DOI":"10.1007\/BF02477740","article-title":"New microtransducer for physiological pressure recordings","volume":"13","year":"1975","journal-title":"Med Biol. Eng. Comput."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"1617","DOI":"10.1364\/OL.17.001617","article-title":"Dispersion of pulsed squeezing for reduction of sensor nonlinearity","volume":"17","author":"Doerr","year":"1992","journal-title":"Opt. Lett."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"222","DOI":"10.1166\/sl.2005.037","article-title":"Nonlinear effect in magnetoelas-Tic sensors","volume":"3","author":"Zeng","year":"2005","journal-title":"Sens. Lett."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"332","DOI":"10.1016\/j.sna.2008.03.012","article-title":"Pressure nonlinearity of micro-machined piezoresistive pressure sensors with thin diaphragms under high residual stresses","volume":"147","author":"Chiou","year":"2008","journal-title":"Sens. Actuat. A-Phys."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"145","DOI":"10.1016\/j.sna.2009.11.016","article-title":"An experimental study of high gain parametric amplification in MEMS","volume":"162","author":"Hu","year":"2010","journal-title":"Sensors Actuators A Phys."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"532","DOI":"10.1016\/j.ijnonlinmec.2010.12.012","article-title":"Computational and quasi-analytical models for non-linear vibrations of resonant MEMS and NEMS sensors","volume":"46","author":"Kacem","year":"2011","journal-title":"Int. J. Nonlinear Mech."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"103","DOI":"10.1016\/j.sna.2015.05.016","article-title":"Nonlinear dynamics study based on uncertainty analysis in electro-thermal excited MEMS resonant sensor","volume":"232","author":"Shi","year":"2015","journal-title":"Sens. Actuator. A Phys."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"624","DOI":"10.1016\/j.ijmecsci.2016.07.035","article-title":"Nonlinear resonant frequency of graphene\/elastic\/piezoelectric laminated films under active electric loading","volume":"115","author":"Li","year":"2016","journal-title":"Int. J. Mech. Sci."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"39","DOI":"10.1016\/j.finel.2016.12.002","article-title":"Geometrically nonlinear dynamic behavior on detection sensitivity of carbon nanotube-based mass sensor using finite element method","volume":"126","author":"Kang","year":"2017","journal-title":"Finite Elem. Anal. Des."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"104","DOI":"10.1016\/j.ijnonlinmec.2018.12.008","article-title":"Nonlinear dynamics and parameter identification of electrostatically coupled resonators","volume":"110","author":"Li","year":"2019","journal-title":"Int. J. Nonlinear Mech."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"470","DOI":"10.1016\/j.apm.2019.03.035","article-title":"Multi-field coupled chaotic vibration for a micro resonant pressure sensor","volume":"72","author":"Fu","year":"2019","journal-title":"Appl. Math. Model."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"749","DOI":"10.1006\/jsvi.2000.2980","article-title":"Evaluation of the non-linear dynamic response to harmonic excitation of a beam with several breathing cracks","volume":"235","author":"Pugno","year":"2000","journal-title":"J. Sound Vib."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"318","DOI":"10.1109\/JMEMS.2012.2221156","article-title":"Parametric excitation, amplification, and tuning of MEMS folded-beam comb drive oscillator","volume":"22","author":"Khirallah","year":"2012","journal-title":"J. Microelectromech. Syst."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"73","DOI":"10.1016\/j.ijmachtools.2011.11.008","article-title":"Stability prediction in straight turning of a flexible workpiece by collocation method","volume":"54\u201355","author":"Urbikain","year":"2012","journal-title":"Int. J. Mach. Tools Manuf."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"5702","DOI":"10.21595\/jve.2017.18226","article-title":"Effects of Casimir force on multi-field coupled nonlinear vibration of orthotropic micro film","volume":"19","author":"Xu","year":"2017","journal-title":"J. Vibroeng."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"317","DOI":"10.1016\/j.jsv.2014.12.002","article-title":"Preventing chatter vibrations in heavy-duty turning operations in large horizontal lathes","volume":"340","author":"Urbikain","year":"2015","journal-title":"J. Sound Vib."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"534","DOI":"10.1016\/j.ymssp.2017.07.034","article-title":"Nonlinearity measurement for low-pressure encapsulated MEMS gyroscopes by transient response","volume":"100","author":"Wei","year":"2018","journal-title":"Mech. Syst. Signal Process."},{"key":"ref_19","first-page":"38","article-title":"Tee compensation of nonlinear thermal bias drift of resonant rate sensor (RRS) using fuzzy logic","volume":"78","author":"Kim","year":"1998","journal-title":"IEEE Trans. Aerosp. Electron. Syst."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"343","DOI":"10.1016\/j.microrel.2005.04.008","article-title":"Temperature compensation of piezoresistive micro-machined porous silicon pressure sensor by ANN","volume":"46","author":"Pramanik","year":"2006","journal-title":"Microelectron. Reliab."},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Araghi, G., and Landry, R.J. (2018, January 23\u201326). Temperature compensation model of MEMS inertial sensors based on neural network. Proceedings of the IEEE\/ION Position, Location and Navigation Symposium (PLANS), Monterey, CA, USA.","DOI":"10.1109\/PLANS.2018.8373395"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"748","DOI":"10.1049\/cje.2019.05.001","article-title":"A high precision software compensation algorithm for silicon piezoresistive pressure sensor","volume":"28","author":"Wu","year":"2019","journal-title":"Chin. J. Electron."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"401","DOI":"10.1016\/j.cnsns.2012.06.025","article-title":"Mathieu equation with application to analysis of dynamic characteristics of resonant inertial sensors","volume":"18","author":"Li","year":"2013","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"451","DOI":"10.1108\/SR-05-2017-0081","article-title":"Dynamic characteristics of resonant vibratory gyroscopes based on the damped Mathieu equation","volume":"37","author":"Li","year":"2017","journal-title":"Sens. Rev."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"4417","DOI":"10.1016\/j.jsv.2012.05.010","article-title":"Frequency measurement study of resonant vibratory gyroscopes","volume":"331","author":"Li","year":"2012","journal-title":"J. Sound Vib."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"19","DOI":"10.1016\/j.sna.2012.01.006","article-title":"Non-linear dynamics of an electrothermally excited resonant pressure sensor","volume":"188","author":"Li","year":"2012","journal-title":"Sens. Actuators A Phys."}],"container-title":["Sensors"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1424-8220\/21\/7\/2545\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,13]],"date-time":"2025-10-13T14:11:19Z","timestamp":1760364679000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1424-8220\/21\/7\/2545"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,4,5]]},"references-count":26,"journal-issue":{"issue":"7","published-online":{"date-parts":[[2021,4]]}},"alternative-id":["s21072545"],"URL":"https:\/\/doi.org\/10.3390\/s21072545","relation":{},"ISSN":["1424-8220"],"issn-type":[{"value":"1424-8220","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,4,5]]}}}