{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,16]],"date-time":"2025-10-16T20:42:30Z","timestamp":1760647350541,"version":"build-2065373602"},"reference-count":31,"publisher":"MDPI AG","issue":"21","license":[{"start":{"date-parts":[[2020,11,6]],"date-time":"2020-11-06T00:00:00Z","timestamp":1604620800000},"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":["61873021"],"award-info":[{"award-number":["61873021"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100012149","name":"National Key Scientific Instrument and Equipment Development Projects of China","doi-asserted-by":"publisher","award":["2014YQ350461"],"award-info":[{"award-number":["2014YQ350461"]}],"id":[{"id":"10.13039\/501100012149","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100018621","name":"Program for Changjiang Scholars and Innovative Research Team in University","doi-asserted-by":"publisher","award":["IRT1203"],"award-info":[{"award-number":["IRT1203"]}],"id":[{"id":"10.13039\/501100018621","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Sensors"],"abstract":"<jats:p>Certain nonlinear influences are found in dual-tube Coriolis mass flowmeters (CMFs). According to experimentation, a nonlinearity dominated by frequency-doubling signals can be observed in the measuring signal. In general, such nonlinear effects are simplified as linear systems or neglected through processing. In this paper, a simplified model has been constructed for dual-beam CMFs based on the theory of nonlinear dynamics, with the spring\u2013damper system as the medium for the dual-beam coupled vibrations. Next, the dynamics differential equation of the coupled vibrations is set up on the basis of the Lagrangian equation. Furthermore, numerical solutions are obtained using the Runge\u2013Kutta fourth-order method. The study then fits discrete points of the numerical solutions, which are converted into the frequency domain to observe the existence of frequency-doubling signal components. Our findings show that frequency-doubling components exist in the spectrogram, proving that these nonlinear influences are a result of the motions of coupled vibrations. In this study, non-linear frequency-doubling signal sources are qualitatively analyzed to formulate a theoretical basis for CMFs design.<\/jats:p>","DOI":"10.3390\/s20216340","type":"journal-article","created":{"date-parts":[[2020,11,6]],"date-time":"2020-11-06T09:03:04Z","timestamp":1604653384000},"page":"6340","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Synchronization Theory-Based Analysis of Coupled Vibrations of Dual-Tube Coriolis Mass Flowmeters"],"prefix":"10.3390","volume":"20","author":[{"given":"Zhong-Xiang","family":"Li","sequence":"first","affiliation":[{"name":"School of Instrumentation and Optoelectronic Engineering, Beihang University, Beijing 100191, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Chun","family":"Hu","sequence":"additional","affiliation":[{"name":"School of Electronic and Information Engineering, Beihang University, Beijing 100191, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3998-5989","authenticated-orcid":false,"given":"De-Zhi","family":"Zheng","sequence":"additional","affiliation":[{"name":"School of Instrumentation and Optoelectronic Engineering, Beihang University, Beijing 100191, China"},{"name":"Research Institute for Frontier Science, Beihang University, Beijing 100191, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Shang-Chun","family":"Fan","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":[[2020,11,6]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"198","DOI":"10.1016\/S1000-9361(08)60087-4","article-title":"Nonlinear vibration characteristics of Coriolis mass flowmeter","volume":"22","author":"Zheng","year":"2009","journal-title":"Chin. J. Aeronaut."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"3032","DOI":"10.1109\/JSEN.2013.2260859","article-title":"Experimental study and implementation of a novel digital closed-loop control system for coriolis mass flowmeter","volume":"13","author":"Hu","year":"2013","journal-title":"IEEE Sens. J."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"99","DOI":"10.1016\/j.flowmeasinst.2014.08.015","article-title":"Coriolis flowmeters: A review of developments over the past 20 years, and an assessment of the state of the art and likely future directions","volume":"40","author":"Wang","year":"2014","journal-title":"Flow Meas. Instrum."},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Hu, C., Zheng, D., Fan, S., and Gong, J. (2012, January 11\u201313). Research and implementation of amplitude ratio calculating system for coriolis mass flowmeter. Proceedings of the 2012 8th IEEE International Symposium on Instrumentation and Control Technology (ISICT) Proceedings, London, UK.","DOI":"10.1109\/ISICT.2012.6291599"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"207","DOI":"10.1016\/S0955-5986(99)00014-X","article-title":"Understanding the experimental response of Coriolis massflow meters to flow pulsations","volume":"10","author":"Cheesewright","year":"1999","journal-title":"Flow Meas. Instrum."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"1025","DOI":"10.1006\/jfls.1998.0176","article-title":"The effect of flow pulsations on Coriolis mass flow meters","volume":"12","author":"Cheesewright","year":"1998","journal-title":"J. Fluids Struct."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"613","DOI":"10.1006\/jfls.2000.0287","article-title":"The simulation of Coriolis meter response to pulsating flow using a general purpose FE code","volume":"14","author":"Belhadj","year":"2000","journal-title":"J. Fluids Struct."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"345","DOI":"10.1016\/S0955-5986(02)00006-7","article-title":"An analytical estimation of the Coriolis meter\u2019s characteristics based on modal superposition","volume":"12","author":"Kutin","year":"2002","journal-title":"Flow Meas. Instrum."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"201","DOI":"10.1049\/ip-smt:20040151","article-title":"Manufacturing variation of the measuring tube in a Coriolis flowmeter","volume":"151","author":"Wang","year":"2003","journal-title":"IEE Proc. Sci. Meas. Technol."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"785","DOI":"10.1016\/j.jfluidstructs.2005.04.008","article-title":"Coupled finite-volume\/finite-element modelling of the straight-tube Coriolis flowmeter","volume":"20","author":"Bobovnik","year":"2005","journal-title":"J. Fluids Struct."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"71","DOI":"10.1016\/j.flowmeasinst.2014.10.001","article-title":"Numerical analysis of installation effects in Coriolis flowmeters: Single and twin tube configurations","volume":"44","author":"Bobovnik","year":"2015","journal-title":"Flow Meas. Instrum."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"497","DOI":"10.1016\/j.flowmeasinst.2010.07.003","article-title":"Effect of asymmetric actuator and detector position on Coriolis flowmeter and measured phase shift","volume":"21","author":"Enz","year":"2010","journal-title":"Flow Meas. Instrum."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"30","DOI":"10.1016\/j.jsv.2015.05.014","article-title":"Theoretical and experimental investigations of flow pulsation effects in Coriolis mass flowmeters","volume":"352","author":"Svete","year":"2015","journal-title":"J. Sound Vib."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"173","DOI":"10.1007\/s11071-019-04934-6","article-title":"Perturbation-based prediction of vibration phase shift along fluid-conveying pipes due to Coriolis forces, nonuniformity, and nonlinearity","volume":"99","author":"Thomsen","year":"2020","journal-title":"Nonlinear Dyn."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"61","DOI":"10.1007\/s10404-016-1727-0","article-title":"Micro-optofluidic switch realized by 3D printing technology","volume":"20","author":"Cairone","year":"2016","journal-title":"Microfluid. Nanofluidics"},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Gagliano, S., Stella, G., and Bucolo, M. (2020). Real-Time Detection of Slug Velocity in Microchannels. Micromachines, 11.","DOI":"10.3390\/mi11030241"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"6","DOI":"10.1109\/MCAS.2009.933853","article-title":"Microfluidic circuits and systems","volume":"9","author":"Sapuppo","year":"2009","journal-title":"IEEE Circuits Syst. Mag."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"39","DOI":"10.1016\/0955-5986(89)90008-3","article-title":"Contribution to the understanding of the zero shift effects in Coriolis mass flowmeters","volume":"1","author":"Keita","year":"1989","journal-title":"Flow Meas. Instrum."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"3065","DOI":"10.1016\/j.jsv.2010.02.010","article-title":"Phase shift effects for fluid conveying pipes on non-ideal supports","volume":"329","author":"Thomsen","year":"2010","journal-title":"J. Sound Vib."},{"key":"ref_20","unstructured":"Thomsen, J.J., Dahl, J., Fuglede, N., and Enz, S. (2009, January 5\u20139). Predicting phase shift of elastic waves in pipes due to fluid flow and imperfections. Proceedings of the 16th International Congress on Sound and Vibration, Krakow, Poland."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"1151","DOI":"10.3390\/s18041151","article-title":"New analysis scheme of flow-acoustic coupling for gas ultrasonic flowmeter with vortex near the transducer","volume":"18","author":"Sun","year":"2018","journal-title":"Sensors"},{"key":"ref_22","first-page":"1469","article-title":"Dynamic stability of tube conveying fluid","volume":"97","author":"Chen","year":"1971","journal-title":"J. Eng. Mech."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"5096","DOI":"10.1016\/j.jsv.2011.05.022","article-title":"Predicting phase shift effects for vibrating fluid-conveying pipes due to Coriolis forces and fluid pulsations","volume":"330","author":"Enz","year":"2011","journal-title":"J. Sound Vib."},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Kabaci\u0144ski, M., and Pospolita, J. (2020). Effect of Distortion in Velocity Profile on Flow Measurements Using Averaging Flow Sensors. Sensors, 20.","DOI":"10.3390\/s20102839"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"359","DOI":"10.1016\/j.flowmeasinst.2006.07.006","article-title":"Theory of errors in Coriolis flowmeter readings due to compressibility of the fluid being metered","volume":"17","author":"Hemp","year":"2006","journal-title":"Flow Meas. Instrum."},{"key":"ref_26","doi-asserted-by":"crossref","unstructured":"Torres, F., Uranga, A., Riverola, M., Sobreviela, G., and Barniol, N. (2016). Enhancement of frequency stability using synchronization of a cantilever array for mems-based sensors. Sensors, 16.","DOI":"10.3390\/s16101690"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"367","DOI":"10.1016\/S0378-4754(01)00378-0","article-title":"Self-synchronization and controlled synchronization: General definition and example design","volume":"58","author":"Blekhman","year":"2002","journal-title":"Math. Comput. Simul."},{"key":"ref_28","doi-asserted-by":"crossref","unstructured":"Xin, X., Muraoka, Y., and Hara, S. (2016, January 6\u20138). Analysis of synchronization of n metronomes on a cart via describing function method: New results beyond two metronomes. Proceedings of the 2016 American Control Conference (ACC), Boston, MA, USA.","DOI":"10.1109\/ACC.2016.7526710"},{"key":"ref_29","doi-asserted-by":"crossref","unstructured":"Nour, M.A., and Hussain, M.M. (2020). A Review of the Real-Time Monitoring of Fluid-Properties in Tubular Architectures for Industrial Applications. Sensors, 20.","DOI":"10.3390\/s20143907"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"622","DOI":"10.1016\/j.chaos.2016.09.001","article-title":"Nonlinear dynamics and contact interactions of the structures composed of beam-beam and beam-closed cylindrical shell members","volume":"91","author":"Krysko","year":"2016","journal-title":"Chaos Solitons Fractals"},{"key":"ref_31","first-page":"8","article-title":"Flexural vibration analysis of nonuniform double-beam system with general boundary and coupling conditions","volume":"2018","author":"Chen","year":"2018","journal-title":"Shock Vib."}],"container-title":["Sensors"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1424-8220\/20\/21\/6340\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T10:30:16Z","timestamp":1760178616000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1424-8220\/20\/21\/6340"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,11,6]]},"references-count":31,"journal-issue":{"issue":"21","published-online":{"date-parts":[[2020,11]]}},"alternative-id":["s20216340"],"URL":"https:\/\/doi.org\/10.3390\/s20216340","relation":{},"ISSN":["1424-8220"],"issn-type":[{"type":"electronic","value":"1424-8220"}],"subject":[],"published":{"date-parts":[[2020,11,6]]}}}