{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,13]],"date-time":"2026-02-13T16:11:08Z","timestamp":1770999068541,"version":"3.50.1"},"reference-count":33,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2023,4,4]],"date-time":"2023-04-04T00:00:00Z","timestamp":1680566400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100002858","name":"China Postdoctoral Science Foundation","doi-asserted-by":"publisher","award":["2021M690352"],"award-info":[{"award-number":["2021M690352"]}],"id":[{"id":"10.13039\/501100002858","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100002858","name":"China Postdoctoral Science Foundation","doi-asserted-by":"publisher","award":["2021AAD0102"],"award-info":[{"award-number":["2021AAD0102"]}],"id":[{"id":"10.13039\/501100002858","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Science and Technology Major Project of Liuzhou City","award":["2021M690352"],"award-info":[{"award-number":["2021M690352"]}]},{"name":"Science and Technology Major Project of Liuzhou City","award":["2021AAD0102"],"award-info":[{"award-number":["2021AAD0102"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Sensors"],"abstract":"<jats:p>Helical springs with nonlinear geometric parameters nowadays have shown great advantages over classical linear springs, especially due to their superior performance in diminishing dynamic responses in high-speed situations. However, existing studies are mostly available for springs with linear properties, and the sole FE spring models using solid elements occupy significant computational resources. This study presents an FE spring model based on Timoshenko beam theory, which allows for high-speed dynamic simulations of nonlinear springs using a beehive valve spring sample. The dynamic results are also compared with the results of the FE model using solid elements and the results of the engine head test and indicate that the proposed FE model can accurately predict dynamic spring forces and the phenomenon of coil clash when simulating the beehive spring at engine speeds of both 5600 and 8000 RPM. The results also indicate that rapid coil impact brings significant spike forces. It should also be noted that the FE spring model using beam elements displays sufficient accuracy in predicting the dynamic responses of nonlinear springs while occupying much fewer computational resources than the FE model using solid elements.<\/jats:p>","DOI":"10.3390\/s23073737","type":"journal-article","created":{"date-parts":[[2023,4,5]],"date-time":"2023-04-05T02:11:41Z","timestamp":1680660701000},"page":"3737","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":10,"title":["Dynamic Finite Element Model Based on Timoshenko Beam Theory for Simulating High-Speed Nonlinear Helical Springs"],"prefix":"10.3390","volume":"23","author":[{"given":"Jianwei","family":"Zhao","sequence":"first","affiliation":[{"name":"Institute of Engineering Technology, University of Science and Technology Beijing, Beijing 100083, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9163-0133","authenticated-orcid":false,"given":"Zewen","family":"Gu","sequence":"additional","affiliation":[{"name":"Department of Engineering Mechanics, College of Pipeline and Civil Engineering, China University of Petroleum, Qingdao 266580, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Quan","family":"Yang","sequence":"additional","affiliation":[{"name":"Institute of Engineering Technology, University of Science and Technology Beijing, Beijing 100083, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jian","family":"Shao","sequence":"additional","affiliation":[{"name":"Institute of Engineering Technology, University of Science and Technology Beijing, Beijing 100083, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2635-9188","authenticated-orcid":false,"given":"Xiaonan","family":"Hou","sequence":"additional","affiliation":[{"name":"Department of Engineering, Lancaster University, Engineering Building, Lancaster LA1 4YW, UK"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2023,4,4]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"105476","DOI":"10.1016\/j.ijmecsci.2020.105476","article-title":"Non-linear finite element model for dynamic analysis of high-speed valve train and coil collisions","volume":"173","author":"Gu","year":"2020","journal-title":"Int. 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