{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,29]],"date-time":"2026-04-29T21:02:48Z","timestamp":1777496568251,"version":"3.51.4"},"reference-count":32,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2022,3,25]],"date-time":"2022-03-25T00:00:00Z","timestamp":1648166400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>A cubic-position negative-velocity (CPNV) feedback controller is proposed in this research in order to suppress the nontrivial oscillations of the 1\/3 order subharmonic resonance of a mass-damper-spring model. Based on the Krylov\u2013Bogoliubov (KB) averaging method, the model\u2019s equation of motion is approximately solved and tested for stability. The nontrivial solutions region is plotted to determine where these solutions occur and try to quench them. The controller parameters can play crucial roles in eliminating such regions, keeping only the trivial solutions, and improving the transient response of the car\u2019s oscillations. Different response curves and relations are included in this study to provide the reader a wide overview of the control process.<\/jats:p>","DOI":"10.3390\/sym14040685","type":"journal-article","created":{"date-parts":[[2022,3,27]],"date-time":"2022-03-27T21:31:25Z","timestamp":1648416685000},"page":"685","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":15,"title":["1\/3 Order Subharmonic Resonance Control of a Mass-Damper-Spring Model via Cubic-Position Negative-Velocity Feedback"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-7176-2489","authenticated-orcid":false,"given":"Ali","family":"Kandil","sequence":"first","affiliation":[{"name":"Department of Physics and Engineering Mathematics, Faculty of Electronic Engineering, Menoufia University, Menouf 32952, Egypt"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0365-0282","authenticated-orcid":false,"given":"Yasser S.","family":"Hamed","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Statistics, College of Science, Taif University, P.O. Box 11099, Taif 21944, Saudi Arabia"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2908-1807","authenticated-orcid":false,"given":"Khadijah M.","family":"Abualnaja","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Statistics, College of Science, Taif University, P.O. Box 11099, Taif 21944, Saudi Arabia"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0387-921X","authenticated-orcid":false,"given":"Jan","family":"Awrejcewicz","sequence":"additional","affiliation":[{"name":"Department of Automation, Biomechanics and Mechatronics, Faculty of Mechanical Engineering, Lodz University of Technology, 90-924 Lodz, Poland"}]},{"given":"Maksymilian","family":"Bednarek","sequence":"additional","affiliation":[{"name":"Department of Automation, Biomechanics and Mechatronics, Faculty of Mechanical Engineering, Lodz University of Technology, 90-924 Lodz, Poland"}]}],"member":"1968","published-online":{"date-parts":[[2022,3,25]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"457","DOI":"10.1016\/0022-460X(83)90347-4","article-title":"The response of single degree of freedom systems with quadratic and cubic non-linearities to a subharmonic excitation","volume":"89","author":"Nayfeh","year":"1983","journal-title":"J. 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