{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,4]],"date-time":"2026-05-04T16:47:16Z","timestamp":1777913236626,"version":"3.51.4"},"reference-count":46,"publisher":"MDPI AG","issue":"11","license":[{"start":{"date-parts":[[2021,11,2]],"date-time":"2021-11-02T00:00:00Z","timestamp":1635811200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100002383","name":"King Saud University","doi-asserted-by":"publisher","award":["\u2022\tThe authors extend their appreciation to King Saud University for funding this work through Researchers Supporting Project number (RSP-2021\/164), King Saud University, Riyadh, Saudi Arabia"],"award-info":[{"award-number":["\u2022\tThe authors extend their appreciation to King Saud University for funding this work through Researchers Supporting Project number (RSP-2021\/164), King Saud University, Riyadh, Saudi Arabia"]}],"id":[{"id":"10.13039\/501100002383","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100009894","name":"Lodz university of technology","doi-asserted-by":"publisher","award":["\u2022\tThis work has been supported by the National Science Centre, Poland, under the grant OPUS 14 No. 2017\/27\/B\/ST8\/01330."],"award-info":[{"award-number":["\u2022\tThis work has been supported by the National Science Centre, Poland, under the grant OPUS 14 No. 2017\/27\/B\/ST8\/01330."]}],"id":[{"id":"10.13039\/501100009894","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>The performance of a nonlinear position-velocity controller in stabilising the lateral vibrations of a rotor-active magnetic-bearings system (RAMBS) is investigated. Cubic nonlinear position-velocity and linear position-velocity controllers are introduced to stabilise RAMBS lateral oscillations. According to the proposed control law, the nonlinear system model is established and then investigated with perturbation analysis. Nonlinear algebraic equations that govern the steady-state oscillation amplitudes and the corresponding phases are derived. Depending on the obtained algebraic equations, the different frequency response curves and bifurcation diagrams are plotted for the studied model. Sensitivity analysis for the linear and nonlinear controllers\u2019 gains is explored. Obtained analytical results demonstrated that the studied model had symmetric bifurcation behaviours in both the horizontal and vertical directions. In addition, the integration of the cubic position controller made the control algorithm more flexible to reshape system dynamical behaviours from the hardening spring characteristic to the softening spring characteristic (or vice versa) to avoid resonance conditions. Moreover, the optimal design of the cubic position gain and\/or cubic velocity gain could stabilise the unstable motion and eliminate the nonlinear effects of the system even at large disc eccentricities. Lastly, numerical validations for all acquired results are performed, where the presented simulations show accurate correspondence between numerical and analytical investigations.<\/jats:p>","DOI":"10.3390\/sym13112069","type":"journal-article","created":{"date-parts":[[2021,11,2]],"date-time":"2021-11-02T22:14:52Z","timestamp":1635891292000},"page":"2069","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":26,"title":["On the Performance of a Nonlinear Position-Velocity Controller to Stabilize Rotor-Active Magnetic-Bearings System"],"prefix":"10.3390","volume":"13","author":[{"given":"Sabry M.","family":"El-Shourbagy","sequence":"first","affiliation":[{"name":"Department of Basic Science, Higher Technological Institute, Tenth of Ramadan City 44629, Egypt"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3275-2392","authenticated-orcid":false,"given":"Nasser A.","family":"Saeed","sequence":"additional","affiliation":[{"name":"Department of Physics and Engineering Mathematics, Faculty of Electronic Engineering, Menoufia University, Menouf 32952, Egypt"}]},{"given":"Magdi","family":"Kamel","sequence":"additional","affiliation":[{"name":"Department of Physics and Engineering Mathematics, Faculty of Electronic Engineering, Menoufia University, Menouf 32952, Egypt"}]},{"given":"Kamal R.","family":"Raslan","sequence":"additional","affiliation":[{"name":"Mathematics Department, Faculty of Science, Al-Azhar University, Nasr-City, Cairo 11651, Egypt"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6967-7747","authenticated-orcid":false,"given":"Emad","family":"Abouel Nasr","sequence":"additional","affiliation":[{"name":"Industrial Engineering Department, College of Engineering, King Saud University, P.O. Box 800, Riyadh 11421, 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, 90924 Lodz, Poland"}]}],"member":"1968","published-online":{"date-parts":[[2021,11,2]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"133","DOI":"10.1006\/jsvi.2000.2916","article-title":"Bifurcation behavior of a rotor supported by active magnetic bearings","volume":"235","author":"Ji","year":"2000","journal-title":"J. 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