{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,25]],"date-time":"2026-03-25T06:48:57Z","timestamp":1774421337932,"version":"3.50.1"},"reference-count":31,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2021,8,16]],"date-time":"2021-08-16T00:00:00Z","timestamp":1629072000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"\u2022\tThis work has been supported by the National Science Centre, Poland, under the grant OPUS 14 No. 2017\/27\/B\/ST8\/01330","award":["OPUS 14 No. 2017\/27\/B\/ST8\/01330"],"award-info":[{"award-number":["OPUS 14 No. 2017\/27\/B\/ST8\/01330"]}]},{"name":"The 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":["RSP-2021\/164"],"award-info":[{"award-number":["RSP-2021\/164"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>This article is dedicated to investigating the nonlinear dynamical behaviors of the 8-pole rotor active magnetic bearing system. The rub and impact forces between the rotating disc and the pole-legs are included in the studied model for the first time. A new control scheme based on modifying the 8-pole positions has been introduced. The proposed control methodology is designed such that four poles only are located in the horizontal and vertical directions (i.e., in +X,+Y,\u2212X,\u2212Y directions), while the other four poles are inserted in a way such that each pole makes 45\u00b0 with two of the axes +X,+Y,\u2212X,\u2212Y. The control currents in the horizontal and vertical poles are suggested to be proportional to both the velocity and displacement of the rotor in the horizontal and vertical directions, respectively, while the control currents in the inclined poles are proposed to be dependent on the combination of both the displacement and velocity of the rotor in the horizontal and vertical directions. Accordingly, the whole-system mathematical model is derived. The derived discontinuous dynamical system is analyzed employing perturbation methods, Poincare maps, bifurcation diagrams, whirling orbits, and frequency spectrum. The obtained results demonstrated that the controller proportional control gain can play a significant role in changing the vibratory behaviors of the system, where the proposed control method can behave either as a cartesian control strategy or as a radial control one depending on the magnitude of the proportional gain. In addition, it is found that the rotor system can vibrate with periodic, periodic-n, quasiperiodic, or chaotic motion when the rub and\/or impact forces occur. Moreover, it is reported for the first time that the rotor-AMB can oscillate symmetrically in X and Y directions either in full annular rub mode or quasiperiodic partial rub mode depending on the impact stiffness coefficient and the dynamic friction coefficient.<\/jats:p>","DOI":"10.3390\/sym13081502","type":"journal-article","created":{"date-parts":[[2021,8,16]],"date-time":"2021-08-16T21:28:04Z","timestamp":1629149284000},"page":"1502","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":31,"title":["Nonlinear Dynamics and Motion Bifurcations of the Rotor Active Magnetic Bearings System with a New Control Scheme and Rub-Impact Force"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-3275-2392","authenticated-orcid":false,"given":"Nasser A.","family":"Saeed","sequence":"first","affiliation":[{"name":"Department of Physics and Engineering Mathematics, Faculty of Electronic Engineering, Menoufia University, Menouf 32952, Egypt"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8115-4882","authenticated-orcid":false,"given":"Emad","family":"Mahrous","sequence":"additional","affiliation":[{"name":"Electrical Engineering Department, College of Engineering, King Saud University, P.O. Box 800, Riyadh 11421, Saudi Arabia"}]},{"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":[[2021,8,16]]},"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. Sound Vib."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/s11071-013-0967-8","article-title":"Nonlinear oscillations of rotor active magnetic bearings system","volume":"74","author":"Saeed","year":"2013","journal-title":"Nonlinear Dyn."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"599","DOI":"10.1006\/jsvi.2000.3257","article-title":"Non-linear oscillations of a rotor in active magnetic bearings","volume":"240","author":"Ji","year":"2001","journal-title":"J. Sound Vib."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"829","DOI":"10.1016\/S0020-7462(01)00136-6","article-title":"Non-linear oscillations of a rotor-magnetic bearing system under superharmonic resonance conditions","volume":"38","author":"Ji","year":"2003","journal-title":"Int. J. Nonlinear. Mech."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"054503","DOI":"10.1115\/1.4033659","article-title":"Elliptic Motions and Control of Rotors Suspending in Active Magnetic Bearings","volume":"11","author":"Yang","year":"2016","journal-title":"J. Comput. Nonlinear Dyn."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"331","DOI":"10.1007\/s11071-005-7959-2","article-title":"Periodic and chaotic motions of a rotor-active magnetic bearing with quadratic and cubic terms and time-varying stiffness","volume":"41","author":"Zhang","year":"2005","journal-title":"Nonlinear Dyn."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"175","DOI":"10.1016\/j.chaos.2005.04.003","article-title":"Multi-pulse chaotic motions of a rotor-active magnetic bearing system with time-varying stiffness","volume":"27","author":"Zhang","year":"2006","journal-title":"Chaos Solitons Fractals"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"586","DOI":"10.1016\/j.chaos.2006.05.095","article-title":"Global bifurcations and chaos for a rotor-active magnetic bearing system with time-varying stiffness","volume":"35","author":"Zhang","year":"2008","journal-title":"Chaos Solitons Fractals"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"743","DOI":"10.1007\/s11071-013-1166-3","article-title":"Saturation-based active controller for vibration suppression of a four-degree-of-freedom rotor-AMBs","volume":"76","author":"Eissa","year":"2014","journal-title":"Nonlinear Dyn."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"1191","DOI":"10.1007\/s11071-019-05256-3","article-title":"Lateral vibration control and stabilization of the quasiperiodic oscillations for rotor-active magnetic bearings system","volume":"98","author":"Saeed","year":"2019","journal-title":"Nonlinear Dyn."},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Wu, R., Zhang, W., and Yao, M.H. (2017, January 6\u20139). Nonlinear vibration of a rotor-active magnetic bearing system with 16-pole legs. Proceedings of the International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, Cleveland, OH, USA.","DOI":"10.1115\/DETC2017-67103"},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Wu, R., Zhang, W., and Yao, M.H. (2017, January 6\u20139). Analysis of nonlinear dynamics of a rotor-active magnetic bearing system with 16-pole legs. Proceedings of the International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, Cleveland, OH, USA.","DOI":"10.1115\/DETC2017-67105"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"113","DOI":"10.1016\/j.ymssp.2017.07.033","article-title":"Nonlinear dynamics near resonances of a rotor-active magnetic bearings system with 16-pole legs and time varying stiffness","volume":"100","author":"Wu","year":"2018","journal-title":"Mech. Syst. Signal Process."},{"key":"ref_14","first-page":"5282904","article-title":"Nonlinear Vibrations of a Rotor-Active Magnetic Bearing System with 16-Pole Legs and Two Degrees of Freedom","volume":"2020","author":"Zhang","year":"2020","journal-title":"Shock. Vib."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.apm.2020.11.005","article-title":"Two different control strategies for 16-pole rotor active magnetic bearings system with constant stiffness coefficients","volume":"92","author":"Saeed","year":"2021","journal-title":"Appl. Math. Model."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"104051","DOI":"10.1016\/j.euromechsol.2020.104051","article-title":"On the nonlinear dynamics of constant stiffness coefficients 16-pole rotor active magnetic bearings system","volume":"84","author":"Kandil","year":"2020","journal-title":"Eur. J. Mech. A\/Solids"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"138859","DOI":"10.1109\/ACCESS.2020.3012447","article-title":"Radial Versus Cartesian Control Strategies to Stabilize the Nonlinear Whirling Motion of the Six-Pole Rotor-AMBs","volume":"8","author":"Saeed","year":"2020","journal-title":"IEEE Access"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"2299","DOI":"10.1007\/s11071-020-05911-0","article-title":"Nonlinear dynamics of the six-pole rotor-AMBs under two different control configurations","volume":"101","author":"Saeed","year":"2020","journal-title":"Nonlinear Dyn."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"1127","DOI":"10.1177\/1077546307074577","article-title":"Vibration suppression of nonlinear rotor systems using a dynamic damper","volume":"13","author":"Ishida","year":"2007","journal-title":"J. Vib. Control."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"109","DOI":"10.1016\/j.ijnonlinmec.2016.10.003","article-title":"Nonlinear PD-controller to suppress the nonlinear oscillations of horizontally supported Jeffcott-rotor system","volume":"87","author":"Saeed","year":"2016","journal-title":"Int. J. Nonlinear Mech."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"457","DOI":"10.1007\/s11071-017-3675-y","article-title":"Active magnetic bearing-based tuned controller to suppress lateral vibrations of a nonlinear Jeffcott rotor system","volume":"90","author":"Saeed","year":"2017","journal-title":"Nonlinear Dyn."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"523","DOI":"10.1016\/j.apm.2017.02.019","article-title":"Time-delayed control to suppress the nonlinear vibrations of a horizontally suspended Jeffcott-rotor system","volume":"44","author":"Saeed","year":"2017","journal-title":"Appl. Math. Model."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"103878","DOI":"10.1016\/j.euromechsol.2019.103878","article-title":"On the steady-state forward and backward whirling motion of asymmetric nonlinear rotor system","volume":"80","author":"Saeed","year":"2019","journal-title":"Eur. J. Mech. A Solids"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"1899","DOI":"10.1007\/s00419-019-01551-y","article-title":"On vibration behavior and motion bifurcation of a nonlinear asymmetric rotating shaft","volume":"89","author":"Saeed","year":"2019","journal-title":"Arch. Appl. Mech."},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Ishida, Y., and Yamamoto, T. (2012). Linear and Nonlinear Rotordynamics: A Modern Treatment with Applications, Wiley-VCH Verlag GmbH & Co. KGaA. [2nd ed.].","DOI":"10.1002\/9783527651894"},{"key":"ref_26","unstructured":"Schweitzer, G., and Maslen, E.H. (2009). Magnetic Bearings: Theory, Design, and Application to Rotating Machinery, Springer."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"792","DOI":"10.1016\/j.apm.2021.01.008","article-title":"Analysis of the rub-impact forces between a controlled nonlinear rotating shaft system and the electromagnet pole legs","volume":"93","author":"Saeed","year":"2021","journal-title":"Appl. Math. Model."},{"key":"ref_28","doi-asserted-by":"crossref","unstructured":"Saeed, N.A., El-Bendary, S.I., Sayed, M., Mohamed, M.S., and Elagan, S.K. (2021). On the oscillatory behaviours and rub-impact forces of a horizontally supported asymmetric rotor system under position-velocity feedback controller. Lat. Am. J. Solids Struct., 18.","DOI":"10.1590\/1679-78256410"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"86","DOI":"10.1016\/j.jsv.2014.05.049","article-title":"Modelling and experimental verification of an asymmetric Jeffcott rotor with radial clearance","volume":"334","author":"Hamaneh","year":"2015","journal-title":"J. Sound Vib."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"2549","DOI":"10.1016\/j.ymssp.2011.04.004","article-title":"Experimental validation of impact energy model for the rub-impact assessment in a rotor system","volume":"25","author":"Cong","year":"2011","journal-title":"Mech. Syst. Signal Process."},{"key":"ref_31","doi-asserted-by":"crossref","unstructured":"Nayfeh, A.H., and Mook, D.T. (1995). Nonlinear Oscillations, Wiley.","DOI":"10.1002\/9783527617586"}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/13\/8\/1502\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T06:47:18Z","timestamp":1760165238000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/13\/8\/1502"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,8,16]]},"references-count":31,"journal-issue":{"issue":"8","published-online":{"date-parts":[[2021,8]]}},"alternative-id":["sym13081502"],"URL":"https:\/\/doi.org\/10.3390\/sym13081502","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,8,16]]}}}