{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,29]],"date-time":"2026-04-29T21:03:00Z","timestamp":1777496580034,"version":"3.51.4"},"reference-count":37,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2022,6,16]],"date-time":"2022-06-16T00:00:00Z","timestamp":1655337600000},"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>This research is concerned with extracting the approximate solutions of a controlled mass-damper-spring model via the harmonic balance method. The stability of these solutions was checked with the aid of Floquet theory. A nonlinear saturation controller (NSC), a linear variable differential transformer (LVDT) and a servo-controlled linear actuator (SCLA), were applied to suppress the undesired oscillations of the harmonically-excited car. 2D and 3D graphical plots are included based upon the equations resulting from the harmonic balance method. Moreover, a numerical simulation was established using the fourth order Rung\u2013Kutta technique in order to confirm the overall controlled behavior of the studied model.<\/jats:p>","DOI":"10.3390\/sym14061247","type":"journal-article","created":{"date-parts":[[2022,6,19]],"date-time":"2022-06-19T21:19:26Z","timestamp":1655673566000},"page":"1247","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":11,"title":["Harmonic Balance Method to Analyze the Steady-State Response of a Controlled Mass-Damper-Spring Model"],"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 Mathematics, Institute of Basic and Applied Sciences, Egypt-Japan University of Science and Technology (E-JUST), New Borg El-Arab City 21934, Egypt"},{"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":"Y. 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\/0000-0003-0387-921X","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"}]}],"member":"1968","published-online":{"date-parts":[[2022,6,16]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"255","DOI":"10.1006\/jsvi.1993.1295","article-title":"On the steady state response and stability of non-linear oscillators using harmonic balance","volume":"166","author":"Hamdan","year":"1993","journal-title":"J. Sound Vib."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"105","DOI":"10.1007\/BF00045453","article-title":"On the local stability analysis of the approximate harmonic balance solutions","volume":"10","author":"Hassan","year":"1996","journal-title":"Nonlinear Dyn."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"737","DOI":"10.1006\/jsvi.1996.0673","article-title":"On the period of large amplitude free vibration of conservative autonomous oscillators with static and inertia type cubic non-linearities","volume":"199","author":"Hamdan","year":"1997","journal-title":"J. Sound Vib."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"205","DOI":"10.1023\/A:1008326307271","article-title":"Response of an oscillator moving along a parabola to an external excitation","volume":"23","author":"Maccari","year":"2000","journal-title":"Nonlinear Dyn."},{"key":"ref_5","first-page":"859","article-title":"Bifurcations and chaos of an immersed cantilever beam in a fluid and carrying an intermediate mass","volume":"253","author":"Hamdan","year":"2003","journal-title":"J. Sound Vib."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"221","DOI":"10.1115\/1.2198875","article-title":"Subharmonic-response computation and stability analysis for a nonlinear oscillator using a split-frequency harmonic balance method. J. Comput","volume":"1","author":"Dunne","year":"2006","journal-title":"Nonlinear Dyn."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"167","DOI":"10.1007\/s11071-006-9066-4","article-title":"Steady-State dynamics of a linear structure weakly coupled to an essentially nonlinear oscillator","volume":"47","author":"Malatkar","year":"2007","journal-title":"Nonlinear Dyn."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"918","DOI":"10.1016\/j.jsv.2008.03.041","article-title":"A novel nonlinear approach to suppress resonant vibrations","volume":"317","author":"Zhang","year":"2008","journal-title":"J. Sound Vib."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"707","DOI":"10.1016\/j.jsv.2008.11.034","article-title":"On the force transmissibility of a vibration isolator with quasi-zero-stiffness","volume":"322","author":"Carrella","year":"2009","journal-title":"J. Sound Vib."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"591","DOI":"10.1016\/j.physd.2010.01.006","article-title":"On the interaction of the responses at the resonance frequencies of a nonlinear two degrees-of-freedom system","volume":"239","author":"Gatti","year":"2010","journal-title":"Phys. D Nonlinear Phenom."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"395","DOI":"10.1007\/s11071-010-9811-6","article-title":"Analysis of the cutting tool on a lathe","volume":"63","author":"Nayfeh","year":"2011","journal-title":"Nonlinear Dyn."},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Luo, A.C.J., and O\u2019Connor, D. (2012, January 6\u201311). Stable and unstable periodic solutions to the Mathieu-Duffing oscillator. Proceedings of the IEEE 4th International Conference on Nonlinear Science and Complexity, NSC, Budapest, Hungary.","DOI":"10.1109\/NSC.2012.6304754"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"99","DOI":"10.1016\/j.ijmecsci.2013.02.009","article-title":"Experimental investigation of a vibration isolation system using negative stiffness structure","volume":"70","author":"Le","year":"2013","journal-title":"Int. J. Mech. Sci."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"2044","DOI":"10.1016\/j.jsv.2009.12.020","article-title":"Suppression of the primary resonance vibrations of a forced nonlinear system using a dynamic vibration absorber","volume":"329","author":"Ji","year":"2010","journal-title":"J. Sound Vib."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"411","DOI":"10.1016\/j.mechrescom.2011.05.014","article-title":"Suppression of super-harmonic resonance response using a linear vibration absorber","volume":"38","author":"Ji","year":"2011","journal-title":"Mech. Res. Commun."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"2161","DOI":"10.1007\/s11071-014-1588-6","article-title":"Secondary resonances of a quadratic nonlinear oscillator following two-to-one resonant Hopf bifurcations","volume":"78","author":"Ji","year":"2014","journal-title":"Nonlinear Dyn."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"299","DOI":"10.1016\/j.jsv.2015.03.049","article-title":"Two families of super-harmonic resonances in a time-delayed nonlinear oscillator","volume":"349","author":"Ji","year":"2015","journal-title":"J. Sound Vib."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"605","DOI":"10.1007\/s11071-014-1323-3","article-title":"Active vibration control of a nonlinear magnetic levitation system via Nonlinear Saturation Controller (NSC)","volume":"77","author":"Kamel","year":"2014","journal-title":"Nonlinear Dyn."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"23","DOI":"10.1016\/j.ijnonlinmec.2015.02.012","article-title":"Vibration suppression of a nonlinear magnetic levitation system via time delayed nonlinear saturation controller","volume":"72","author":"Eissa","year":"2015","journal-title":"Int. J. Non. Linear. Mech."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"61","DOI":"10.1016\/j.jsv.2014.11.038","article-title":"The isolation characteristics of an archetypal dynamical model with stable-quasi-zero-stiffness","volume":"340","author":"Hao","year":"2015","journal-title":"J. Sound Vib."},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"EL-Sayed, A.T., and Bauomy, H.S. (2015). Vibration suppression of subharmonic resonance response using a nonlinear vibration absorber. J. Vib. Acoust. Trans. ASME, 137.","DOI":"10.1115\/1.4029268"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"515","DOI":"10.1007\/s40430-015-0352-2","article-title":"Analysis on jump and bifurcation phenomena in the forced vibration of nonlinear cantilever beam using HBM","volume":"38","author":"Motallebi","year":"2016","journal-title":"J. Brazilian Soc. Mech. Sci. Eng."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"468","DOI":"10.1016\/j.cnsns.2015.11.026","article-title":"Steady state, oscillations and chaotic behavior of a gas inside a cylinder with a mobile piston controlled by PI and nonlinear control","volume":"36","year":"2016","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"82","DOI":"10.1016\/j.ijnonlinmec.2016.04.014","article-title":"Nonlinear dynamic analysis of a quarter vehicle system with external periodic excitation","volume":"84","author":"Zhou","year":"2016","journal-title":"Int. J. Non. Linear. Mech."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"1727","DOI":"10.1007\/s11071-017-3761-1","article-title":"Improvement of positive position feedback controller for suppressing compressor blade oscillations","volume":"90","author":"Kandil","year":"2017","journal-title":"Nonlinear Dyn."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"1924","DOI":"10.1007\/s42452-020-03614-0","article-title":"Study of Hopf curves in the time delayed active control of a 2DOF nonlinear dynamical system","volume":"2","author":"Kandil","year":"2020","journal-title":"SN Appl. Sci."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"14","DOI":"10.1016\/j.cnsns.2016.06.029","article-title":"Effects of asymmetrical damping on a 2 DOF quarter-car model under harmonic excitation","volume":"43","author":"Silveira","year":"2017","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"1488","DOI":"10.1177\/1077546316661680","article-title":"Suppressing the nonlinear vibrations of a compressor blade via a nonlinear saturation controller","volume":"24","author":"Kandil","year":"2018","journal-title":"JVC\/J. Vib. Control."},{"key":"ref_29","doi-asserted-by":"crossref","unstructured":"Hamed, Y.S., and Kandil, A. (2021). Influence of time delay on controlling the non-linear oscillations of a rotating blade. Symmetry, 13.","DOI":"10.3390\/sym13010085"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"75","DOI":"10.1007\/s11071-019-05172-6","article-title":"Dynamic and steady analysis of a 2-DOF vehicle system by modified incremental harmonic balance method","volume":"98","author":"Zhou","year":"2019","journal-title":"Nonlinear Dyn."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"73857","DOI":"10.1109\/ACCESS.2021.3080457","article-title":"Tuned Positive Position Feedback Control of an Active Magnetic Bearings System with 16-Poles and Constant Stiffness","volume":"9","author":"Kandil","year":"2021","journal-title":"IEEE Access"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"133694","DOI":"10.1109\/ACCESS.2021.3114356","article-title":"Rotor Active Magnetic Bearings System Control via a Tuned Nonlinear Saturation Oscillator","volume":"9","author":"Kandil","year":"2021","journal-title":"IEEE Access"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"1755","DOI":"10.1007\/s11071-021-06790-9","article-title":"Highly sensitive measurements of perturbations in stiffness of a resonator by virtual coupling with a virtual resonator","volume":"107","author":"Uemori","year":"2021","journal-title":"Nonlinear Dyn."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"153012","DOI":"10.1109\/ACCESS.2021.3126868","article-title":"2D and 3D visualizations of the mass-damper-spring model dynamics controlled by a servo-controlled linear actuator","volume":"9","author":"Kandil","year":"2021","journal-title":"IEEE Access"},{"key":"ref_35","doi-asserted-by":"crossref","unstructured":"Nayfeh, A.H., and Mook, D.T. (1995). Nonlinear Oscillations, Wiley.","DOI":"10.1002\/9783527617586"},{"key":"ref_36","doi-asserted-by":"crossref","unstructured":"Krack, M., and Gross, J. (2019). Harmonic Balance for Nonlinear Vibration Problems, Springer Nature.","DOI":"10.1007\/978-3-030-14023-6"},{"key":"ref_37","doi-asserted-by":"crossref","unstructured":"Nayfeh, A.H., and Balachandran, B. (1995). Applied Nonlinear Dynamics, Wiley.","DOI":"10.1002\/9783527617548"}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/6\/1247\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T23:32:43Z","timestamp":1760139163000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/6\/1247"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,6,16]]},"references-count":37,"journal-issue":{"issue":"6","published-online":{"date-parts":[[2022,6]]}},"alternative-id":["sym14061247"],"URL":"https:\/\/doi.org\/10.3390\/sym14061247","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,6,16]]}}}