{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T03:38:54Z","timestamp":1760240334967,"version":"build-2065373602"},"reference-count":47,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2019,5,16]],"date-time":"2019-05-16T00:00:00Z","timestamp":1557964800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["11772218"],"award-info":[{"award-number":["11772218"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Tianjin Research Program of Application Foundation and Advanced Technology","award":["17JCYBJC18900"],"award-info":[{"award-number":["17JCYBJC18900"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>In this article, we present a new accurate iterative and asymptotic method to construct analytical periodic solutions for a strongly nonlinear system, even if it is not Z2-symmetric. This method is applicable not only to a conservative system but also to a non-conservative system with a limit cycle response. Distinct from the general harmonic balance method, it depends on balancing a few trigonometric terms (at most five terms) in the energy equation of the nonlinear system. According to this iterative approach, the dynamic frequency is a trigonometric function that varies with time t, which represents the influence of derivatives of the higher harmonic terms in a compact form and leads to a significant reduction of calculation workload. Two examples were solved and numerical solutions are presented to illustrate the effectiveness and convenience of the method. Based on the present method, we also outline a modified energy balance method to further simplify the procedure of higher order computation. Finally, a nonlinear strength index is introduced to automatically identify the strength of nonlinearity and classify the suitable strategies.<\/jats:p>","DOI":"10.3390\/sym11050676","type":"journal-article","created":{"date-parts":[[2019,5,16]],"date-time":"2019-05-16T11:21:22Z","timestamp":1558005682000},"page":"676","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["Periodic Solution of the Strongly Nonlinear Asymmetry System with the Dynamic Frequency Method"],"prefix":"10.3390","volume":"11","author":[{"given":"Zhiwei","family":"Zhang","sequence":"first","affiliation":[{"name":"Tianjin Key Laboratory of Nonlinear Dynamics and Chaos Control, Tianjin University, Tianjin 300350, China"}]},{"given":"Yingjie","family":"Wang","sequence":"additional","affiliation":[{"name":"Tianjin Key Laboratory of Nonlinear Dynamics and Chaos Control, Tianjin University, Tianjin 300350, China"}]},{"given":"Wei","family":"Wang","sequence":"additional","affiliation":[{"name":"Tianjin Key Laboratory of Nonlinear Dynamics and Chaos Control, Tianjin University, Tianjin 300350, China"},{"name":"School of Computing and Engineering, Huddersfield University, Huddersfield HD1 3DH, UK"}]},{"given":"Ruilan","family":"Tian","sequence":"additional","affiliation":[{"name":"Department of Mechanics Engineering, Shijiazhuang Tiedao University, Shijiazhuang 050043, China"}]}],"member":"1968","published-online":{"date-parts":[[2019,5,16]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"459","DOI":"10.18514\/MMN.2012.562","article-title":"Periodic successive approximations and interval halving","volume":"13","year":"2012","journal-title":"Miskolc Math. Notes"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"689","DOI":"10.1016\/j.amc.2014.11.021","article-title":"A new approach to non-local boundary value problems for ordinary differential systems","volume":"250","author":"Jana","year":"2015","journal-title":"Appl. Math. Comput."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"3179","DOI":"10.1016\/S0362-546X(96)00355-0","article-title":"Numerical-analytic successive approximation method for nonlinear boundary value problems","volume":"30","year":"1997","journal-title":"Nonlinear Anal. Theor."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"121","DOI":"10.1007\/s10998-008-5121-3","article-title":"On two numerical-analytic methods for the investigation of periodic solutions","volume":"56","year":"2008","journal-title":"Period. Math. Hung."},{"key":"ref_5","unstructured":"Mitropolskii, Y.A., and Moseekov, B.I. (1976). Asymptotic Solutions of Partial Differential Equations, Vyshcha Shkola."},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Mickens, R.E. (1996). Oscillations in Planar Dynamic Systems, World Scientific.","DOI":"10.1142\/2778"},{"key":"ref_7","unstructured":"Bogolubov, N.N., and Mitropolski, Y.A. (1961). Asymptotic Methods in the Theory of Nonlinear Oscillations, CRC Press."},{"key":"ref_8","unstructured":"Krylov, N.M., and Bogoliubov, N.N. (1947). Introduction to Nonlinear Mechanics, Princeton University Press."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"211","DOI":"10.1016\/S0378-4371(98)00041-7","article-title":"Approximate solutions of a class of complex nonlinear dynamical systems","volume":"253","author":"Mahmoud","year":"1998","journal-title":"Phys. A"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"67","DOI":"10.1007\/BF02750660","article-title":"A new method of expansion in mathematical physics","volume":"36","author":"Sandri","year":"1965","journal-title":"Il Nuovo Cimento"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"211","DOI":"10.1002\/asna.18821031404","article-title":"Ueder die integration einer fur die storungstheorie wichtigen differential gleichung","volume":"103","author":"Lindstedt","year":"1882","journal-title":"Astron. Nach."},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Nayfeh, A.H., and Mook, D.T. (1995). Nonlinear Oscillations, John Wiley & Sons.","DOI":"10.1002\/9783527617586"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"1447","DOI":"10.1016\/j.jsv.2007.10.010","article-title":"Solution of the relativistic (an) harmonic oscillator using the harmonic balance method","volume":"311","author":"Belendez","year":"2007","journal-title":"J. Sound Vib."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"591","DOI":"10.1016\/S0022-460X(87)81390-1","article-title":"A generalization of the method of harmonic balance","volume":"116","author":"Bejarano","year":"1987","journal-title":"J. Sound Vib."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"535","DOI":"10.1016\/0022-0396(72)90024-1","article-title":"On the approximation of nonlinear oscillations","volume":"12","author":"Stokes","year":"1972","journal-title":"J. Differ. Equ."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"67","DOI":"10.1016\/j.jde.2012.09.011","article-title":"A theoretical basis for the harmonic balance method","volume":"254","author":"Gasull","year":"2013","journal-title":"J. Differ. Equ."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"1343","DOI":"10.1016\/j.ijnonlinmec.2003.08.008","article-title":"A modified and compact form of Krylov-Bogoliubov-Mitropolskii unified method for solving an nth order nonlinear differential equation","volume":"39","author":"Alam","year":"2004","journal-title":"Int. J. Nonlinear Mech."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"141","DOI":"10.1115\/1.2447304","article-title":"A modified lindstedt-poincare method for strongly mixed-parity nonlinear oscillators","volume":"2","author":"Sun","year":"2007","journal-title":"Trans. ASME. J. Comput. Nonlinear Dyn."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"243","DOI":"10.1023\/A:1015620928121","article-title":"A perturbation-incremental method for strongly nonlinear autonomous oscillators with many degrees of freedom","volume":"28","author":"Chung","year":"2002","journal-title":"Nonlinear Dyn."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"854","DOI":"10.1016\/j.amc.2004.09.066","article-title":"An analytic approach to solve multiple solutions of a strongly nonlinear problem","volume":"169","author":"Li","year":"2005","journal-title":"Appl. Math. Comput."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"751","DOI":"10.1006\/jsvi.1996.0313","article-title":"A modified Lindstedt-Poincare method for a strongly non-linear two degree-of-freedom system","volume":"193","author":"Chen","year":"1996","journal-title":"J. Sound Vib."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1006\/jsvi.1993.1193","article-title":"A perturbation technique that works even when the nonlinearity is not small","volume":"164","author":"Senator","year":"1993","journal-title":"J. Sound Vib."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"1115","DOI":"10.1016\/j.jsv.2004.06.009","article-title":"Improved Lindstedt-Poincare method for the solution of the nonlinear problems","volume":"283","author":"Amore","year":"2005","journal-title":"J. Sound Vib."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"2251","DOI":"10.1016\/S0960-0779(99)00144-7","article-title":"Prediction of homoclinic bifurcation: The elliptic averaging method","volume":"11","author":"Belhaq","year":"2000","journal-title":"Chaos Solitons Fractals."},{"key":"ref_25","first-page":"130","article-title":"Asymptotic solutions and bifurcation analysis of the strongly nonlinear oscillation system with two degrees of freedom","volume":"27","author":"Wang","year":"2008","journal-title":"J. Vib. Shock"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"107","DOI":"10.1016\/S0093-6413(02)00237-9","article-title":"Preliminary report on the energy balance for nonlinear oscillations","volume":"29","author":"He","year":"2002","journal-title":"Mech. Res. Commun."},{"key":"ref_27","first-page":"3423","article-title":"Application of improved amplitude frequency formulation to nonlinear differential equation of motion equations","volume":"23","author":"Davod","year":"2009","journal-title":"Mod. Phys. Lett. B"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"889","DOI":"10.1007\/s00521-014-1576-2","article-title":"Improved accuracy of He\u2019s energy balance method for analysis of conservative nonlinear oscillator","volume":"25","author":"Khan","year":"2014","journal-title":"Neural Comput. Appl."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"3222","DOI":"10.1016\/j.camwa.2010.03.013","article-title":"Frequency analysis of strongly nonlinear generalized Duffing oscillators using He\u2019s frequency-amplitude formulation and He\u2019s energy balance method","volume":"59","author":"Younesian","year":"2010","journal-title":"Comput. Math. Appl."},{"key":"ref_30","first-page":"620591","article-title":"Energy method to obtain approximate solutions of strongly nonlinear oscillators","volume":"2012","year":"2013","journal-title":"Math. Probl. Eng."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"417","DOI":"10.1007\/s11071-009-9489-9","article-title":"Homoclinic and heteroclinic solutions of cubic strongly nonlinear autonomous oscillators by the hyperbolic perturbation method","volume":"58","author":"Chen","year":"2009","journal-title":"Nonlinear Dyn."},{"key":"ref_32","unstructured":"Stoker, J.J. (1950). Nonlinear Vibrations in Mechanical and Electrical Systems, Interscience Publishers."},{"key":"ref_33","first-page":"185","article-title":"Iteration procedure for determining approximate solutions to nonlinear oscillator equations","volume":"116","author":"Mickens","year":"1987","journal-title":"J. Vib. Shock"},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"69","DOI":"10.1007\/s00707-004-0112-3","article-title":"An iteration approach to nonlinear oscillations of conservative single-degree-of-freedom systems","volume":"170","author":"Li","year":"2004","journal-title":"Acta Mech."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"33","DOI":"10.1016\/j.bulsci.2014.08.002","article-title":"The period function and the harmonic balance method","volume":"139","author":"Gasull","year":"2015","journal-title":"Bull. Sci. Math."},{"key":"ref_36","unstructured":"Mickens, R.E. (1996). Turly Nonlinear Oscillations, World Scientific."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"959","DOI":"10.1115\/1.3157762","article-title":"Amplitude incremental variational principle for nonlinear vibration of elastic system","volume":"48","author":"Lau","year":"1981","journal-title":"ASME J. Appl. Mech."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"167","DOI":"10.1023\/A:1013067311749","article-title":"A method for obtaining approximate analytic periods for a class of nonlinear oscillators","volume":"36","author":"Wu","year":"2001","journal-title":"Meccanica"},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"951","DOI":"10.1115\/1.1406960","article-title":"A new approach to nonlinear oscillations","volume":"68","author":"Wu","year":"2001","journal-title":"ASME J. Appl. Mech."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"1051","DOI":"10.1007\/s11071-012-0326-1","article-title":"On the harmonic balance with linearization for asymmetric single degree of freedom non-linear oscillators","volume":"69","year":"2012","journal-title":"Nonlinear Dyn."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"223","DOI":"10.1007\/BF01833743","article-title":"Energy method for computing periodic solutions of strongly nonlinear systems-autonomous systems","volume":"9","author":"Li","year":"1996","journal-title":"Nonlinear Dyn."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"3417","DOI":"10.1016\/j.physleta.2017.08.049","article-title":"On certain properties of nonlinear oscillator with coordinate-dependent mass","volume":"381","author":"Lev","year":"2017","journal-title":"Phys. Lett. A."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"270","DOI":"10.1016\/j.rinp.2018.06.015","article-title":"Homotopy perturbation method for nonlinear oscillators with coordinatedependent mass","volume":"10","author":"Wu","year":"2018","journal-title":"Results Phys."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"2014","DOI":"10.1080\/00207160802562564","article-title":"Approximate solutions to Van der Pol damped nonlinear oscillators by means of He\u2019s energy balance method","volume":"9","author":"Ganji","year":"2010","journal-title":"Int. J. Comput. Math."},{"key":"ref_45","first-page":"29","article-title":"An analytical technique for solving nonlinear oscillators of the motion of a rigid rod rocking bock and tapered beams","volume":"2","author":"Ismail","year":"2016","journal-title":"J. Appl. Comput. Mech."},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"3484","DOI":"10.1016\/j.jsv.2010.03.005","article-title":"Galerkin methods for natural frequencies of high-speed axially moving beams","volume":"329","author":"Ding","year":"2010","journal-title":"J. Sound Vib."},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"2426","DOI":"10.1016\/j.jsv.2011.12.036","article-title":"Convergence of galerkin truncation for dynamic response of nite beams on nonlinear foundations under a moving load","volume":"331","author":"Ding","year":"2012","journal-title":"J. Sound Vib."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/11\/5\/676\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T12:52:35Z","timestamp":1760187155000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/11\/5\/676"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,5,16]]},"references-count":47,"journal-issue":{"issue":"5","published-online":{"date-parts":[[2019,5]]}},"alternative-id":["sym11050676"],"URL":"https:\/\/doi.org\/10.3390\/sym11050676","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2019,5,16]]}}}