{"title":"修正的广义谐函数扰动法及其在分析广义达芬-谐波-雷利-李纳振荡器中的应用","authors":"","doi":"10.1016/j.ijnonlinmec.2024.104832","DOIUrl":null,"url":null,"abstract":"<div><p>A modified generalized harmonic function perturbation method is proposed in this paper. Compared with the classical version of this method, the modified version can execute its procedures pure symbolically without the need to assign any system parameters even for some complicated nonlinear oscillators. This means that the relations between amplitude of limit cycles and system parameters can be derived analytically from the proposed method. Meanwhile, the analytical expression of characteristic quantity of limit cycles can be also obtained. Via these analytical expressions, the evolutional process of limit cycles can be studied quantitatively in amplitude domain. It demonstrates the entire live period of each limit cycle from its generation to bifurcation to destination. To show the feasibility of the proposed method, a complicated oscillator named generalized Duffing–Harmonic–Rayleigh–Liénard oscillator is investigated in this paper. First, the two analytical expressions mentioned above are derived and the global evolution of its limit cycles are analyzed quantitatively. Second, the critical value of homoclinic and heteroclinic bifurcation parameters are also predicted via this two analytical expressions. Moreover, the analytical approximate solutions of both limit cycles and homo-heteroclinic orbits are calculated. To prove the accuracy, all the above results obtained via the proposed methods are confirmed by the Runge–Kutta method, which show a good accordance. Therefore, the proposed method can be considered as an effective modification for a classical perturbation method. It provides another feasible and reliable analytical quantitative method for analyzing global dynamics of strongly nonlinear oscillators.</p></div>","PeriodicalId":50303,"journal":{"name":"International Journal of Non-Linear Mechanics","volume":null,"pages":null},"PeriodicalIF":2.8000,"publicationDate":"2024-07-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"A modified generalized harmonic function perturbation method and its application in analyzing generalized Duffing–Harmonic–Rayleigh–Liénard oscillator\",\"authors\":\"\",\"doi\":\"10.1016/j.ijnonlinmec.2024.104832\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>A modified generalized harmonic function perturbation method is proposed in this paper. Compared with the classical version of this method, the modified version can execute its procedures pure symbolically without the need to assign any system parameters even for some complicated nonlinear oscillators. This means that the relations between amplitude of limit cycles and system parameters can be derived analytically from the proposed method. Meanwhile, the analytical expression of characteristic quantity of limit cycles can be also obtained. Via these analytical expressions, the evolutional process of limit cycles can be studied quantitatively in amplitude domain. It demonstrates the entire live period of each limit cycle from its generation to bifurcation to destination. To show the feasibility of the proposed method, a complicated oscillator named generalized Duffing–Harmonic–Rayleigh–Liénard oscillator is investigated in this paper. First, the two analytical expressions mentioned above are derived and the global evolution of its limit cycles are analyzed quantitatively. Second, the critical value of homoclinic and heteroclinic bifurcation parameters are also predicted via this two analytical expressions. Moreover, the analytical approximate solutions of both limit cycles and homo-heteroclinic orbits are calculated. To prove the accuracy, all the above results obtained via the proposed methods are confirmed by the Runge–Kutta method, which show a good accordance. Therefore, the proposed method can be considered as an effective modification for a classical perturbation method. It provides another feasible and reliable analytical quantitative method for analyzing global dynamics of strongly nonlinear oscillators.</p></div>\",\"PeriodicalId\":50303,\"journal\":{\"name\":\"International Journal of Non-Linear Mechanics\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":2.8000,\"publicationDate\":\"2024-07-14\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"International Journal of Non-Linear Mechanics\",\"FirstCategoryId\":\"5\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S0020746224001975\",\"RegionNum\":3,\"RegionCategory\":\"工程技术\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"MECHANICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"International Journal of Non-Linear Mechanics","FirstCategoryId":"5","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0020746224001975","RegionNum":3,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MECHANICS","Score":null,"Total":0}
A modified generalized harmonic function perturbation method and its application in analyzing generalized Duffing–Harmonic–Rayleigh–Liénard oscillator
A modified generalized harmonic function perturbation method is proposed in this paper. Compared with the classical version of this method, the modified version can execute its procedures pure symbolically without the need to assign any system parameters even for some complicated nonlinear oscillators. This means that the relations between amplitude of limit cycles and system parameters can be derived analytically from the proposed method. Meanwhile, the analytical expression of characteristic quantity of limit cycles can be also obtained. Via these analytical expressions, the evolutional process of limit cycles can be studied quantitatively in amplitude domain. It demonstrates the entire live period of each limit cycle from its generation to bifurcation to destination. To show the feasibility of the proposed method, a complicated oscillator named generalized Duffing–Harmonic–Rayleigh–Liénard oscillator is investigated in this paper. First, the two analytical expressions mentioned above are derived and the global evolution of its limit cycles are analyzed quantitatively. Second, the critical value of homoclinic and heteroclinic bifurcation parameters are also predicted via this two analytical expressions. Moreover, the analytical approximate solutions of both limit cycles and homo-heteroclinic orbits are calculated. To prove the accuracy, all the above results obtained via the proposed methods are confirmed by the Runge–Kutta method, which show a good accordance. Therefore, the proposed method can be considered as an effective modification for a classical perturbation method. It provides another feasible and reliable analytical quantitative method for analyzing global dynamics of strongly nonlinear oscillators.
期刊介绍:
The International Journal of Non-Linear Mechanics provides a specific medium for dissemination of high-quality research results in the various areas of theoretical, applied, and experimental mechanics of solids, fluids, structures, and systems where the phenomena are inherently non-linear.
The journal brings together original results in non-linear problems in elasticity, plasticity, dynamics, vibrations, wave-propagation, rheology, fluid-structure interaction systems, stability, biomechanics, micro- and nano-structures, materials, metamaterials, and in other diverse areas.
Papers may be analytical, computational or experimental in nature. Treatments of non-linear differential equations wherein solutions and properties of solutions are emphasized but physical aspects are not adequately relevant, will not be considered for possible publication. Both deterministic and stochastic approaches are fostered. Contributions pertaining to both established and emerging fields are encouraged.