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

IF 2.8 3区 工程技术 Q2 MECHANICS International Journal of Non-Linear Mechanics Pub Date : 2024-07-14 DOI:10.1016/j.ijnonlinmec.2024.104832
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引用次数: 0

摘要

本文提出了一种改进的广义谐函数扰动法。与该方法的经典版本相比,改进版本可以纯符号方式执行其程序,即使对于一些复杂的非线性振荡器,也无需指定任何系统参数。这意味着极限周期振幅与系统参数之间的关系可以通过所提出的方法分析得出。同时,还可以得到极限周期特征量的解析表达式。通过这些分析表达式,可以在振幅域定量研究极限循环的演变过程。它展示了每个极限周期从产生到分叉再到终点的整个生命周期。为了证明所提方法的可行性,本文研究了一个名为广义达芬-谐波-雷利-李纳振荡器的复杂振荡器。首先,推导了上述两个分析表达式,并定量分析了其极限循环的全局演化。其次,还通过这两个分析表达式预测了同轴和异轴分岔参数的临界值。此外,还计算了极限循环和同-异轨道的解析近似解。为了证明上述方法的准确性,我们用 Runge-Kutta 方法对上述方法得出的所有结果进行了验证,结果表明它们非常吻合。因此,所提出的方法可以看作是对经典扰动方法的有效修正。它为分析强非线性振荡器的全局动力学提供了另一种可行而可靠的定量分析方法。
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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.

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来源期刊
CiteScore
5.50
自引率
9.40%
发文量
192
审稿时长
67 days
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.
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