FORCED NONLINEAR OSCILLATOR IN A FRACTAL SPACE

IF 10.1 2区工程技术 Q1 ENGINEERING, MECHANICAL

Facta Universitatis-Series Mechanical Engineering Pub Date : 2022-04-08 DOI:10.22190/fume220118004h

Ji-Huan He, G. Moatimid, M. Zekry

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引用次数: 46

Abstract

A critical hurdle of a nonlinear vibration system in a fractal space is the inefficiency in modelling the system. Specifically, the differential equation models cannot elucidate the effect of porosity size and distribution of the periodic property. This paper establishes a fractal-differential model for this purpose, and a fractal Duffing-Van der Pol oscillator (DVdP) with two-scale fractal derivatives and a forced term is considered as an example to reveal the basic properties of the fractal oscillator. Utilizing the two-scale transforms and He-Laplace method, an analytic approximate solution may be attained. Unfortunately, this solution is not physically preferred. It has to be modified along with the nonlinear frequency analysis, and the stability criterion for the equation under consideration is obtained. On the other hand, the linearized stability theory is employed in the autonomous arrangement. Consequently, the phase portraits around the equilibrium points are sketched. For the non-autonomous organization, the stability criteria are analyzed via the multiple time scales technique. Numerical estimations are designed to confirm graphically the analytical approximate solutions as well as the stability configuration. It is revealed that the exciting external force parameter plays a destabilizing role. Furthermore, both of the frequency of the excited force and the stiffness parameter, execute a dual role in the stability picture.

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分形空间中的强迫非线性振子

分形空间中非线性振动系统的一个关键障碍是系统建模的低效率。具体来说，微分方程模型不能解释孔隙度大小和周期性分布的影响。本文为此建立了分形-微分模型，并以具有两尺度分形导数和强制项的分形Duffing-Van der Pol振子(DVdP)为例，揭示了分形振子的基本性质。利用双尺度变换和He-Laplace方法，可以得到解析近似解。不幸的是，这种解决方案在物理上不是首选的。在进行非线性频率分析的同时对其进行修正，得到了所考虑方程的稳定性判据。另一方面，将线性化稳定性理论应用于自主布置。因此，画出了平衡点周围的相图。对于非自治组织，采用多时间尺度技术分析了稳定性准则。设计了数值估计，以图形化地确定解析近似解和稳定性构型。结果表明，激振外力参数起着不稳定的作用。此外，激振力的频率和刚度参数在稳定性图中起双重作用。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Facta Universitatis-Series Mechanical Engineering ENGINEERING, MECHANICAL-

CiteScore

14.40

自引率

2.50%

发文量

审稿时长

6 weeks

期刊介绍： Facta Universitatis, Series: Mechanical Engineering (FU Mech Eng) is an open-access, peer-reviewed international journal published by the University of Niš in the Republic of Serbia. It publishes high-quality, refereed papers three times a year, encompassing original theoretical and/or practice-oriented research as well as extended versions of previously published conference papers. The journal's scope covers the entire spectrum of Mechanical Engineering. Papers undergo rigorous peer review to ensure originality, relevance, and readability, maintaining high publication standards while offering a timely, comprehensive, and balanced review process.