A powerful and simple frequency formula to nonlinear fractal oscillators

Wang, Kang‐Le; Wei, Chengzhou

doi:10.1177/1461348420947832

Cited by 35 publications

(6 citation statements)

References 25 publications

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“…The spring-pendulum systems have many applications in satellites, submarines, aircraft, and energy harvesting device [14]. The recent research frontier in the nonlinear vibration theory is the fractal vibration; hence the description of a vibrating system in a fractal space [15][16][17][18].…”

Section: Introductionmentioning

confidence: 99%

Simplified Hamiltonian-Based Frequency-Amplitude Formulation for Nonlinear Vibration Systems

2022

FU Mech Eng

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The Hamiltonian-based frequency formulation has been hailed as an unprecedented success for it gives a straightforward insight into a complex nonlinear vibration system with simple calculation. This paper gives a systematical analysis of the formulation, and two simplified formulations are suggested. The cubic-quintic Duffing oscillator is used as an example to show extremely simple calculation and remarkable accuracy. It can be used as a paradigm for many other applications, and the one-step solving process has cleaned up the road of the nonlinear vibration theory.

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Section: Introductionmentioning

confidence: 99%

Simplified Hamiltonian-Based Frequency-Amplitude Formulation for Nonlinear Vibration Systems

2022

FU Mech Eng

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“…He's frequency formulation could be extended to fractal oscillators [34] as well as nonconservative oscillators [35][36][37]. In engineering, a simple calculation is always appreciated; the simpler the calculation, the better.…”

Section: Discussionmentioning

confidence: 99%

A Simple Frequency Formulation for the Tangent Oscillator

Yang

et al. 2021

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The frequency of a nonlinear vibration system is nonlinearly related to its amplitude, and this relationship is critical in the design of a packaging system and a microelectromechanical system (MEMS). This paper proposes a straightforward frequency prediction method for nonlinear oscillators with arbitrary initial conditions. The tangent oscillator, the hyperbolic tangent oscillator, a singular oscillator, and a MEMS oscillator are chosen to elucidate the simple solving process. The results, when compared with those obtained by the homotopy perturbation method, exhibit a good agreement. This paper introduces a very convenient procedure for attaining quick and accurate insight into the vibration property of a nonlinear vibration system.

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“…Moreover, Wang [55] used He's frequency formula to solve another nonlinear fractal oscillator. In [56,57], the same author also proposed a clear frequency formulation to investigate the nonlinear oscillators with fractal space. In addition, Tian [58] introduced another frequency formulation to deal with a system of fractal oscillators.…”

Section: Introductionmentioning

confidence: 99%

An innovative technique to solve a fractal damping Duffing-jerk oscillator

El‐Dib

Elgazery

Khattab

et al. 2023

Commun. Theor. Phys.

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The idea of the present article is to look into the nonlinear dynamics and vibration of a damping Duffing-jerk oscillator in fractal space exhibiting the non-perturbative approach. Using a new analytical technique, namely, the modification of He's a fractal derivative that converts the fractal derivative to the traditional derivative in continuous space, this study provides an effective and easy-to-apply procedure that is dependent on He's fractal derivative approach. The analytic approximate solution has significant match with the results of the numerical simulation as the fractal parameter is very closer to unity, which proves the reliability of the method. Stability behavior is discussed and illustrated graphically. These new powerful analytical tools are developed in an attempt to obtain effective analytical tools, valid for any fractal nonlinear problems.

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A powerful and simple frequency formula to nonlinear fractal oscillators

Cited by 35 publications

References 25 publications

Simplified Hamiltonian-Based Frequency-Amplitude Formulation for Nonlinear Vibration Systems

Simplified Hamiltonian-Based Frequency-Amplitude Formulation for Nonlinear Vibration Systems

A Simple Frequency Formulation for the Tangent Oscillator

An innovative technique to solve a fractal damping Duffing-jerk oscillator

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