An efficient approach to converting the damping fractal models to the traditional system

Journal of Low Frequency Noise, Vibration and Active Control

Elgazery

Alyousef

2023

Self Cite

The primary goal of the present study is to investigate how to obtain a periodic solution for a fractal Mathieu–Duffing oscillator. To achieve this, the fractal oscillator in the fractal space has been transformed into a damping Mathieu–Duffing equation in the continuous space by employing a new modification of He’s definition of the fractal derivative. The required analytical periodic solution has been based on the rank upgrade technique (RUT) presented. The RUT successfully generates a periodic solution without sacrificing the damping coefficient by creating an alternate equation, aside from any challenges in managing the impact of the linear damping component. The homotopy perturbation method (HPM) has been used to find the required periodic solution for the alternate equation. A comparison of the numerical solutions of the original equation and the alternative equation showed good agreement. The stability behavior in the non-resonance case as well as in the sub-harmonic resonance case has also been discussed. Further, another method, “the non-perturbative approach”, that deals with the obtained equation has been introduced.

Section: Discussionmentioning

confidence: 99%

Section: Parametric Excitation In Fractal Spacementioning

confidence: 99%

An instrumental insight for a periodic solution of a fractal Mathieu–Duffing equation

Journal of Low Frequency Noise, Vibration and Active Control

Elgazery

Alyousef

2023

Self Cite

“…Also, this approach was applied to the parametric Gaylord's oscillator with a discussion of the resonance response with the non-perturbative approach. 24 Recently, a new perspective on He's formula has discussed a delayed dynamical system by El-Dib et al 25 Further, application to He's frequency formula for a class of fractal vibration systems has been proposed by Tian 26 and El-Dib et al 27,28 The fractal Toda oscillator has been established by the non-perturbative method and He's frequency formula has been applied to determine the approximate analytical solution. 8,18 Recently, El-Dib 18 has established an extended frequency-amplitude in the differentiative form to cover the family of the Duffing oscillator with nonlinearity having the higher powers…”

Section: Introductionmentioning

confidence: 99%

Successive approximate solutions for nonlinear oscillation and improvement of the solution accuracy with efficient non-perturbative technique

Journal of Low Frequency Noise, Vibration and Active Control

Alyousef

2023

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In the present study, several successive approximate solutions of the nonlinear oscillator are derived by using the efficient frequency formula. A systematical analysis of the formulation of the nonlinear frequency helps to establish a general periodic solution. Each approximation represents, individually, the solution of the nonlinear oscillator. For the optimal design and accurate prediction of structural behavior, a new optimizer is demonstrated for efficient solutions. The classical Duffing frequency formula has been modified. The numerical calculations show high agreement with the exact frequency. The justifiability of the obtained solutions is confirmed by comparison with the numerical solution. It is shown that the enhanced solution is accurate for large amplitudes and is not restricted to oscillations that have small amplitudes. The new approach can provide a perfect approximation for the nonlinear oscillation.

“…El-Dib et al [65] also provided an analytical solution for the Klein-Gordon model with time fraction derivatives. For more recent and significant contributions, see [28,[66][67][68].…”

Section: Introductionmentioning

confidence: 99%

“…Consequently, fractal oscillation behaves differently than traditional behavior since there are important applications for the Duffing-Jerk equation in our everyday activities as mentioned in the literature. Recently, El-Dib and Elgazery [66,67] have suggested an efficient novel approach to convert the fractal model into a traditional one. Hence, the current study aims to extend El-Dib and Elgazery's studies to introduce a novel technique to solve a fractal damping Duffing-jerk oscillator.…”

Section: Introductionmentioning

confidence: 99%

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

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.