The stability conditions of the cubic damping Van der Pol-Duffing oscillator using the HPM with the frequency-expansion technology

El‐Dib, Yusry O.

doi:10.17512/jamcm.2018.3.03

Cited by 5 publications

(5 citation statements)

References 22 publications

(31 reference statements)

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“…12. El-Dib 13 formulated a periodic solution for a strong time-delayed Duffing oscillator. In this investigation, a second-order analytic solution with a damping part was obtained using the multiple scales homotopy perturbation technique.…”

Section: Introductionmentioning

confidence: 99%

A novel technique to obtain a time-delayed vibration control analytical solution with simulation of He’s formula

El‐Dib

Elgazery

Gad

2023

Journal of Low Frequency Noise, Vibration and Active Control

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The impetus for the current investigation relates to the implementation of an efficient novel technique to obtain a time-delayed vibration control analytical solution. The current technique follows simple and easy-to-apply criteria. This technique is based on converting the nonlinear time-delayed Van der Pol (VDP)-Duffing oscillator to an equivalent linear one. Details of the conversion to an equivalent linear ordinary differential equation are mentioned. The convergence between the numerical outcomes and the analytic solution is achieved and gives a satisfying accuracy of the equivalent result.

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Section: Introductionmentioning

confidence: 99%

A novel technique to obtain a time-delayed vibration control analytical solution with simulation of He’s formula

El‐Dib

Elgazery

Gad

2023

Journal of Low Frequency Noise, Vibration and Active Control

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“…Satisfying conditions (66) should ensure that the roots of equation ( 64) are real. But this condition conflicts with the last condition in (65). So the frequency equation ( 64) is improper.…”

Section: Some Fallacies In the Study Of Non-conservative Issuesmentioning

confidence: 96%

“…Ex10: Consider the following generalized Van der Pol type oscillator 65 The homotopy equation isConsidering the frequency analysis so that we define the following frequency expansionAssuming that the function y(t;ρ) has been expanded asEmploying (151) and (152) with (150) and equating the identical powers of ρ to zero yieldSolution of the zero-order problem leads toSubstituting (155) into (154), the requirement of no secular term in y1(t) needsandSolution of (154) without secular terms becomesIf the first-order approximation is enough, then setting ρ→1 in the expansions (151) and (152) yields the approximate solution and the frequency, respectivelyIt is observed that the above oscillation becomes in the form of the conservative behavior whenBy combining (159) and (161), we show the periodic solution can occur only when the amplitude A satisfies the following relationEx11: In the present example, we have develo...…”

Section: The Non-conservative Oscillators Through the Modified Homoto...mentioning

confidence: 99%

A heuristic review on the homotopy perturbation method for non-conservative oscillators

El‐Dib

2021

Journal of Low Frequency Noise, Vibration and Active Control

102

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The homotopy perturbation method (HPM) was proposed by Ji-Huan. He was a rising star in analytical methods, and all traditional analytical methods had abdicated their crowns. It is straightforward and effective for many nonlinear problems; it deforms a complex problem into a linear system; however, it is still developing quickly. The method is difficult to deal with non-conservative oscillators, though many modifications have appeared. This review article features its last achievement in the nonlinear vibration theory with an emphasis on coupled damping nonlinear oscillators and includes the following categories: (1) Some fallacies in the study of non-conservative issues; (2) non-conservative Duffing oscillator with three expansions; (3)the non-conservative oscillators through the modified homotopy expansion; (4) the HPM for fractional non-conservative oscillators; (5) the homotopy perturbation method for delay non-conservative oscillators; and (6) quasi-exact solution based on He’s frequency formula. Each category is heuristically explained by examples, which can be used as paradigms for other applications. The emphasis of this article is put mainly on Ji-Huan He’s ideas and the present authors’ previous work on the HPM, so the citation might not be exhaustive.

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“…Through the examination of different design configurations and optimization methodologies made possible by this methodology, the field of MEMS is ultimately advanced, and the functionality and efficiency of MEMSs in a variety of applications are improved. Highly nonlinear problems have been solved using the homotopy perturbation method (HPM) [35][36][37][38][39][40]. This method provides the answer in the form of a series that quickly approaches the approximate answer.…”

Section: Introductionmentioning

confidence: 99%

The Homotopy Perturbation Method for Electrically Actuated Microbeams in Mems Systems Subjected to Van Der Waals Force and Multiwalled Carbon Nanotubes

Amir,

Haider,

Ashraf

2024

Acta Mechanica Et Automatica

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This paper presents a summary of a study that uses the Aboodh transformation and homotopy perturbation approach to analyze the behavior of electrically actuated microbeams in microelectromechanical systems that incorporate multiwalled carbon nanotubes and are subjected to the van der Waals force. All of the equations were transformed into linear form using the HPM approach. Electrically operated microbeams, a popular structure in MEMS, are the subject of this work. Because of their interaction with a nearby surface, these microbeams are sensitive to a variety of forces, such as the van der Waals force and body forces. MWCNTs are also incorporated into the MEMSs in this study because of their special mechanical, thermal, and electrical characteristics. The suggested method uses the HPM to model how electrically activated microbeams behave when MWCNTs and the van der Waals force are present. The nonlinear equations controlling the dynamics of the system can be roughly solved thanks to the HPM. The HPM offers a precise and effective way to analyze the microbeam’s reaction to these outside stimuli by converting the nonlinear equations into linear forms. The study’s findings shed important light on how electrically activated microbeams behave in MEMSs. A more thorough examination of the system’s performance is made possible with the addition of MWCNTs and the van der Waals force. With its ability to approximate solutions and characterize system behavior, the HPM is a potent instrument that improves comprehension of the physics at play and facilitates the design and optimization of MEMS devices. The aforementioned method’s accuracy is verified by comparing it with published data that directly aligns with Anjum et al.’s findings. We have faith in this method’s accuracy and its current application.

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The stability conditions of the cubic damping Van der Pol-Duffing oscillator using the HPM with the frequency-expansion technology

Cited by 5 publications

References 22 publications

A novel technique to obtain a time-delayed vibration control analytical solution with simulation of He’s formula

A novel technique to obtain a time-delayed vibration control analytical solution with simulation of He’s formula

A heuristic review on the homotopy perturbation method for non-conservative oscillators

The Homotopy Perturbation Method for Electrically Actuated Microbeams in Mems Systems Subjected to Van Der Waals Force and Multiwalled Carbon Nanotubes

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