Periodic Property and Instability of a Rotating Pendulum System

He, Ji‐Huan; Amer, T. S.; Elnaggar, Shimaa; Galal, A. A.

doi:10.3390/axioms10030191

Cited by 71 publications

(45 citation statements)

References 40 publications

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“…By employing the two expansions (10) and (11) into equation (20) and setting all coefficients of like power of ρ to zero and inserting the zero-order solution (14) into the first-order problems gives…”

Section: Let Us Consider First the Conservative Duffing Oscillatormentioning

confidence: 99%

“…Employing (34) and (11) with (33), equating the identical powers of ρ to zero yields the zero-order solution y 0 ðtÞ ¼ A cos Vt (35) The first-order problem has the form…”

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

confidence: 99%

“…Substituting the regular expansion (11) in the previous example into the above equation and rearranged in terms of powers of ρ, we obtain the following equations…”

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

confidence: 99%

“…10 The instability of a system also caught much attention to avoid any damage. 11 In general, solving nonlinear differential equations is more complicated than linear differential equations. This paper focuses on a heuristic review on the HPM for non-conservative oscillators by the HPM.…”

Section: Introductionmentioning

confidence: 99%

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

El‐Dib

2021

Journal of Low Frequency Noise, Vibration and Active Control

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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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Section: Let Us Consider First the Conservative Duffing Oscillatormentioning

confidence: 99%

“…Employing (34) and (11) with (33), equating the identical powers of ρ to zero yields the zero-order solution y 0 ðtÞ ¼ A cos Vt (35) The first-order problem has the form…”

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

confidence: 99%

“…Substituting the regular expansion (11) in the previous example into the above equation and rearranged in terms of powers of ρ, we obtain the following equations…”

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

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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

El‐Dib

2021

Journal of Low Frequency Noise, Vibration and Active Control

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“…In this section, we will demonstrate the properties of nonlinear amplitude for the system of Equations (37) and investigate their stabilities [28][29][30]. Then, the transformations below are taken into account…”

Section: Non-linear Analysismentioning

confidence: 99%

Analyzing the Stability for the Motion of an Unstretched Double Pendulum near Resonance

et al. 2021

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This work looks at the nonlinear dynamical motion of an unstretched two degrees of freedom double pendulum in which its pivot point follows an elliptic route with steady angular velocity. These pendulums have different lengths and are attached with different masses. Lagrange’s equations are employed to derive the governing kinematic system of motion. The multiple scales technique is utilized to find the desired approximate solutions up to the third order of approximation. Resonance cases have been classified, and modulation equations are formulated. Solvability requirements for the steady-state solutions are specified. The obtained solutions and resonance curves are represented graphically. The nonlinear stability approach is used to check the impact of the various parameters on the dynamical motion. The comparison between the attained analytic solutions and the numerical ones reveals a high degree of consistency between them and reflects an excellent accuracy of the used approach. The importance of the mentioned model points to its applications in a wide range of fields such as ships motion, swaying buildings, transportation devices and rotor dynamics.

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Nonlinear suppression using time-delayed controller to excited Van der Pol–Duffing oscillator: analytical solution techniques

Moatimid

Amer

2022

Arch Appl Mech

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To suppress the nonlinearity of an excited Van der Pol–Duffing oscillator (VdPD), time-delayed position and velocity are used throughout this study. The time delay is supplemental to prevent the nonlinear vibration of the considered system. The topic of this work is extremely current because technologies with a time delay have been the subject of several studies in the latest days. The classical homotopy perturbation method (HPM) is utilized to extract an approximate systematic explanation for the system at hand. Furthermore, a modification of the HPM reveals a more accurate approximate solution. This accuracy is tested through a comparison with the numerical solution. The practical approximate analytical methodology makes the work possible to qualitatively evaluate the results. The time histories of the obtained solutions are drawn for various values of the natural frequency and the time delay parameters. Discussion of the results is presented in light of the plotted curves. On the other hand, the multiple scale procedure examines the organized nonlinear prototypical approach. The influence of the diverse regulatory restrictions on the organization’s vibration performances is explored. Two important cases of resonance, the sub-harmonic and super-harmonic, are examined according to the cubic nonlinearity. The modulation equations achieved for these cases are examined graphically according to the impact of the used parameters.

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Periodic Property and Instability of a Rotating Pendulum System

Cited by 71 publications

References 40 publications

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

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

Analyzing the Stability for the Motion of an Unstretched Double Pendulum near Resonance

Nonlinear suppression using time-delayed controller to excited Van der Pol–Duffing oscillator: analytical solution techniques

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