An analytical solution of the fractal toda oscillator

Feng, Guang-Qing; Niu, Jing-Yan

doi:10.1016/j.rinp.2023.106208

Cited by 15 publications

(9 citation statements)

References 41 publications

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“…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%

“…The application of Galerkin’s method and the least square technique 2 has successfully evaluated the frequency ω2(A). There are many analytical available methods in the literature 3–8 to estimate the frequency. Following the classical procedure of He’s formula, 1 Younesiana et al 9 have considered the generalized Duffing equation having F(u)=∑n=0Na2n+1u2n+1.…”

Section: Introductionmentioning

confidence: 99%

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

El‐Dib

Alyousef

2023

Journal of Low Frequency Noise, Vibration and Active Control

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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.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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

El‐Dib

Alyousef

2023

Journal of Low Frequency Noise, Vibration and Active Control

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show abstract

“…The theory of fractal variational is used in Ref. 25 to establish the fractal Toda oscillator, and the non-perturbative method is employed to obtain the appropriate analytical solution.…”

Section: Introductionmentioning

confidence: 99%

Bifurcations of Liouville tori of coupled sextic anharmonic oscillators

El-Sabaa

Amer

Gad

et al. 2023

Journal of Low Frequency Noise, Vibration and Active Control

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In the current paper, the problem of sextic anharmonic oscillators is investigated. There are three integrable cases of this problem. Emphasis is placed on two integrable cases, and a full description of each one is provided. The separated functions of the first and second integrability cases are transformed from a higher degree to the third and fourth degrees. Respectively, the periodic solution is obtained using Jacobi’s elliptic functions. The topology of phase space and Liouville tori’s bifurcations are discussed. The phase portrait is studied to determine singular points and classify their types in addition to the graphic representation for each of them. Finally, the numerical illustrations are introduced using the Poincaré surface section to emphasize the problem’s integrability.

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“…In Ref. 24, the fractal Toda oscillator is established using the theory of fractal variation and the proper analytical solution is obtained using the non-perturbative approach.…”

Section: Introductionmentioning

confidence: 99%

Analysis of the stick-slip behavior of coupled oscillators with dry friction

Amer

Shaker

Dahab

2023

Journal of Low Frequency Noise, Vibration and Active Control

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In this study, a model of two nonlinear spring masses subjected to a smooth friction-velocity curve is examined. The second order governing system of motion is obtained in view of the system friction forces. This system is transformed to another suitable one of first order to investigate the points of equilibrium using Hurwitz’s theory. The excitation and critical stick-slip (SS) speeds are determined in accordance with the characteristic equation for the Jacobian matrix of the system. The stability and behavior of the system motion, along with the behavior of the SS movement, are examined. The effect of various parameters, such as excitation speed, damping and fraction coefficients, linear and nonlinear stiffness of springs, and masses that affect the motion and stability of the system is analyzed and studied. This research has numerous practical applications in a variety of industries, including airports, modern car brakes, well drilling, explaining and understanding seismic events, mechanical transportation, friction in bridges, and civil systems, which all exhibit impacted friction in dynamics and stick-slip phenomena.

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An analytical solution of the fractal toda oscillator

Cited by 15 publications

References 41 publications

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

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

Bifurcations of Liouville tori of coupled sextic anharmonic oscillators

Analysis of the stick-slip behavior of coupled oscillators with dry friction

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