Asymptotic analysis and accurate approximate solutions for strongly nonlinear conservative symmetric oscillators

Wu, Baisheng; Liu, Weijia; Chen, Xin; Lim, C.W.

doi:10.1016/j.apm.2017.05.004

Cited by 15 publications

(7 citation statements)

References 26 publications

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“…We next show the construct of the second order approximation to (5). According to GRHBM, the second order approximated solution to (5) can be expressed as…”

Section: Analysis Of the Strongly Nonlinear Oscillator (1) By Grhbmmentioning

confidence: 99%

“…Nonlinear oscillations have wide applications in physics, mathematics, mechanics and engineering areas. [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15] Generally, the nonlinear differential equations (NDEs) can be used to model the nonlinear oscillators. Due to the strong nonlinearity of NDEs, the study of accurate approximations has been paid much attention.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Global residue harmonic balance method for strongly nonlinear oscillator with cubic and harmonic restoring force

2022

Journal of Low Frequency Noise, Vibration and Active Control

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This paper focuses on the numerical investigation of a strongly nonlinear oscillator with cubic and harmonic restoring force. We transform this oscillator as a free damped cubic-quintic Duffing oscillator equation by Taylor approximation. The approximated solutions with high accuracy are provided by using the global residue harmonic balance method (GRHBM) without any discretization or restrict assumptions. The sensitive analysis of the approximation or the frequency with respect to the amplitude is considered in detail. Numerical comparisons with Runge–Kutta method and harmonic balance method are given to show the efficiency and stability of GRHBM.

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“…We next show the construct of the second order approximation to (5). According to GRHBM, the second order approximated solution to (5) can be expressed as…”

Section: Analysis Of the Strongly Nonlinear Oscillator (1) By Grhbmmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Global residue harmonic balance method for strongly nonlinear oscillator with cubic and harmonic restoring force

2022

Journal of Low Frequency Noise, Vibration and Active Control

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“…The nature of each equilibrium state is determined due to the influence of the governing parameters. Besides, a quantitative analysis is presented to construct analytical approximate solutions for periodic motions around each stable equilibrium point by means of the Newton Harmonic Balance (NHB) method [23,24]. This approach has been successfully used to derive accurate approximate solutions for various strongly nonlinear problems in structural and mechanical engineering [11, 25−28].…”

Section: Motivationmentioning

confidence: 99%

Analysis of Large-Amplitude Oscillations in Triple-Well Non-Natural Systems

Lai

Yang

Gao

2019

Journal of Computational and Nonlinear Dynamics

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In this paper, the large-amplitude oscillation of a triple-well non-natural system, covering both qualitative and quantitative analysis, is investigated. The nonlinear system is governed by a quadratic velocity term and an odd-parity restoring force having cubic and quintic nonlinearities. Many mathematical models in mechanical and structural engineering applications can give rise to this nonlinear problem. In terms of qualitative analysis, the equilibrium points and its trajectories due to the change of the governing parameters are studied. It is interesting that there exist heteroclinic and homoclinic orbits under different equilibrium states. By adjusting the parameter values, the dynamic behavior of this conservative system is shifted accordingly. As exact solutions for this problem expressed in terms of an integral form must be solved numerically, an analytical approximation method can be used to construct accurate solutions to the oscillation around the stable equilibrium points of this system. This method is based on the harmonic balance method incorporated with Newton's method, in which a series of linear algebraic equations can be derived to replace coupled and complicated nonlinear algebraic equations. According to this harmonic balance-based approach, only the use of Fourier series expansions of known functions is required. Accurate analytical approximate solutions can be derived using lower order harmonic balance procedures. The proposed analytical method can offer good agreement with the corresponding numerical results for the whole range of oscillation amplitudes.

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“…Nonlinear oscillations have wide applications across various fields such as physics [1][2][3][4][5], mathematics [6][7][8][9], mechanics, and engineering [10][11][12][13][14][15]. In order to describe these oscillators accurately, nonlinear differential equations (NDEs) are often employed.…”

Section: Introductionmentioning

confidence: 99%

Generic numerical and analytical methods for solving nonlinear oscillators

Kontomaris,

Mazi,

Chliveros

et al. 2024

Phys. Scr.

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Nonlinear oscillations are a challenging problem due to the high complexity of the underlying differential equations. This paper focuses on the nonlinear oscillator with cubic and harmonic restoring force as it represents a very general case. In particular, many well-known oscillators (e.g., the Duffing equation, the capillary oscillator, the cubic-quintic Duffing oscillator, the simple pendulum, etc.) are subcases of it. Accurate solutions in terms of Fourier series functions for different oscillation amplitudes were derived using an energy-based numerical method and fitting processes. In addition, an analytical approach was developed by finding the derivative of the restoring force function at the point x=c0A, where A is the amplitude of the oscillation, and c0 is a constant. Using this approach, the error was negligible (~0.02−0.22%) even for A → ∞. The methods and solutions for the nonlinear oscillator with cubic and harmonic restoring force can be equally applied to all its subcases.

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Asymptotic analysis and accurate approximate solutions for strongly nonlinear conservative symmetric oscillators

Cited by 15 publications

References 26 publications

Global residue harmonic balance method for strongly nonlinear oscillator with cubic and harmonic restoring force

Global residue harmonic balance method for strongly nonlinear oscillator with cubic and harmonic restoring force

Analysis of Large-Amplitude Oscillations in Triple-Well Non-Natural Systems

Generic numerical and analytical methods for solving nonlinear oscillators

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