Global residue harmonic balance method to periodic solutions of a class of strongly nonlinear oscillators

Ju, Peijun; Xue, Xi

doi:10.1016/j.apm.2014.05.026

Cited by 23 publications

(17 citation statements)

References 13 publications

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“…In this paper, we will obtain higher-order analytical approximations to the periodic solution of the above nonlinear oscillator by using the global residue harmonic balance method, which was introduced in [10][11][12]. This method is very effective and convenient for nonlinear oscillators for all the former residual errors are used in the solving process of present order approximation.…”

Section: Introductionmentioning

confidence: 98%

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Global residue harmonic balance method for a nonlinear oscillator with discontinuity

2015

Applied Mathematical Modelling

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a b s t r a c tWe use a new accurate numerical method, the global residue harmonic balance method, to obtain the periodic solutions of a nonlinear oscillator with discontinuity. The absolute value of the relative error for the second-order approximate frequency is as low as 0.005887%. Comparison of the result obtained using the proposed method with the exact ones and existing results reveal that the highly accurate, simplicity and efficiency of the proposed procedure.

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Section: Introductionmentioning

confidence: 98%

“…There are some attempts in developing new analytical approach for these problems. For example, the modified homotopy perturbation method [1,2], Newton harmonic balance method [3], the variational iteration method [4,5], and various types of modified harmonic balance methods [6][7][8][9][10][11][12].…”

Section: Introductionmentioning

confidence: 99%

Global residue harmonic balance method for a nonlinear oscillator with discontinuity

2015

Applied Mathematical Modelling

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“…This method has some advantages in calculation in case of high dimension, but it can lead to nonphysical solutions [12][13][14]. Recently, other new methods based on HBM were developed by Cochelin et al [15][16][17] and Ju et al [18][19][20].…”

Section: Introductionmentioning

confidence: 99%

Frequency dependent iteration method for forced nonlinear oscillators

Hoang

Duhamel

Forêt

et al. 2017

Applied Mathematical Modelling

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A new iteration method for nonlinear vibrations has been developed by decomposing the periodic solution in two parts corresponding to low and high harmonics. For a nonlinear forced oscillator, the iteration schema is proposed with different formulations for these two parts. Then, the schema is deduced by using the harmonic balance technique. This method has proven to converge to the periodic solutions provided that a convergence condition is satisfied. The convergence is also demonstrated analytically for linear oscillators. Moreover, the new method has been applied to Duffing oscillators as an example. The numerical results show that each iteration schema converges in a domain of the excitation frequency and it can converge to different solutions of the nonlinear oscillator.

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“…Latterly, Guo et al [36,37] have presented the residue harmonic balance solution procedure to approximate the periodic behavior of different oscillation systems and they have obtained some more accurate results. Ju and Xue [38,39] proposed the global residue harmonic balance method to study strongly nonlinear systems. Comparing the obtained solutions with the exact one, they discovered that the approximate results excellently agree with the exact one.…”

Section: Introductionmentioning

confidence: 99%

The Spreading Residue Harmonic Balance Method for Strongly Nonlinear Vibrations of a Restrained Cantilever Beam

Qian¹,

Pan²,

Chen³

et al. 2017

Advances in Mathematical Physics

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The exact solutions of the nonlinear vibration systems are extremely complicated to be received, so it is crucial to analyze their approximate solutions. This paper employs the spreading residue harmonic balance method (SRHBM) to derive analytical approximate solutions for the fifth-order nonlinear problem, which corresponds to the strongly nonlinear vibration of an elastically restrained beam with a lumped mass. When the SRHBM is used, the residual terms are added to improve the accuracy of approximate solutions. Illustrative examples are provided along with verifying the accuracy of the present method and are compared with the HAM solutions, the EBM solutions, and exact solutions in tables. At the same time, the phase diagrams and time history curves are drawn by the mathematical software. Through analysis and discussion, the results obtained here demonstrate that the SRHBM is an effective and robust technique for nonlinear dynamical systems. In addition, the SRHBM can be widely applied to a variety of nonlinear dynamic systems.

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Global residue harmonic balance method to periodic solutions of a class of strongly nonlinear oscillators

Cited by 23 publications

References 13 publications

Global residue harmonic balance method for a nonlinear oscillator with discontinuity

Global residue harmonic balance method for a nonlinear oscillator with discontinuity

Frequency dependent iteration method for forced nonlinear oscillators

The Spreading Residue Harmonic Balance Method for Strongly Nonlinear Vibrations of a Restrained Cantilever Beam

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