Residue harmonic balance solution procedure to nonlinear delay differential systems

Guo, Zhao; Xiao-yan, MA

doi:10.1016/j.amc.2014.03.090

Cited by 13 publications

(13 citation statements)

References 33 publications

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“…Moreover, they predicted solutions of many autonomous delay differential systems by the former method. 26 Simultaneously, Xiao et al 27 studied the fractional order van der Pol oscillator by RHBM. Then, the global RHBM 28,29 was presented by Ju et al and used to solve the approximate solutions of the strongly nonlinear systems.…”

Section: Introductionmentioning

confidence: 99%

The spreading residue harmonic balance method for studying the doubly clamped beam-type N/MEMS subjected to the van der Waals attraction

Qian

Pan

Qiang

et al. 2018

Journal of Low Frequency Noise, Vibration and Active Control

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Since the doubly clamped beam-type N/MEMS subjected to the van der Waals attraction is transformed into a mathematical model through Galerkin method, which is a nonlinear differential equation, the spreading residue harmonic balance method is introduced to solve the approximate solution of the nonlinear problem in this paper. The spreading residue harmonic balance method is developed on the basis of the residue harmonic balance method. The disadvantage of the approach is small parameter assumption is needed. In order to improve the accuracy of the approximate solution, the residual of the former order harmonic approximation is introduced. Besides, we compare the second-order spreading residue harmonic balance method solution with the numerical one by the Runge-Kutta method. This proves the availability and validity of spreading residue harmonic balance method.

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Section: Introductionmentioning

confidence: 99%

The spreading residue harmonic balance method for studying the doubly clamped beam-type N/MEMS subjected to the van der Waals attraction

Qian

Pan

Qiang

et al. 2018

Journal of Low Frequency Noise, Vibration and Active Control

View full text Add to dashboard Cite

show abstract

“…Even if there exists such a parameter, the analytical solution determined by the perturbation method has a small validity. Thus, many approximation methods were developed for solving strongly nonlinear oscillators, including modified Lindstedt-Poincare method [4][5], harmonic balance method [6][7][8][9][10], residue harmonic balance method [11], global residue harmonic balance method [12], iterative homotopy harmonic balance method [13], amplitude-frequency formulation [14] and homotopy analysis method [15]. Moreover energy balance method is used technique for solving strongly nonlinear oscillators [16][17][18][19][20][21].…”

Section: Introductionmentioning

confidence: 99%

A New Analytical Approach for Solving Van der Pol Oscillator

Mondal¹,

Molla²,

Alam³

2019

SJAMS

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The Van der Pol oscillator is a nonlinear damping and non-conservative oscillator. Energy is generated at low amplitude and dissipated at high amplitude. This nonlinear oscillator was first introduced by Dutch electrical engineer and physicist B. Van der Pol and it was originally used to investigate vacuum tubes. Nowadays, it is used in both physical and biological sciences. It is also used in sociology and even in economics. It has a limit cycle and in earlier it was determined by the classical perturbation methods when the nonlinear term is small. Then the harmonic balance method was used to determine the limit cycle for stronger nonlinear case. Moreover, many researchers have been analyzed this oscillator by various numerical approaches. In this article, a new analytical approach based on harmonic balance method is presented to determine the limit cycle as well as approximate solutions of this nonlinear oscillator. The frequency as well as the limit cycle obtained by new approach has been compared with those obtained by other existing methods. The present method gives better result than other existing results and also close to the corresponding numerical result (considered to the exact result). Moreover, the present method is simpler than the existing harmonic balance method.

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“…In recent years, several analytical methods such as homotopy perturbation [1], harmonic balance [2], residue harmonic balance [3], The Hamiltonian approach [4], homotopy analysis [5], max-min approach [6], coupling of homotopy variation [7], iterative homotopy harmonic balance method [8], global residue harmonic balance [9], Fourier series solutions with finite harmonic terms [10], amplitude-frequency formulation [11][12][13], parameter-expansion method [14][15][16][17][18][19], multi-step homotopy analysis method [20], multiple-scales homotopy perturbation method [21][22][23] and the Frobenius-homotopy method [24] have been developed for solving strongly nonlinear oscillators.…”

Section: Introductionmentioning

confidence: 99%

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

El‐Dib¹

2018

J. Appl. Math. Comput. Mech.

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In this paper, we perform the frequency-expansion formula for the nonlinear cubic damping van der Pol's equation, and the nonlinear frequency is derived. Stability conditions are performed, for the first time ever, by the nonlinear frequency technology and for the nonlinear oscillator. In terms of the van der Pol's coefficients the stability conditions have been performed. Further, the stability conditions are performed in the case of the complex damping coefficients. Moreover, the study has been extended to include the influence of a forcing van der Pol' oscillator. Stability conditions have been derived at each resonance case. Redoing the perturbation theory for the van der Pol oscillator illustrates more of a resonance formulation such as sub-harmonic resonance and super-harmonic resonance. More approximate nonlinear dispersion relations of quartic and quintic forms in the squaring of the extended frequency are derived, respectively.

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Residue harmonic balance solution procedure to nonlinear delay differential systems

Cited by 13 publications

References 33 publications

The spreading residue harmonic balance method for studying the doubly clamped beam-type N/MEMS subjected to the van der Waals attraction

The spreading residue harmonic balance method for studying the doubly clamped beam-type N/MEMS subjected to the van der Waals attraction

A New Analytical Approach for Solving Van der Pol Oscillator

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

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