Modified harmonic balance method for solving strongly nonlinear oscillators

Sharif, Nazmul; Razzak, Abdur; Alam, M. Z.

doi:10.1080/09720502.2019.1624304

Cited by 8 publications

(7 citation statements)

References 33 publications

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“…According to Melnikov's theory, [10][11][12] if Melnikov's function has a simple zero point which does not depend on ε , that is, there exists t 0 such that M t ± = ( ) , 0 0 then for a sufficiently small ε , the Poincaré mapping of equation 7is chaotic in the sense of Smale horseshoe transformation. For a certain frequency, if…”

Section: Melnikov Methods For Chaotic Motion Analysis Of the Systemmentioning

confidence: 99%

“…This method can be used to analyze and determine chaos in the sense of Smale in quasi Hamiltonian systems. Compared with the traditional numerical simulation methods of chaotic motion, [9][10][11][12][13][14][15] Melnikov method can give analytical conditions with less computation as the necessary conditions for existence of chaotic motions in the systems. For a Hamiltonian system, if the system has homoclinic or heteroclinic orbits, the stable and unstable manifolds of the fixed point of the system corresponding to the periodic orbit will split under the disturbance of weak periodic external forces.…”

Section: Introductionmentioning

confidence: 99%

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Chaotic motion of a nonlinear near resonance centrifuge

Guo

2020

Noise & Vibration Worldwide

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The equilibrium point and stability of the motion equation of the nonlinear near resonance centrifuge is studied, and the critical conditions for chaotic motions of the system under external excitation are studied by Melnikov method. The expression of Melnikov function and the boundary value between chaotic and non-chaotic regions are given. According to the range of parameters, the numerical simulations are carried out. The results show that the critical parameters of chaotic motion determined using Melnikov method are consistent with that obtained by the numerical simulation. This method effectively judges the occurrence of chaotic motion.

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Section: Melnikov Methods For Chaotic Motion Analysis Of the Systemmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Chaotic motion of a nonlinear near resonance centrifuge

Guo

2020

Noise & Vibration Worldwide

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“…In the present article, we have used the simple modified form of the harmonic balance method 36 to solve a strongly nonlinear oscillator with cubic nonlinearity and harmonic restoring force modeled by 30 where a and b are known constants and initial conditions are x(0)=A,x˙(0)=0.…”

Section: Introductionmentioning

confidence: 99%

“…Solution obtained by the global residue harmonic balance method needed second or larger approximations to improve the result. Therefore, a simple and efficient harmonic balance method 36 presented in this article to overcome the above-mentioned problems. Only the first approximate solution of the present method provides better results than the other existing methods.…”

Section: Introductionmentioning

confidence: 99%

A simple modified harmonic balance method for strongly nonlinear oscillator with cubic non-linearity and harmonic restoring force

Sharif,

Molla,

Alim

2023

Journal of Low Frequency Noise, Vibration and Active Control

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In this article, a very simple modified form of the harmonic balance method is used to solve a strongly nonlinear oscillator with cubic nonlinearity and harmonic restoring force. Taylor series expansion up to third term is considered for the harmonic restoring force. The first approximate solutions of the present method pleasantly agree with the numerical solution obtained by Runge–Kutta fourth order method. Accuracy and simplicity of the present method solution is established when compared with the other method solutions. The present method can be utilized to other nonlinear oscillators.

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“…Sinusoidal Input Describing Functions (HOSIDFs) 52 and many modified versions of HBM and IHB [53][54][55][56][57][58][59] in the 1990s, 2000s and 2010s.…”

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confidence: 99%

Determination of the characteristic curves of a nonlinear first order system from Fourier analysis

Gonzalez

2023

Sci Rep

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Based on Fourier analysis, we develop an expression for modeling and simulating nonlinear first order systems. This expression is associated to a nonlinear first order differential equation $$y=f(x)+g(x)x'$$ y = f ( x ) + g ( x ) x ′ , where $$x=x(t)$$ x = x ( t ) is the dynamical variable, $$y=y(t)$$ y = y ( t ) is the driving force, and the f and g functions are the characteristic curves which are associated to dissipative and memory elements, respectively. The model is obtained from the sinusoidal response, specifically by calculating the Fourier analysis of y(t) for $$x(t)=A_1\sin (\omega t)+A_0$$ x ( t ) = A 1 sin ( ω t ) + A 0 , where the model parameters are the Fourier coefficients of the response, and the values of $$A_0$$ A 0 , $$A_1$$ A 1 and $$A_1'=A_1\omega$$ A 1 ′ = A 1 ω . The same expression is used for two kinds of time-domain simulations: to calculate other driving force $$\hat{y}$$ y ^ based on a dynamical variable $$\hat{x}$$ x ^ ; and, to calculate the dynamical variable $$\hat{x}$$ x ^ based on a driving force $$\hat{y}$$ y ^ . In both cases, the dynamical variable must remain in the range $$\hat{x}\in [A_0-A_1,A_0+A_1]$$ x ^ ∈ [ A 0 - A 1 , A 0 + A 1 ] . By analyzing this expression, we found an equivalence between the Fourier coefficients and the polynomial regressions of the characteristic curves of f and g. This result allows us to obtain the system modeling and simulation based on the amplitude and phase Fourier spectrum obtained from the Fast Fourier Transform (FFT) of the sampled $$y_n$$ y n version of y(t). It is shown that this technique has a low computational complexity, and it is expected to be suitable for real-time applications for system modeling, simulation and control, in particular when the explicit expressions of the characteristic curves are unknown. Fourier analysis is a fundamental tool in electronics, mathematics and physics, but to the best of the author’s knowledge, no work has found this clear evidence of the connection between the Fourier analysis and a first order differential equation. The aim of this work is to initiate a systematic study on this topic.

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Modified harmonic balance method for solving strongly nonlinear oscillators

Cited by 8 publications

References 33 publications

Chaotic motion of a nonlinear near resonance centrifuge

Chaotic motion of a nonlinear near resonance centrifuge

A simple modified harmonic balance method for strongly nonlinear oscillator with cubic non-linearity and harmonic restoring force

Determination of the characteristic curves of a nonlinear first order system from Fourier analysis

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