Improvement of Harmonic Balance Using Jacobian Elliptic Functions

Yamgoué, Serge Bruno; Nana, B.; Lekeufack, Olivier Tiokeng

doi:10.4236/jamp.2015.36081

Cited by 3 publications

(3 citation statements)

References 14 publications

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“…(3.24). In this regard we follow the simple rule suggested in [27]. It recommends that the harmonics to include in the increment to the ansatz of a given stage should be higher than or equal to the least harmonic of the residual terms of that stage.…”

Section: An Example Of Asymmetric Problemmentioning

confidence: 99%

Harmonic Balance for Non-Periodic Hyperbolic Solutions of Nonlinear Ordinary Differential Equations

Yamgoué

Lekeufack

Kofané

2017

Mathematical Modelling and Analysis

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In this paper, we propose a new approach for obtaining explicit analytical approximations to the homoclinic or heteroclinic solutions of a general class of strongly nonlinear ordinary differential equations describing conservative singledegree-of-freedom systems. Through a simple and explicit change of the independent variable that we introduce, these equations are transformed to others for which the original homoclinic or heteroclinic solutions are mapped into periodic solutions that satisfy some boundary conditions. Recent simplified methods of harmonic balance can then be exploited to construct highly accurate analytic approximations to these solutions. Here, we adopt the combination of Newton linearization with the harmonic balance to construct the approximates in incremental steps, thereby proposing both appropriate initial approximates and increments that together satisfy the required boundary conditions. Three examples including a septic Duffing oscillator, a controlled mechanical pendulum and a perturbed KdV equations are presented to illustrate the great accuracy and simplicity of the new approach.

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Section: An Example Of Asymmetric Problemmentioning

confidence: 99%

Harmonic Balance for Non-Periodic Hyperbolic Solutions of Nonlinear Ordinary Differential Equations

Yamgoué

Lekeufack

Kofané

2017

Mathematical Modelling and Analysis

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“…The above improved harmonic balance methods have been widely applied by researchers in different fields of physics and engineering applications [43][44][45][46][47][48][49]. Yamgoue et al [50] improved the HB method using the Jacobian elliptic functions and applied the proposed method to the single degree-of-freedom nonlinear oscillators. Wang et al [51] improved the IHB method by applying the equivalent piecewise linearization approach.…”

Section: Introductionmentioning

confidence: 99%

Improved harmonic balance method for analyzing asymmetric restoring force functions in nonlinear vibration of mechanical systems

Mohammadian,

Ismail

2024

Phys. Scr.

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The non-linear differential equation governing some mechanical systems, such as the non-linear vibrations of FG beams resting on a nonlinear foundation, behaves similarly to the oscillations of a non-linear oscillator with an asymmetric restoring force function that includes both odd and non-odd non-linear terms. The objective of this research is to obtain higher-order approximate analytical solutions for such problems by introducing an enhanced harmonic balance method. By building upon the original problem, two new symmetric systems with odd non-linear terms are introduced, and their higher-order approximate analytical solutions are derived using a novel approach. In proposed method, the restoring force function is represented by its Fourier series expansion. Unlike previous papers, linearizing the equation or taking the first derivative of the Fourier series in the subsequent iteration is unnecessary. However, to enhance the solution’s accuracy, the remaining error from each iteration is utilized in the next one. Finally, by combining the results from the two introduced systems, the analytical period and corresponding periodic solution of the original problem can be obtained. This method is applied to the governing differential equation of FG beams resting on non-linear foundations, a physical non-natural oscillator, and conservative Toda oscillator. The key advantages of this approach are its simplicity and its ability to provide highly precise solutions for both small and large amplitudes in a single iteration.

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“…Numerous approximate analytical techniques are now available in the literature as the results of these efforts. They include: the Newton harmonic balance [3]- [8], the linear-delta expansion [9], the Homotopy analysis method [10], the method of cubication [11]- [13], the rational harmonic balance method [14], the global error minimization [15], the recent method of quintication [16], to name just a few. Although some of these techniques are very competitive with respect to accuracy, rate of convergence and ease of use, there is always a room for some improvements.…”

Section: Introductionmentioning

confidence: 99%

Generalization of the Global Error Minimization for Constructing Analytical Solutions to Nonlinear Evolution Equations

Yamgoué¹,

Nana²

2015

JAMP

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The global error minimization is a variational method for obtaining approximate analytical solutions to nonlinear oscillator equations which works as follows. Given an ordinary differential equation, a trial solution containing unknowns is selected. The method then converts the problem to an equivalent minimization problem by averaging the squared residual of the differential equation for the selected trial solution. Clearly, the method fails if the integral which defines the average is undefined or infinite for the selected trial. This is precisely the case for such non-periodic solutions as heteroclinic (front or kink) and some homoclinic (dark-solitons) solutions. Based on the fact that these types of solutions have vanishing velocity at infinity, we propose to remedy to this shortcoming of the method by averaging the product of the residual and the derivative of the trial solution. In this way, the method can apply for the approximation of all relevant type of solutions of nonlinear evolution equations. The approach is simple, straightforward and accurate as its original formulation. Its effectiveness is demonstrated using a Helmholtz-Duffing oscillator.

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Improvement of Harmonic Balance Using Jacobian Elliptic Functions

Cited by 3 publications

References 14 publications

Harmonic Balance for Non-Periodic Hyperbolic Solutions of Nonlinear Ordinary Differential Equations

Harmonic Balance for Non-Periodic Hyperbolic Solutions of Nonlinear Ordinary Differential Equations

Improved harmonic balance method for analyzing asymmetric restoring force functions in nonlinear vibration of mechanical systems

Generalization of the Global Error Minimization for Constructing Analytical Solutions to Nonlinear Evolution Equations

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