An Approximate Equivalent Linearization Technique for Nonlinear Oscillations

Denman, Harry H.

doi:10.1115/1.3564651

Cited by 12 publications

(18 citation statements)

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“…Following Denman [2] and Jonckheere [3], the function f(Ay) is expanding in terms of Chebyshev polynomials of the first kind…”

Section: Formulation and Solution Methodsmentioning

confidence: 99%

“…This linearization scheme uses the Chebyshev series expansion of the restoring force is used instead of the Taylor one [2][3][4] and an approximate frequency depending on the oscillation amplitude is obtained. In this paper we present a generalization of this technique in which the original second-order nonlinear differential equation is replaced with the well-known Duffing equation, and this cubic equation is exactly solved.…”

Section: Introductionmentioning

confidence: 99%

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Cubication of conservative nonlinear oscillators

Álvarez¹,

Fernández²,

Pascual³

2009

Eur. J. Phys.

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“…Following Denman [2] and Jonckheere [3], the function f(Ay) is expanding in terms of Chebyshev polynomials of the first kind…”

Section: Formulation and Solution Methodsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Cubication of conservative nonlinear oscillators

Álvarez¹,

Fernández²,

Pascual³

2009

Eur. J. Phys.

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“…Denman [12] and Jonckheere [13] proposed the determination of the period of nonlinear oscillators by means the Chebyshev polynomials. Taking this into account, it is possible to expand the function f(y) in terms of Chebyshev polynomials of the first kind € T n (x) as follows [13] A. Beléndez, D. I. Méndez, E. Fernández, S. Marini and I. Pascual, "An explicit approximate solution to the Duffing-harmonic oscillator by a cubication method", Physics Letters A, Vol.…”

Section: Solution Proceduresmentioning

confidence: 99%

“…This oscillator is a conservative non-linear oscillatory system modelled by a potential having a rational form for the potential energy [11]. To do this, the Chebychev series expansion of the restoring force is used [12][13][14] to obtain an approximate frequency-amplitude relation as a function of the complete elliptic integral of the first kind. As we can see, the results presented in this paper reveal that the method considered here is very effective and convenient for the Duffing-harmonic oscillator.…”

Section: Introductionmentioning

confidence: 99%

An explicit approximate solution to the Duffing-harmonic oscillator by a cubication method

et al. 2009

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The nonlinear oscillations of a Duffing-harmonic oscillator are investigated by an approximated method based on the 'cubication' of the initial nonlinear differential equation. These explicit formulas are valid for all values of the initial amplitude and we conclude this cubication method works very well for the whole range of initial amplitudes. Excellent agreement of the approximate frequencies and periodic solutions with the exact ones is demonstrated and discussed and the relative error for the approximate frequency is as low as 0.071%. Unlike other approximate methods applied to this oscillator, which are not capable to reproduce exactly the behaviour of the approximate frequency when A tends to zero, the cubication method used in this paper predicts exactly the behaviour of the approximate frequency not only when A tends to infinity, but also when A tends to zero. Finally, a closedform expression for the approximate frequency is obtained in terms of elementary functions.To do this, the relationship between the complete elliptic integral of the first kind and the arithmetic-geometric mean as well as Legendre's formula to approximately obtain this mean are used.

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“…Following Denman [11] and Jonckheere [12], the nonlinear function f (Ay) is expanded in terms of the Chebyshev polynomials of the first kind T n (y) as…”

Section: Solution Methodsmentioning

confidence: 99%

Solutions for Conservative Nonlinear Oscillators Using an Approximate Method Based on Chebyshev Series Expansion of the Restoring Force

et al. 2016

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Approximate solutions for small and large amplitude oscillations of conservative systems with odd nonlinearity are obtained using a "cubication" method. In this procedure, the Chebyshev polynomial expansion is used to replace the nonlinear function by a third-order polynomial equation. The original second-order differential equation, which governs the dynamics of the system, is replaced by the Duffing equation, whose exact frequency and solution are expressed in terms of the complete elliptic integral of the first kind and the Jacobi elliptic function cn, respectively. Then, the exact solution for the Duffing equation is the approximate solution for the original nonlinear differential equation. The coefficients for the linear and cubic terms of the approximate Duffing equation -obtained by "cubication" of the original second-order differential equation -depend on the initial oscillation amplitude. Six examples of different types of common conservative nonlinear oscillators are analysed to illustrate this scheme. The results obtained using the cubication method are compared with those obtained using other approximate methods such as the harmonic, linearized and rational balance methods as well as the homotopy perturbation method. Comparison of the approximate frequencies and solutions with the exact ones shows good agreement.

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An Approximate Equivalent Linearization Technique for Nonlinear Oscillations

Cited by 12 publications

References 0 publications

Cubication of conservative nonlinear oscillators

Cubication of conservative nonlinear oscillators

An explicit approximate solution to the Duffing-harmonic oscillator by a cubication method

Solutions for Conservative Nonlinear Oscillators Using an Approximate Method Based on Chebyshev Series Expansion of the Restoring Force

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