A finite extensibility nonlinear oscillator

Febbo, Mariano

doi:10.1016/j.amc.2011.01.011

Cited by 13 publications

(41 citation statements)

References 37 publications

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“…(18) and (19) are very simple and they can be easily computed with the help of symbolic computation software such as MATHEMATICA.…”

Section: Solution Methodsmentioning

confidence: 99%

“…As in all previous examples considered, we compare the approximate frequency obtained using the cubication method with other methods of approximation. Febbo [19] analysed carefully this nonlinear oscillator and obtained the analytical approximate frequency using the secondorder LHBM. He obtained a relative error less than 4% for oscillation amplitudes lower than 0.9.…”

Section: Example 6 Finite Extensibility Nonlinear Oscillator Fenomentioning

confidence: 99%

“…(28) and (29) into Eqs. (18) and (19) yields the following analytical approximations for the frequency and periodic solution of the anti-symmetric quadratic nonlinear oscillator:…”

Section: Example 1 Anti-symmetric Constant Force Oscillatormentioning

confidence: 99%

“…(24) and (25). The value for the "L 2 -norm" is also analyzed in order to obtain a global estimation of the accuracy of the approximate solution, where L 2 is calculated as follows [19]:…”

Section: Example 1 Anti-symmetric Constant Force Oscillatormentioning

confidence: 99%

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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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“…(18) and (19) are very simple and they can be easily computed with the help of symbolic computation software such as MATHEMATICA.…”

Section: Solution Methodsmentioning

confidence: 99%

Section: Example 6 Finite Extensibility Nonlinear Oscillator Fenomentioning

confidence: 99%

“…(28) and (29) into Eqs. (18) and (19) yields the following analytical approximations for the frequency and periodic solution of the anti-symmetric quadratic nonlinear oscillator:…”

Section: Example 1 Anti-symmetric Constant Force Oscillatormentioning

confidence: 99%

“…(24) and (25). The value for the "L 2 -norm" is also analyzed in order to obtain a global estimation of the accuracy of the approximate solution, where L 2 is calculated as follows [19]:…”

Section: Example 1 Anti-symmetric Constant Force Oscillatormentioning

confidence: 99%

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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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“…From Tables 1 and 2 we conclude that there are two dominant harmonics, a 1 and a 3 , and therefore consider only the cases in which one or two harmonics are present in the approximate solution, which correspond to the first and the second approximation, respectively. In order to have a global estimation of the accuracy of each approximate solution, L 2 norm errors were calculated as follows [13] …”

Section: Some Analytical Approximate Solutionsmentioning

confidence: 99%

Exact and approximate solutions for the anti-symmetric quadratic truly nonlinear oscillator

Garde

Pascual

Beléndez

et al. 2014

Applied Mathematics and Computation

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The exact solution of the anti-symmetric quadratic truly nonlinear oscillator is derived from the first integral of the nonlinear differential equation which governs the behaviour of this oscillator. This exact solution is expressed as a piecewise function including Jacobi elliptic cosine functions. The Fourier series expansion of the exact solution is also analyzed and its coefficients are computed numerically. We also show that these Fourier coefficients decrease rapidly and, consequently, using just a few of them provides an accurate analytical representation of the exact periodic solution. Some approximate solutions containing only two harmonics as well as a rational harmonic representation are obtained and compared with the exact solution.

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