In this article, a new technique is introduced for establishing analytical approximate solutions to conservative oscillators with strong odd nonlinearity using the variational iteration method and the Fourier series. The illustrated examples show that only a few iterations can provide very accurate approximate solutions for the whole range of oscillation amplitude even for longer time ranges.
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Technical Briefs
References
1.
Nayfeh
, A. H.
, 1973
, Perturbation Methods,Wiley-Interscience
, New York
.2.
Nayfeh
, A. H.
, and Mook
, D. T.
, 1979
, Nonlinear Oscillations
, John Wiley & Sons, Inc., Wiley Classics Library Edition Published 1995
.3.
He
, J. H.
, 1998
, “Approximate Analytical Solution for Seepage Flow With Fractional Derivatives in Porous Media
,” Comput. Methods Appl. Mech. Eng.
, 167
, pp. 57
–68
.10.1016/S0045-7825(98)00108-X4.
He
, J. H.
, 1999
, “Variational Iteration Method - A Kind of Non-Linear Analytical Technique: Some Examples
,” Int. J. Nonlinear Mech.
, 34
, pp. 699
–708
.10.1016/S0020-7462(98)00048-15.
He
, J. H.
, and Wu
, X.
, 2007
, “Variational Iteration Method: New Development and Applications
,” Comput. Math. Appl.
, 54
, pp. 881
–894
.10.1016/j.camwa.2006.12.0836.
He
, J. H.
, 2006
, “Some Asymptotic Methods for Strongly Nonlinear Equations
,” Int. J. Mod. Phys. B
, 20
, pp. 1141
–1199
.10.1142/S02179792060337967.
He
, J. H.
, 2007
, “Variational Iteration Method-Some Recent Results and New Interpretations
,” J. Comput. Appl. Math.
, 207
, pp. 3
–17
.10.1016/j.cam.2006.07.0098.
He
, J. H.
, 1999
, “Homotopy Perturbation Technique
,” Comput. Methods Appl. Mech. Eng.
, 178
, pp. 257
–262
.10.1016/S0045-7825(99)00018-39.
He
, J. H.
, 2000
, “A New Perturbation Technique Which is also Valid for Large Parameters
,” J. Sound Vib.
, 229
, pp. 1257
–1263
.10.1006/jsvi.1999.250910.
Liao
, S. J.
, 2003
, Beyond Perturbation: Introduction to the Homotopy Analysis Method
, CRC Press, Boca Raton, Chapman & Hall
.11.
Liao
, S. J.
, 2004
, “On the Homotopy Analysis Method for Nonlinear Problems
,” Appl. Math. Comput.
, 147
, pp. 499
–513
.10.1016/S0096-3003(02)00790-712.
He
, J. H.
, 2002
, “Preliminary Report on the Energy Balance for Nonlinear Oscillations
,” Mech. Res. Commun.
, 29
, pp. 107
–111
.10.1016/S0093-6413(02)00237-913.
Marinca
, V.
, and Herisanu
, N.
, 2008
, “Periodic Solutions of Duffing Equation With Strong Non-Linearity
,” Chaos, Solitons Fractals
, 37
, pp. 144
–149
.10.1016/j.chaos.2006.08.03314.
Chen
, Y. M.
, Meng
, G.
, and Liu
, J. K.
, 2010
, “Variational Iteration Method for Conservative Oscillators With Complicated Nonlinearities
,” Appl. Math. Comput.
, 15
, pp. 802
–809
. Available at: http://www.mcajournal.org/index.php?option=com_content&view=article&id=97&Itemid=190.15.
Wu
, B. S.
, Sun
, W. P.
, and Lim
, C. W.
, 2006
, “An Analytical Approximate Technique for a Class of Strongly Non-Linear Oscillators
,” Int. J. Non-Linear Mech.
, 41
, pp. 766
–774
.10.1016/j.ijnonlinmec.2006.01.00616.
Xu
, L.
, 2007
, “Application of He’s Parameter-Expansion Method to an Oscillation of a Mass Attached to a Stretched Elastic Wire
,” Phys. Lett. A
, 368
, pp. 259
–262
.10.1016/j.physleta.2007.04.00417.
Mickens
, R. E.
, 2001
, “Mathematical and Numerical Study of Duffing-Harmonic Oscillator
,” J. Sound Vib.
, 244
, pp. 563
–567
.10.1006/jsvi.2000.350218.
Lim
, C. W.
, and Wu
, B. S.
, 2003
, “A New Analytical Approach to the Duffing-Harmonic Oscillator
,” Phys. Lett. A
, 311
, pp. 365
–373
.10.1016/S0375-9601(03)00513-919.
Tiwari
, S. B.
, Rao
, B. N.
, Swamy
, N. S.
, Sai
, K. S.
, and Nataraja
, H. R.
, 2005
, “Analytical Study on a Duffing-Harmonic Oscillator
,” J. Sound Vib.
, 285
, pp. 1217
–1222
.10.1016/j.jsv.2004.11.00120.
Ozis
, T.
, and Yıldırım
, A.
, 2007
, “Determination of the Frequency-Amplitude Relation for a Duffing-Harmonic Oscillator by the Energy Balance Method
,” Comput. Math. Appl.
, 54
, pp. 1184
–1187
.10.1016/j.camwa.2006.12.06421.
Fesanghary
, M.
, Pirbodaghi
, T.
, Asghari
, M.
, and Sojoudi
, H.
, 2009
, “A New Analytical Approximation to the Duffing-Harmonic Oscillator
,” Chaos, Solitons Fractals
, 42
, pp. 571
–576
.10.1016/j.chaos.2009.01.024Copyright © 2013 by ASME
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