The variational iteration method for nonlinear oscillators with discontinuities

Rafei, M.; Ganji, D.D.; Daniali, Hamid Reza Mohammadi; Pashaei, H.

doi:10.1016/j.jsv.2007.04.020

Cited by 89 publications

(41 citation statements)

References 35 publications

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“…The most commons and most widely studied methods of all approximation methods for nonlinear differential equations are perturbation methods [1,2]. Some of other techniques include variational and variational iteration methods [3][4][5][6][7][8][9][10][11][12][13][14][15], expfunction [16][17][18][19], homotopy perturbation [20][21][22][23][24][25][26][27][28][29][30][31][32][33][34][35], equivalent linearization [36,37], standard and modified Lindstedt-Poincaré [38][39][40][41][42][43][44], artificial parameter [45,46], parameter expanding [47,50], harmonic balance methods …”

Section: Introductionmentioning

confidence: 99%

A Novel Rational Harmonic Balance Approach for Periodic Solutions of Conservative Nonlinear Oscillators

Gimeno¹,

Álvarez²,

Gallego³

et al. 2009

International Journal of Nonlinear Sciences and Numerical Simulation

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An analytical approximate procedure for a class of conservative single degree-of-freedom nonlinear oscillators with odd non-linearity is proposed. This technique is based on the generalized harmonic balance method in which analytical approximate solutions have rational forms. Unlike the classical harmonic balance techniques, in this new procedure the approximate solution and the restoring force are expanded in Fourier series prior to substituting them in the nonlinear differential equation. This approach gives us not only a truly periodic solution but also the frequency of the motion as a function of the amplitude of oscillation. Four nonlinear oscillators are presented to illustrate the usefulness and effectiveness of the proposed technique. The most significant features of this method are its simplicity and its excellent accuracy for the whole range of oscillation amplitude values and the results reveal that this technique is very effective and convenient for solving a class of conservative nonlinear oscillatory systems.

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Section: Introductionmentioning

confidence: 99%

A Novel Rational Harmonic Balance Approach for Periodic Solutions of Conservative Nonlinear Oscillators

Gimeno¹,

Álvarez²,

Gallego³

et al. 2009

International Journal of Nonlinear Sciences and Numerical Simulation

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“…(6) and (20), the approximate analytical periodic solutions u EBM (t) computed by Eqs. (6) and (19), are plotted in Figs. 1(a)-(c).…”

Section: Applications Of Strongly Nonlinear Vibration Systemsmentioning

confidence: 99%

“…Recently, some approximate variational methods, including approximate energy method [15-17, 37, 38], variational iteration method [18][19][20][21][22] and variational approach [23-29, ] and other methods [39], to solution, bifurcation, limit cycle and period solutions of nonlinear equations have been paid much attention. This paper presents energy balance method (EBM) to study periodic solutions of strongly nonlinear systems.…”

Section: Introductionmentioning

confidence: 99%

Approximate Analytical Solutions to Nonlinear Oscillations of Non-Natural Systems Using He's Energy Balance Method

Ganji¹,

Karimpour²,

Ganji³

2008

PIER M

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Abstract-This paper applies He's Energy balance method (EBM) to study periodic solutions of strongly nonlinear systems such as nonlinear vibrations and oscillations. The method is applied to two nonlinear differential equations. Some examples are given to illustrate the effectiveness and convenience of the method. The results are compared with exact solutions which lead us showing a good accuracy. The method can be easily extended to other nonlinear systems and can therefore be found widely applicable in engineering and other science.

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“…Recently, some of them were modified (Cheung et al, 1991) to investigate strong nonlinear problems. Homotopy perturbation (Belendez, 2007;Ganji and Sadighi, 2006;Belendez et al, 2008;Ozis and Yildirim, 2007), iteration method (Haque et al, 2013;Jamshidi and Ganji, 2010;Lim et al, 2006;Baghani et al, 2012;Rafei et al, 2007) are useful for obtaining approximate periodic solution with large amplitude of oscillations; however, they are applicable only for odd nonlinearity problems. Harmonic balance method (Mickens, 1986;Lim et al, 2005;Belendez et al, 2006;Wu et al, 2006;Mickens, 2007;Alam et al, 2007;Belendez et al, 2009;Lai et al, 2009;Hosen et al, 2012) is a powerful method in which truncated Fourier series is used.…”

Section: Introductionmentioning

confidence: 99%

The variational iteration method for nonlinear oscillators with discontinuities

Cited by 89 publications

References 35 publications

A Novel Rational Harmonic Balance Approach for Periodic Solutions of Conservative Nonlinear Oscillators

A Novel Rational Harmonic Balance Approach for Periodic Solutions of Conservative Nonlinear Oscillators

Approximate Analytical Solutions to Nonlinear Oscillations of Non-Natural Systems Using He's Energy Balance Method

More accurate approximate analytical solution of pendulum with rotating support

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