Full Text:   <5342>

CLC number: O34

On-line Access: 

Received: 2008-09-08

Revision Accepted: 2008-10-27

Crosschecked: 2009-07-21

Cited: 19

Clicked: 5644

Citations:  Bibtex RefMan EndNote GB/T7714

-   Go to

Article info.
Open peer comments

Journal of Zhejiang University SCIENCE A 2009 Vol.10 No.9 P.1263-1268

http://doi.org/10.1631/jzus.A0820651


Solution of nonlinear cubic-quintic Duffing oscillators using He’s Energy Balance Method


Author(s):  D. D. GANJI, M. GORJI, S. SOLEIMANI, M. ESMAEILPOUR

Affiliation(s):  Department of Mechanical Engineering, University of Mazandaran, Babol, Iran

Corresponding email(s):   ddg_davood@yahoo.com

Key Words:  Energy Balance Method (EBM), Cubic-quintic Duffing equation, Oscillator


D. D. GANJI, M. GORJI, S. SOLEIMANI, M. ESMAEILPOUR. Solution of nonlinear cubic-quintic Duffing oscillators using He’s Energy Balance Method[J]. Journal of Zhejiang University Science A, 2009, 10(9): 1263-1268.

@article{title="Solution of nonlinear cubic-quintic Duffing oscillators using He’s Energy Balance Method",
author="D. D. GANJI, M. GORJI, S. SOLEIMANI, M. ESMAEILPOUR",
journal="Journal of Zhejiang University Science A",
volume="10",
number="9",
pages="1263-1268",
year="2009",
publisher="Zhejiang University Press & Springer",
doi="10.1631/jzus.A0820651"
}

%0 Journal Article
%T Solution of nonlinear cubic-quintic Duffing oscillators using He’s Energy Balance Method
%A D. D. GANJI
%A M. GORJI
%A S. SOLEIMANI
%A M. ESMAEILPOUR
%J Journal of Zhejiang University SCIENCE A
%V 10
%N 9
%P 1263-1268
%@ 1673-565X
%D 2009
%I Zhejiang University Press & Springer
%DOI 10.1631/jzus.A0820651

TY - JOUR
T1 - Solution of nonlinear cubic-quintic Duffing oscillators using He’s Energy Balance Method
A1 - D. D. GANJI
A1 - M. GORJI
A1 - S. SOLEIMANI
A1 - M. ESMAEILPOUR
J0 - Journal of Zhejiang University Science A
VL - 10
IS - 9
SP - 1263
EP - 1268
%@ 1673-565X
Y1 - 2009
PB - Zhejiang University Press & Springer
ER -
DOI - 10.1631/jzus.A0820651


Abstract: 
In this study, He’s energy Balance Method (EBM) was applied to solve strong nonlinear Duffing oscillators with cubic-quintic nonlinear restoring force. The complete EBM solution procedure of the cubic-quintic Duffing oscillator equation is presented. For illustration of effectiveness and convenience of the EBM, different cases of cubic-quintic Duffing oscillator with different parameters of α, β and γ were compared with the exact solution. We found that the solutions were valid for small as well as large amplitudes of oscillation. The results show that the EBM is very convenient and precise, so it can be widely applicable in engineering and other sciences.

Darkslateblue:Affiliate; Royal Blue:Author; Turquoise:Article

Reference

[1] Bormashenko, E., Whyman, G., 2008. Variational approach to wetting problems: calculation of a shape of sessile liquid drop deposited on a solid substrate in external field. Chemical Physics Letters, 463(1-3):103-105.

[2] Chen, Y.M., Liu, J.K., 2007. A new method based on the harmonic balance method for nonlinear oscillators. Physics Letters A, 368:371-378.

[3] D′Acunto M., 2006. Determination of limit cycles for a modified van der Pol oscillator. Mechanics Research Communications, 33:93-100.

[4] Ganji, D.D., Rafei, M., 2006. Solitary wave solutions for a generalized Hirota-Satsuma coupled KdV equation by homotopy perturbation method. Physics Letter A, 356:131-137.

[5] Ganji, D.D., Rajabi, A., 2006. Assessment of homotopy-perturbation and perturbation methods in heat radiation equations. International Communications in Heat and Mass Transfer, 33:391-400.

[6] Ganji, D.D. Sadighi, A., 2006. Application of He’s methods to nonlinear coupled systems of reactions. International Journal of Nonlinear Science and Numerical Simulation, 7(4):411-418.

[7] Ganji, D.D., Nourollahi, M., Rostamian, M., 2007. A comparison of variational iteration method with Adomian’s Decomposition Method in some highly nonlinear equations. International Journal of Science and Technology, 2(2):179-188.

[8] Gorji, M., Ganji, D.D., Soleimani, S., 2007. New application of He’s homotopy perturbation method. International Journal of Nonlinear Science and Numerical Simulation, 8(3):319-328.

[9] Ganji, S.S., Ganji, D.D., Ganji, Z.Z., Karimpour, S., 2008. Periodic solution for strongly nonlinear vibration systems by He’s Energy Balance Method. Acta Applied Mathematics, 106(1)79-92.

[10] Gottlieb, H.P.W., 2006. Harmonic balance approach to limit cycles for nonlinear jerk equations. Journal of Sound and Vibration, 297:243-250.

[11] He, J.H., 1999a. Some new approaches to Duffing equation with strongly and high order nonlinearity (II) parameterized perturbation technique. Communications in Nonlinear Science and Numerical Simulation, 4:81-82.

[12] He, J.H., 1999b. Variational iteration method—a kind of nonlinear analytical technique: some examples. International Journal of Nonlinear Mechanics, 34:699-708.

[13] He, J.H., 2000. A review on some new recently developed nonlinear analytical techniques. International Journal of Nonlinear Science and Numerical Simulation, 1:51-70.

[14] He, J.H., 2001. Modified Lindstedt-Poincare, methods for some strongly nonlinear oscillations, Part III: double series expansion. International Journal of Non-linear Science and Numerical Simulation, 2:317-320.

[15] He, J.H., 2002a. Modified Lindstedt-Poincare methods for some strongly nonlinear oscillations, Part I: expansion of a constant. International Journal Non-linear Mechanic, 37:309-314.

[16] He, J.H., 2002b. Preliminary report on the energy balance for nonlinear oscillations. Mechanics Research Communications, 29:107-118.

[17] He, J.H., 2003. Determination of limit cycles for strongly nonlinear oscillators. Physical Review Letters, 90(17), Article No. 174301.

[18] He, J.H., 2006a. Determination of limit cycles for strongly nonlinear oscillators. Physic Review Letter, 90:174-181.

[19] He, J.H., 2006b. Non-perturbative Methods for Strongly Nonlinear Problems. PhD Thesis, de-Verlag im Internet GmbH, Berlin.

[20] He, J.H., 2006c. Some asymptotic methods for strongly nonlinear equations. International Journal of Modern Physic B, 20:1141-1199.

[21] Hu, H., 2006. Solution of a quadratic nonlinear oscillator by the method of harmonic balance. Journal of Sound and Vibration, 293:462-468.

[22] He, J.H., X.H, Wu., 2006. Construction of solitary solution and compaction-like solution by variational iteration method. Chaos Solitons & Fractals, 29:108-113.

[23] He, J.H., 2007. Variational approach for nonlinear oscillators. Chaos Solitons & Fractals, 34:1430-1439.

[24] Hu, H., Tang, J.H., 2006. Solution of a Duffing-harmonic oscillator by the method of harmonic balance. Journal of Sound and Vibration, 294:637-639.

[25] Itovich, G.R., Moiola, J.L., 2005. On period doubling bifurcations of cycles and the harmonic balance method. Chaos Solitons & Fractals, 27:647-665.

[26] Jordan, D.W., Smith, P., 1999. Nonlinear Ordinary Differential Equations: an Introduction to Dynamical systems. Oxford, New York.

[27] Lai, S.K., Lam, C.W., Wu, B.S., Wang, C., Zeng, Q.C., He, X.F., 2008. Newton-harmonic balancing approach for accurate solutions to nonlinear cubic-quintic Duffing oscillators. Applied Mathematical Modeling, 33(2):852-866.

[28] Lim, C.W., Lai, S.K., 2006. Accurate higher-order analytical approximate solutions to non-conservative nonlinear oscillators and application to van der Pol damped oscillators. International Journal of Mechanical Sciences, 48:483-492.

[29] Mickens, R.E., 1996. Oscillations in Planar Dynamic Systems. World Scientific, Singapore.

[30] Nayfeh, A.H., Mook, D.T., 1995. Nonlinear Oscillations. Wiley Classics Library Edition, New York.

[31] Ozis, T., Yıldırım, A., 2007. Determination of the frequency-amplitude relation for a Duffing-harmonic oscillator by the energy balance method. Computers and Mathematics with Applications, 54:1184-1187.

[32] Pashaei, H., Ganji, D.D., Akbarzade, M., 2008. Application of the Energy Balance Method for strongly nonlinear oscillators. Progress in Electromagnetics Research M, 2:47-56.

[33] Rafei, M., Ganji, D.D., Daniali, H., Pashaei, H., 2007. The variational iteration method for nonlinear oscillators with discontinuities. Journal of Sound and Vibration, 305:614-620.

[34] Stoker, J.J., 1992. Nonlinear Vibrations in Mechanical and Electrical Systems. Wiley, New York.

[35] Varedi, S.M., Hosseini, M.J., Rahimi, M., Ganji, D.D., 2007. He’s variational iteration method for solving a semi-linear inverse parabolic Equation. Physics Letters A, 370:275-280.

[36] Wang, S.Q., He, J.H., 2008. Nonlinear oscillator with discontinuity by parameter-expansion method. Chaos Soliton and Fractals, 35:688-691.

[37] Wu, Y., 2007. Variational approach to higher-order water-wave equations. Chaos Solitons & Fractals, 32:195-203.

[38] Xu, L., 2008. Variational approach to solitons of nonlinear dispersive K(m, n) equations. Chaos Solitons & Fractals, 37:137-143.

Open peer comments: Debate/Discuss/Question/Opinion

<1>

D.D. Ganji@University of Mazandaran<ddg_davood@yahoo.com>

2010-07-02 19:16:46

Thank you for your E-mail

Please provide your name, email address and a comment





Journal of Zhejiang University-SCIENCE, 38 Zheda Road, Hangzhou 310027, China
Tel: +86-571-87952783; E-mail: cjzhang@zju.edu.cn
Copyright © 2000 - 2022 Journal of Zhejiang University-SCIENCE