Newton-harmonic balancing approach for accurate solutions to nonlinear cubic-quintic Duffing oscillators

Applied Mathematical Modelling

Volumn 33, Issue 2, 2009, Pages 852-866

  Lai, S K ^a   Lim, C W ^a   Wu, B S ^b   Wang, C ^c   Zeng, Q C ^d   He, X F ^e  

^a CITY UNIVERSITY OF HONG KONG   (Hong Kong)

^b JILIN UNIVERSITY   (China)

^c HUAZHONG UNIVERSITY OF SCIENCE AND TECHNOLOGY   (China)

^d Hong Kong and Shanghai Banking Corporation   (Hong Kong)

^e Dongfang Electric Autocontrol Engineering Company Ltd   (China)

Author keywords

Duffing equation; Harmonic Balance method; Newton's method

Indexed keywords

CONTROL NONLINEARITIES; CONTROL THEORY; HARMONIC ANALYSIS; LINEAR EQUATIONS; NEWTON-RAPHSON METHOD; OSCILLATORS (ELECTRONIC); OSCILLATORS (MECHANICAL); SOLUTIONS;

ACCURATE; ANALYTICAL EXPRESSIONS; ANALYTICAL SOLUTIONS; APPROXIMATE SOLUTIONS; CUBIC NONLINEARITIES; DUFFING EQUATION; DUFFING OSCILLATORS; GOVERNING DIFFERENTIAL EQUATIONS; HARMONIC BALANCE METHOD; HARMONIC BALANCE METHODS; HARMONIC BALANCING; ILLUSTRATIVE EXAMPLES; LARGE AMPLITUDES; LINEAR ALGEBRAIC EQUATIONS; NEW APPROACHES; NEWTON'S METHOD; NEWTON'S METHODS; NON-LINEARITY; NONLINEAR; NONLINEAR ALGEBRAIC EQUATIONS; NONLINEAR RESTORING FORCES; PERIODIC SOLUTIONS; PERTURBATION METHODS; QUINTIC NONLINEARITIES;

NONLINEAR EQUATIONS;

EID: 54049138060     PISSN: 0307904X     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.apm.2007.12.012     Document Type: Article

References (28)

1. Nayfeh A.H., and Mook D.T. Nonlinear Oscillations (1979), Wiley, New York
2. Nayfeh A.H. Problems in Perturbation (1985), Wiley, New York
3. Hagedorn P. Non-linear Oscillations (1988), Clarendon, Oxford translated by Wolfram Stadler
4. Cheung Y.K., Chen S.H., and Lau S.L. Application of the incremental harmonic balance method to cubic non-linearity systems. J. Sound Vib. 140 (1990) 273-286
5. Cheung Y.K., Chen S.H., and Lau S.L. A modified Lindstedt-Poincaré method for certain strongly non-linear oscillators. Int. J. Non-Linear Mech. 26 (1991) 367-378
6. Leung A.Y.T., and Chui S.K. Non-linear vibration of coupled Duffing oscillators by an improved incremental harmonic balance method. J. Sound Vib. 181 (1995) 619-633
7. Mickens R.E. Oscillations in Planar Dynamic Systems (1996), Word Scientific, Singapore
8. Qaisi M.I. A power series approach for the study of periodic motion. J. Sound Vib. 196 (1996) 401-406
9. Liao S.J., and Chwang A.T. Application of homotopy analysis method in nonlinear oscillations. J. Appl. Mech.-Trans. ASME 65 (1998) 914-922
10. Gottlieb H.P.W. Harmonic balance approach to periodic solutions of non-linear Jerk equations. J. Sound Vib. 271 (2004) 671-683
11. Amore P., and Aranda A. Improved Lindstedt-Poincaré method for the solution of nonlinear problems. J. Sound Vib. 283 (2005) 1115-1136
12. Xiong Z.G., and Hu H. Simplified continuous finite element method for a class of nonlinear oscillating equations. J. Sound Vib. 287 (2005) 367-373
13. Hu H. Solutions of nonlinear oscillators with fractional powers by an iteration procedure. J. Sound Vib. 294 (2006) 608-614
14. Hu H. Solutions of a quadratic nonlinear oscillator: iteration procedure. J. Sound Vib. 298 (2006) 1159-1165
15. Xie F., and Gao X. Exact travelling wave solutions for a class of nonlinear partial differential equations. Chaos Soliton Fract 19 (2004) 1113-1117
16. Lim C.W., and Wu B.S. A new analytical approach to the Duffing-harmonic oscillator. Phys. Lett. A 311 (2003) 365-373
17. Hu H. Solutions of the Duffing-harmonic oscillator by an iteration procedure. J. Sound Vib. 298 (2006) 446-452
18. Wagner H. Large-amplitude free vibrations of a beam. J. Appl. Mech.-Trans. ASME 32 (1965) 887-892
19. Hamdan M.N., and Shabaneh N.H. On the large amplitude free vibrations of a restrained uniform beam carrying an intermediate lumped mass. J. Sound Vib. 199 (1997) 711-736
20. Lenci S., Menditto G., and Tarantino A.M. Homoclinic and heteroclinic bifurcations in the non-linear dynamics of a beam resting on an elastic substrate. Int. J. Non-Linear Mech. 34 (1999) 615-632
21. Yan Z. A new sine-Gordon equation expansion algorithm to investigate some special nonlinear differential equations. Chaos Soliton Fract 23 (2005) 767-775
22. Huang D.J., and Zhang H.Q. Link between travelling waves and first order nonlinear ordinary differential equation with a sixth-degree nonlinear term. Chaos Soliton Fract 29 (2006) 928-941
23. Maǐmistov A.I. Propagation of an ultimately short electromagnetic pulse in a nonlinear medium described by the fifth-order Duffing model. Opt. Spectrosc. 94 (2003) 251-257
24. Wu B.S., and Li P.S. A method for obtaining approximate analytic periods for a class of nonlinear oscillators. Meccanica 36 (2001) 167-176
25. Wu B.S., Lim C.W., and Ma Y.F. Analytical approximation to large-amplitude oscillation of a non-linear conservative system. Int. J. Non-Linear Mech. 38 (2003) 1037-1043
26. Lim C.W., and Wu B.S. Accurate higher-order approximations to frequencies of nonlinear oscillators with fractional powers. J. Sound Vib. 281 (2005) 1157-1162
27. Lim C.W., Wu B.S., and He L.H. A new approximate analytical approach for dispersion relation of the nonlinear Klein-Gordon equation. Chaos 11 (2001) 843-848
28. Wu B.S., Sun W.P., and Lim C.W. An analytical approximate technique for a class of strongly nonlinear oscillators. Int. J. Non-Linear Mech. 41 (2006) 766-774