Solutions for Conservative Nonlinear Oscillators Using an Approximate Method Based on Chebyshev Series Expansion of the Restoring Force

• Authors: A. Beléndez , A. Hernández , T. Beléndez , C. Pascual , M. Alvarez , E. Arribas
• Languages of publication: EN

Abstracts

EN

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.

Keywords

EN

02.30.Hq; 02.30.Mv; 05.45.-a

Discipline

• 05.45.-a: Nonlinear dynamics and chaos(see also section 45 Classical mechanics of discrete systems; for chaos in fluid dynamics, see 47.52.+j; for chaos in superconductivity, see 74.40.De)
• 02.30.Mv: Approximations and expansions
• 02.30.Hq: Ordinary differential equations

• Publisher: Institute of Physics, Polish Academy of Sciences
• Journal: Acta Physica Polonica A
• Year: 2016
• Volume: 130
• Issue: 3
• Pages: 667-678

Dates

published

2016-09

received

2015-06-16

(unknown)

2016-08-08

Contributors

author

A. Beléndez

• Departamento de Física, Ingeniería de Sistemas y Teoría de la Señal, Universidad de Alicante, Apartado 99, E-03080 Alicante, Spain
• Instituto Universitario de Física Aplicada a las Ciencias y las Tecnologías, Universidad de Alicante, Apartado 99, E-03080 Alicante, Spain

author

A. Hernández

• Departamento de Física, Ingeniería de Sistemas y Teoría de la Señal, Universidad de Alicante, Apartado 99, E-03080 Alicante, Spain
• Instituto Universitario de Física Aplicada a las Ciencias y las Tecnologías, Universidad de Alicante, Apartado 99, E-03080 Alicante, Spain

author

T. Beléndez

• Departamento de Física, Ingeniería de Sistemas y Teoría de la Señal, Universidad de Alicante, Apartado 99, E-03080 Alicante, Spain
• Instituto Universitario de Física Aplicada a las Ciencias y las Tecnologías, Universidad de Alicante, Apartado 99, E-03080 Alicante, Spain

author

C. Pascual

• Departamento de Física, Ingeniería de Sistemas y Teoría de la Señal, Universidad de Alicante, Apartado 99, E-03080 Alicante, Spain
• Instituto Universitario de Física Aplicada a las Ciencias y las Tecnologías, Universidad de Alicante, Apartado 99, E-03080 Alicante, Spain

author

M. Alvarez

• Departamento de Física, Ingeniería de Sistemas y Teoría de la Señal, Universidad de Alicante, Apartado 99, E-03080 Alicante, Spain
• Instituto Universitario de Física Aplicada a las Ciencias y las Tecnologías, Universidad de Alicante, Apartado 99, E-03080 Alicante, Spain

author

E. Arribas

• Departamento de Física Aplicada, Escuela Superior de Ingeniería Informática, Universidad de Castilla-La Mancha, Avda. España s/n, E-02071 Albacete, Spain

References

• [1] A.H. Nayfeh, Problems in Perturbation, Wiley, New York 1985
• [2] R.E. Mickens, Oscillations in Planar Dynamics Systems, World Sci., Singapore 1996
• [3] M.H. Pashai, I. Khatami, N. Tolou, Math. Prob. Eng. 2008, 956170 (2008), doi: 10.1155/2008/956170
• [4] S.J. Liao, Appl. Math. Mech. 19, 885 (1998), doi: 10.1007/BF02457955
• [5] A. Elías-Zúñiga, O. Martínez-Romero, Math. Prob. Eng. 2013, 620591 (2013), doi: 10.1155/2013/620591
• [6] M.O. Kaya, S. Altay Demirbağ, F. Özen Zengin, Math. Prob. Eng. 2009, 450862 (2009), doi: 10.1155/2009/450862
• [7] V. Marinca, N. Herişanu, Chaos Soliton Fract. 37, 144 (2008), doi: 10.1016/j.chaos.2006.08.033
• [8] S.S. Motsa, P. Sibanda, Math. Prob. Eng. 2012, 693453 (2012), doi: 10.1155/2012/693453
• [9] C. Bota, B. Căruntu, O. Bundău, Math. Prob. Eng. 2014, 513473 (2014), doi: 10.1155/2014/513473
• [10] A. Beléndez, M.L. Alvarez, E. Fernández, I. Pascual, Eur. J. Phys. 30, 973 (2009), doi: 10.1088/0143-0807/30/5/006
• [11] J.H. Denman, J. Appl. Mech. 36, 358 (1969), doi: 10.1115/1.3564651
• [12] R.E. Jonckheere, Z. Angew. Math. Mech. 51, 389 (1971), doi: 10.1002/zamm.19710510508
• [13] Orthogonal Polynomials, in: Handbook of Mathematical Functions with Formulas, Graphics and Mathematical Tables, Eds. M. Abramowitz, I.A. Stegun, 9th ed., Dover, New York 1972, Ch. 22, p. 771
• [14] E.W. Weisstein, Chebyshev Polynomial of the First Kind, from MathWorld - A Wolfram web resource, http://mathworld.wolfram.com/ChebyshevPolynomialoftheFirstKind.html
• [15] A. Beléndez, M.L. Alvarez, E. Fernández, I. Pascual, Eur. J. Phys. 30, 259 (2009), doi: 10.1088/0143-0807/30/2/004
• [16] B.S. Wu, W.P. Sun, C.W. Lim, Int. J. Non-Linear Mech. 41, 766 (2006), doi: 10.1016/j.ijnonlinmec.2006.01.006
• [17] A. Beléndez, E. Gimeno, M.L. Alvarez, S. Gallego, M. Ortuño, D.I. Méndez, Int. Nonlin. Sci. Num. Simulat. 10, 13 (2009), doi: 10.1515/IJNSNS.2009.10.1.13
• [18] A. Beléndez, A. Hernández, T. Beléndez, C. Neipp, A. Márquez, Phys. Lett. A 372, 2010 (2008), doi: 10.1016/j.physleta.2007.10.081
• [19] M. Febbo, Appl. Math. Comput. 217, 6464 (2011), doi: 10.1016/j.amc.2011.01.011
• [20] I. Kovacic, Z. Rakaric, L. Cveticanin, Appl. Math. Comput. 217, 3944 (2010), doi: 10.1016/j.amc.2010.09.058
• [21] A. Beléndez, E. Gimeno, M.L. Alvarez, M.S. Yebra, D.I. Méndez, Int. J. Comput. Math. 87, 1497 (2010), doi: 10.1080/00207160802380942
• [22] A. Beléndez, C. Pascual, T. Beléndez, A. Hernández, Nonlinear Anal. Real World Appl. 10, 416 (2009), doi: 10.1016/j.nonrwa.2007.10.002
• [23] R.E. Mickens, J. Sound Vibrat. 292, 964 (2006), doi: 10.1016/j.jsv.2005.08.020
• [24] C.W. Lim, B.S. Wu, J. Sound Vibrat. 281, 1157 (2005), doi: 10.1016/j.jsv.2004.04.030
• [25] A. Beléndez, C. Pascual, S. Gallego, M. Ortuño, C. Neipp, Phys. Lett. A 371, 421 (2007), doi: 10.1016/j.physleta.2007.06.042
• [26] A. Beléndez, A. Hernández, T. Beléndez, M.L. Alvarez, S. Gallego, M. Ortuño, C. Neipp, J. Sound Vibrat. 302, 1018 (2007), doi: 10.1016/j.jsv.2006.12.011
• [27] W.P. Sun, B.S. Wu, C.W. Lim, J. Sound Vibrat. 300, 1042 (2007), doi: 10.1016/j.jsv.2006.08.025
• [28] E. Gimeno, M.L. Álvarez, M.S. Yebra, J. Rosa-Herranz, A. Beléndez, Int. Nonlin. Sci. Num. Simulat. 10, 493 (2009), doi: 10.1515/IJNSNS.2009.10.4.493
• [29] L. Cveticanin, T. Pogány, J. Appl. Math. 2002, 649050 (2012), doi: 10.1155/2012/649050
• [30] A. Beléndez, E. Arribas, J. Francés, I. Pascual, Appl. Math. Comput. 218, 6168 (2012), doi: 10.1016/j.amc.2011.12.012
• [31] A. Beléndez, E. Arribas, J. Francés, I. Pascual, Math. Comput. Mod. 54, 3204 (2011), doi: 10.1016/j.mcm.2011.06.024
• [32] E.W. Weisstein, Hypergeometric Function, from MathWorld - A Wolfram Web Resource, http://mathworld.wolfram.com/HypergeometricFunction.html
• [33] A. Beléndez, T. Beléndez, F.J. Martínez, C. Pascual, M.L. Alvarez, E. Arribas, Nonlinear Dyn. 85, 1 (2016), doi: 10.1007/s11071-016-2986-8
• [34] A. Beléndez, D.I. Méndez, E. Fernández, S. Marini, I. Pascual, Phys. Lett. A 373, 2805 (2009), doi: 10.1016/j.physleta.2009.05.074
• [35] A. Beléndez, G. Bernabeu, J. Francés, D.I. Méndez, S. Marini, Math. Comput. Mod. 52, 637 (2010), doi: 10.1016/j.mcm.2010.04.010
• [36] A. Beléndez, M.L. Alvarez, J. Francés, S. Bleda, T. Beléndez, A. Nájera, E. Arribas, J. Appl. Math. 2012, 286290 (2012), doi: 10.1155/2012/286290
• [37] A. Elías-Zúñiga, O. Martínez-Romero, R.K. Córdoba-Díaz, Math. Prob. Eng. 2012, 618750 (2012), doi: 10.1155/2012/618750
• [38] A. Elías-Zúñiga, O. Martínez-Romero, Math. Prob. Eng. 2013, 842423 (2013), doi: 10.1155/2013/842423
• [39] A. Elías-Zúñiga, Appl. Math. Comput. 243, 849 (2014), doi: 10.1016/j.amc.2014.05.085

Identifiers

DOI

10.12693/APhysPolA.130.667