Table of Contents Author Guidelines Submit a Manuscript
Journal of Applied Mathematics
Volume 2012, Article ID 286290, 16 pages
http://dx.doi.org/10.1155/2012/286290
Research Article

Analytical Approximate Solutions for the Cubic-Quintic Duffing Oscillator in Terms of Elementary Functions

1Departamento de Física, Ingeniería de Sistemas y Teoría de la Señal, Universidad de Alicante, Apartado 99, 03080 Alicante, Spain
2Instituto Universitario de Física Aplicada a las Ciencias y las Tecnologías, Universidad de Alicante, Apartado 99, 03080 Alicante, Spain
3Departamento de Ciencias Médicas, Facultad de Medicina, Universidad de Castilla-La Mancha, C/Almansa No. 14, 02006 Albacete, Spain
4Departamento de Física Aplicada, Escuela Superior de Ingeniería Informática, Universidad de Castilla-La Mancha, Avenida de España s/n, 02071 Albacete, Spain

Received 28 June 2012; Accepted 13 August 2012

Academic Editor: Livija Cveticanin

Copyright © 2012 A. Beléndez et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. S. K. Lai, C. W. Lim, B. S. Wu, C. Wang, Q. C. Zeng, and X. F. He, “Newton-harmonic balancing approach for accurate solutions to nonlinear cubic-quintic Duffing oscillators,” Applied Mathematical Modelling. Simulation and Computation for Engineering and Environmental Systems, vol. 33, no. 2, pp. 852–866, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  2. J. I. Ramos, “On Linstedt-Poincaré technique for the quintic Duffing equation,” Applied Mathematics and Computation, vol. 193, no. 2, pp. 303–310, 2007. View at Publisher · View at Google Scholar
  3. C. W. Lim, R. Xu, S. K. Lai, Y. M. Yu, and Q. Yang, “Nonlinear free vibration of an elastically-restrained beam with a point mass via the Newton-harmonic balancing approach,” International Journal of Nonlinear Sciences and Numerical Simulation, vol. 10, no. 5, pp. 661–674, 2009. View at Publisher · View at Google Scholar
  4. A. Beléndez, G. Bernabeu, J. Francés, D. I. Méndez, and S. Marini, “An accurate closed-form approximate solution for the quintic Duffing oscillator equation,” Mathematical and Computer Modelling, vol. 52, no. 3-4, pp. 637–641, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  5. J. H. Denman, “An approximate equivalent linearization technique for nonlinear oscillations,” Journal of Applied Mechanics, vol. 36, no. 2, pp. 358–360, 1969. View at Publisher · View at Google Scholar
  6. R. E. Jonckheere, “Determination of the period of nonlinear oscillations by means of Chebyshev polynomials,” Zeitschrift fur Angewandte Mathematik und Mechanik, vol. 51, pp. 389–393, 1971. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  7. S. Bravo-Yuste, “Cubication of nonlinear oscillators using the principle of harmonic balance,” International Journal of Non-Linear Mechanics, vol. 27, no. 3, pp. 347–356, 1992. View at Publisher · View at Google Scholar
  8. S. Bravo-Yuste and A. Martín-Sánchez, “A weighted mean-square method of “cubication” for nonlinear oscillators,” Journal of Sound and Vibration, vol. 134, no. 3, pp. 423–433, 1989. View at Publisher · View at Google Scholar
  9. A. Beléndez, M. L. Álvarez, E. Fernández, and I. Pascual, “Cubication of conservative nonlinear oscillators,” European Journal of Physics, vol. 30, no. 5, pp. 973–981, 2009. View at Publisher · View at Google Scholar
  10. A. Beléndez, D. I. Méndez, E. Fernández, S. Marini, and I. Pascual, “An explicit approximate solution to the Duffing-harmonic oscillator by a cubication method,” Physics Letters A, vol. 373, pp. 2805–2809, 2009. View at Publisher · View at Google Scholar
  11. A. Elías-Zúñiga, O. Martínez-Romero, and R. Córdoba-Díaz, “Approximate solution for the Duffing-harmonic oscillator by the enhanced cubication method,” Mathematical Problems in Engineering, vol. 2012, Article ID 618750, 12 pages, 2012. View at Publisher · View at Google Scholar
  12. J. Cai, X. Wu, and Y. P. Li, “An equivalent nonlinearization method for strongly nonlinear oscillations,” Mechanics Research Communications, vol. 32, no. 5, pp. 553–560, 2005. View at Publisher · View at Google Scholar
  13. S. Bullett and J. Stark, “Renormalizing the simple pendulum,” SIAM Review, vol. 35, no. 4, pp. 631–640, 1993. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  14. A. Beléndez, E. Arribas, A. Márquez, M. Ortuño, and S. Gallego, “Approximate expressions for the period of a simple pendulum using a Taylor series expansion,” European Journal of Physics, vol. 32, no. 5, pp. 1303–1310, 2011. View at Publisher · View at Google Scholar
  15. R. E. Mickens, Oscillations in Planar Dynamic Systems, World Scientific, Singapore, Singapore, 1996.
  16. E. W. Weisstein, “Arithmetic-Geometric Mean,” From MathWorld—A Wolfram Web Resource, http://mathworld.wolfram.com/Arithmetic-GeometricMean.html.
  17. C. G. Carvalhaes and P. Suppes, “Approximation for the period of the simple pendulum based on the arithmetic-geometric mean,” American Journal of Physics, vol. 76, no. 12, pp. 1150–1154, 2008. View at Publisher · View at Google Scholar
  18. M. Febbo, “A finite extensibility nonlinear oscillator,” Applied Mathematics and Computation, vol. 217, no. 14, pp. 6464–6475, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH