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Approximate solution for the Duffing-harmonic oscillator by the enhanced cubication method. (English) Zbl 1264.34067
Summary: The cubication and the equivalent nonlinearization methods are used to replace the original Duffing-harmonic oscillator by an approximate Duffing equation in which the coefficients for the linear and cubic terms depend on the initial oscillation amplitude. It is shown that this procedure leads to angular frequency values with a maximum relative error of 0.055%. This value is 21% lower than the relative errors attained by previously developed approximate solutions.

34C15 Nonlinear oscillations and coupled oscillators for ordinary differential equations
34A45 Theoretical approximation of solutions to ordinary differential equations
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[1] A. H. Nayfeh, Perturbation Methods, Wiley-Interscience, New York, NY, USA, 1973. · Zbl 0265.35002
[2] R. E. Mickens, “Mathematical and numerical study of the Duffing-harmonic oscillator,” Journal of Sound and Vibration, vol. 244, no. 3, pp. 563-567, 2001. · Zbl 1237.65082
[3] R. H. Rand, Lecture Notes on Nonlinear Vibrations, 2001, http://www.tam.cornell.edu/randdocs/nlvib36a.pdf.
[4] P. Amore and A. Aranda, “Improved Lindstedt-Poincaré method for the solution of nonlinear problems,” Journal of Sound and Vibration, vol. 283, no. 3-5, pp. 1115-1136, 2005. · Zbl 1237.70097
[5] S. B. Tiwari, B. N. Rao, N. S. Swamy, K. S. Sai, and H. R. Nataraja, “Analytical study on a Duffing-harmonic oscillator,” Journal of Sound and Vibration, vol. 285, no. 4-5, pp. 1217-1222, 2005. · Zbl 1238.34070
[6] H. Hu, “Solution of a quadratic nonlinear oscillator by the method of harmonic balance,” Journal of Sound and Vibration, vol. 293, no. 1-2, pp. 462-468, 2006. · Zbl 1243.34048
[7] C. W. Lim, B. S. Wu, and W. P. Sun, “Higher accuracy analytical approximations to the Duffing-harmonic oscillator,” Journal of Sound and Vibration, vol. 296, no. 4-5, pp. 1039-1045, 2006. · Zbl 1243.34021
[8] T. Özi\cs and A. Yildirim, “Determination of the frequency-amplitude relation for a Duffing-harmonic oscillator by the energy balance method,” Computers & Mathematics with Applications, vol. 54, no. 7-8, pp. 1184-1187, 2007. · Zbl 1147.34321
[9] A. Beléndez, E. Gimeno, M. L. Álvarez, D. I. Méndez, and A. Hernández, “Application of a modified rational harmonic balance method for a class of strongly nonlinear oscillators,” Physics Letters A, vol. 372, no. 39, pp. 6047-6052, 2008. · Zbl 1223.34055
[10] A. Beléndez, C. Pascual, E. Fernandez, C. Neipp, and T. Belendez, “Higherorder approximate solutions to the relativistic and Duffing-harmonic oscillators by modified He’s homotopy methods,” Physica Scripta, vol. 77, Article ID 065004, 14 pages, 2008. · Zbl 1175.70023
[11] A. Beléndez, M. L. Alvarez, E. Fernandez, and I. Pascual, “Cubication of conservative nonlinear oscillators,” European Journal of Physics, vol. 30, pp. 973-981, 2009. · Zbl 1257.65048
[12] 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. · Zbl 1233.70008
[13] 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. · Zbl 1201.34019
[14] J. Cai, X. Wu, and Y. P. Li, “An equivalent nonlinearization method for strongly nonlinear oscillations,” Mechanics Research Communications, vol. 32, pp. 553-560, 2005. · Zbl 1192.70038
[15] S. B. Yuste and Á. M. Sánchez, “A weighted mean-square method of “cubication” for nonlinear oscillators,” Journal of Sound and Vibration, vol. 134, no. 3, pp. 423-433, 1989. · Zbl 1235.70168
[16] S. B. 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. · Zbl 0766.70016
[17] S. C. Sinha and P. Srinivasan, “A weighted mean square method of linearization in non-linear oscillations,” Journal of Sound and Vibration, vol. 16, pp. 139-148, 1971. · Zbl 0218.70009
[18] S. V. S. Narayana Murty and B. Nageswara Rao, “Further comments on ‘harmonic balance comparison of equation of motion and energy methods’,” Journal of Sound and Vibrations, vol. 183, no. 3, pp. 563-565, 1995.
[19] G. Radhakrishnan, B. N. Rao, and M. S. Sarma, “On the uniqueness of angular frequency using harmonic balance from the equation of motion and the energy relation,” Journal of Sound and Vibration, vol. 200, no. 3, pp. 367-370, 1997. · Zbl 1235.70161
[20] Z. Guo, A. Y. T. Leung, and H. X. Yang, “Iterative homotopy harmonic balancing approach for conservative oscillator with strong odd-nonlinearity,” Applied Mathematical Modelling, vol. 35, no. 4, pp. 1717-1728, 2011. · Zbl 1217.65163
[21] A. Elías-Zúñiga, C. A. Rodríguez, and O. Martínez Romero, “On the solution of strong nonlinear oscillators by applying a rational elliptic balance method,” Computers & Mathematics with Applications, vol. 60, no. 5, pp. 1409-1420, 2010. · Zbl 1201.34056
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