Application of a modified rational harmonic balance method for a class of strongly nonlinear oscillators. (English) Zbl 1223.34055

Summary: An analytical approximate technique for conservative nonlinear oscillators is proposed. This method is a modification of the rational harmonic balance method in which analytical approximate solutions have rational form. This approach gives us the frequency of the motion as a function of the amplitude of oscillation. We find that this method works very well for the whole range of parameters, and excellent agreement of the approximate frequencies with the exact one has been demonstrated and discussed. The most significant features of this method are its simplicity and its excellent accuracy for the whole range of oscillation amplitude values and the results reveal that this technique is very effective and convenient for solving conservative truly nonlinear oscillatory systems with complex nonlinearities.


34C15 Nonlinear oscillations and coupled oscillators for ordinary differential equations
70K20 Stability for nonlinear problems in mechanics
Full Text: DOI Link


[1] Mickens, R.E., Oscillations in planar dynamics systems, (1996), World Scientific Singapore · Zbl 1232.34045
[2] He, J.H., Non-perturbative methods for strongly nonlinear problems, (2006), dissertation.de-Verlag im Internet GmbH Berlin
[3] Amore, P.; Raya, A.; Fernández, F.M., Eur. J. phys., 26, 1057, (2005)
[4] He, J.H., Int. J. non-linear mech., 37, 309, (2002)
[5] Darvishi, M.T.; Karami, A.; Shin, B.C., Phys. lett. A, 372, 5381, (2008)
[6] Wang, S.Q.; He, J.H., Chaos solitons fractals, 35, 688, (2008)
[7] He, J.H., Chaos solitons fractals, 34, 1430, (2007)
[8] He, J.H.; Wu, X.H., Chaos solitons fractals, 29, 108, (2006)
[9] He, J.H., Int. J. non-linear sci. numer. simul., 6, 207, (2005)
[10] Beléndez, A.; Hernández, A.; Beléndez, T.; Fernández, E.; Álvarez, M.L.; Neipp, C., Int. J. non-linear sci. numer. simul., 8, 79, (2007)
[11] Gorji, M.; Ganji, D.D.; Soleimani, S., Int. J. non-linear sci. numer. simul., 8, 319, (2007)
[12] Beléndez, A.; Hernández, A.; Beléndez, T.; Neipp, C.; Márquez, A., Eur. J. phys., 28, 93, (2007)
[13] Beléndez, A.; Hernández, A.; Márquez, A.; Beléndez, T.; Neipp, C., Eur. J. phys., 27, 539, (2006)
[14] Rafei, M.; Ganji, D.D., Int. J. non-linear sci. numer. simul., 7, 321, (2006)
[15] Hu, H.; Tang, J.H., J. sound vibration, 294, 637, (2006)
[16] Lim, C.W.; Wu, B.S., Phys. lett. A, 311, 365, (2003) · Zbl 1055.70009
[17] Beléndez, A.; Hernández, A.; Beléndez, T.; Álvarez, M.L.; Gallego, S.; Ortuño, M.; Neipp, C., J. sound vibration, 302, 1018, (2007)
[18] He, J.H., Int. J. mod. phys. B, 20, 1141, (2006)
[19] Mickens, R.E.; Semwogerere, D., J. sound vibration, 195, 528, (1996)
[20] Beléndez, A.; Méndez, D.I.; Beléndez, T.; Hernández, A.; Álvarez, M.L., J. sound vibration, 314, 775, (2008)
[21] Ramos, J.I., J. sound vibration, (2008)
[22] Beléndez, A.; Gimeno, E.; Fernández, E.; Méndez, D.I.; Álvarez, M.L., Phys. scr., 77, 065004, (2008)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.