volume | PIER Journals

Abstract

A He's Energy balance method (EBM) is used to calculate the periodic solutions of nonlinear oscillators with fractional potential. Some examples are given to illustrate the effectiveness and convenience of the method. We find this EBM works very well for the whole range of initial amplitudes, and the excellent agreement of the approximate frequencies and periodic solutions with the Exact or other analytical solutions has been demonstrated and discussed. Comparison of the result obtained using this method with that obtained by Exact or other analytical solutions reveal that the EBM is very effective and convenient and can therefore be found widely applicable in engineering and other science.

1. Nayfeh, A. H., Perturbation Methods, Wiley-Interscience, 1973.

2. Mickens, R. E., Oscillations in Planar Dynamic Systems, Scientific, 1966.

3. Jordan, D. W. and P. Smith, Nonlinear Ordinary Differential Equations, Clarendon Press, 1987.

4. He, J. H., Non-perturbative methods for strongly nonlinear problems, Dissertation.de-Verlag im Internet GmbH, 2006.

5. He, J. H., "Homotopy perturbation method for bifurcation on nonlinear problems ," International Journal of Non-Linear Science and Numerical Simulation, Vol. 6, 207-208, 2005.

6. Ganji, D. D. and A. Sadighi, "Application of He's homotopyprturbation method to nonlinear coupled systems of reactiondiffusion equations," Int. J. Nonl. Sci. and Num. Simu, Vol. 7, No. 4, 411-418, 2006.

7. He, J. H., "The homotopy perturbation method for nonlinear oscillators with discontinuities," Applied Mathematics and Computation, Vol. 151, 287-292, 2004.
doi:10.1016/S0096-3003(03)00341-2

8. Belendez, A., C. Pascual, S. Gallego, M. Ortuno, and C. Neipp, "Application of a modified He's homotopy perturbation method to obtain higher-order approximations of an force nonlinear oscillator ," Physics Letters A, Vol. 371, 421-426, 2007.
doi:10.1016/j.physleta.2007.06.042

9. Alizadeh, S. R. S., G. Domairry, and S. Karimpour, "An approximation of the analytical solution of the linear and nonlinear integro-differential equations by homotopy perturbation method ," Acta Applicandae Mathematicae, doi: 10.1007/s10440-008-9261-z.

10. He, J. H., "Modified Lindstedt-Poincare methods for some strongly nonlinear oscillations, Part I: Expansion of a constant," International Journal Non-linear Mechanic, Vol. 37, 309-314, 2002.
doi:10.1016/S0020-7462(00)00116-5

11. He, J. H., "Modified Lindstedt-Poincare methods for some strongly nonlinear oscillations. Part III: Double series expansion," International Journal Non-linear Science and Numerical Simulation, Vol. 2, 317-320, 2001.

12. Ozis, T. and A. Yildirim, "Determination of periodic solution for a u^1/3 force by He's modified Lindstedt-Poincare method," Journal of Sound and Vibration, Vol. 301, 415-419, 2007.
doi:10.1016/j.jsv.2006.10.001

13. Wang, S. Q. and J. H. He, "Nonlinear oscillator with discontinuity by parameter-expansion method," Chaos & Soliton and Fractals , Vol. 35, 688-691, 2008.
doi:10.1016/j.chaos.2007.07.055

14. He, J. H., "Some asymptotic methods for strongly nonlinear equations," International Journal Modern Physic B, Vol. 20, 1141-1199, 2006.
doi:10.1142/S0217979206033796

15. Wang, S. Q. and J. H. He, "Nonlinear oscillator with discontinuity by parameter-expanding method," Chaos, Solitons & Fractals, Vol. 29, 108-113, 2006; Vol. 35, 688-691, 2008.

16. Shou, D. H. and J. H. He, "Application of parameter-expanding method to strongly nonlinear oscillators," International Journal of Nonlinear Sciences and Numerical Simulation, Vol. 8, No. 1, 121-124, 2007.

17. He, J. H., "Some new approaches to Duffing equation with strongly and high order nonlinearity (II) parameterized perturbation technique ," Communications in Nonlinear Science and Numerical Simulation, Vol. 4, 81-82, 1999.
doi:10.1016/S1007-5704(99)90065-5

18. He, J. H., "A review on some new recently developed nonlinear analytical techniques," International Journal of Nonlinear Science and Numerical Simulation, Vol. 1, 51-70, 2000.

19. Rong, H. W., X. D. Wang, W. Xu, and T. Fang, "Saturation and resonance of nonlinear system under bounded noise excitation ," Journal of Sound and Vibration, Vol. 291, 48-59, 2006.
doi:10.1016/j.jsv.2005.05.021

20. Das, S. K., P. C. Ray, and G. Pohit, "Free vibration analysis of a rotating beam with nonlinear spring and mass system," Journal of Sound and Vibration, Vol. 301, 165-188, 2007.
doi:10.1016/j.jsv.2006.09.028

21. Okuizumi, N. and K. Kimura, "Multiple time scale analysis of hysteretic systems subjected to harmonic excitation," Journal of Sound and Vibration, Vol. 272, 675-701, 2004.
doi:10.1016/S0022-460X(03)00404-8

22. Marathe, A., "Anindya Chatterjee, wave attenuation in nonlinear periodic structures using harmonic balance and multiple scales ," Journal of Sound and Vibration, Vol. 289, 871-888, 2006.
doi:10.1016/j.jsv.2005.02.047

23. Gottlieb, H. P. W., "Harmonic balance approach to limit cycles for nonlinear jerk equations," Journal of Sound and Vibration, Vol. 297, 243-250, 2006.
doi:10.1016/j.jsv.2006.03.047

24. Yuan, Z. W., F. L. Chu, and Y. L. Lin, "External and internal coupling effects of rotor's bending and torsional vibrations under unbalances," Journal of Sound and Vibration, Vol. 299, 339-347, 2007.
doi:10.1016/j.jsv.2006.06.054

25. Wiener, N., Nonlinear Problems in Random Theory, Wiley, 1958.

26. Penga, Z. K., Z. Q. Langa, S. A. Billingsa, and G. R. Tomlinson, "Comparisons between harmonic balance and nonlinear output frequency response function in nonlinear system analysis," Journal of Sound and Vibration, Vol. 311, 56-73, 2008.
doi:10.1016/j.jsv.2007.08.035

27. He, J. H., "Determination of limit cycles for strongly nonlinear oscillators," Physic Review Letter, Vol. 90, 174-181, 2006.

28. Ganji, S. S., S. Karimpour, D. D. Ganji, and Z. Z. Ganji, "Periodic solution for strongly nonlinear vibration systems by energy balance method ," Acta Applicandae Mathematicae , doi: 10.1007/s10440-008-9283-6..

29. He, J. H., "Preliminary report on the energy balance for nonlinear oscillations," Mechanics Research Communications, Vol. 29, 107-118, 2002.
doi:10.1016/S0093-6413(02)00237-9

30. Ozis, T. and A. Yildirim, "Determination of the frequencyamplitude relation for a Duffing-harmonic oscillator by the energy balance method ," Computers and Mathematics with Applications, Vol. 54, 1184-1187, 2007.
doi:10.1016/j.camwa.2006.12.064

31. Porwal, R. and N. S. Vyas, "Determination of the frequencyamplitude relation for a Duffing-harmonic oscillator by the energy balance method," Computers and Mathematics with Applications, Vol. 54, 1184-1187, 2007.

32. He, J. H., "Variational iteration method — A kind of nonlinear analytical technique: Some examples ," Int J Nonlinear Mech., Vol. 34, 699-708, 1999.
doi:10.1016/S0020-7462(98)00048-1

33. Rafei, M., D. D. Ganji, H. Daniali, and H. Pashaei, "The variational iteration method for nonlinear oscillators with discontinuities," Journal of Sound and Vibration, Vol. 305, 614-620, 2007.
doi:10.1016/j.jsv.2007.04.020

34. He, J. H. and X. H. Wu, "Construction of solitary solution and compaction-like solution by variational iteration method," Chaos, Solitons & Fractals , Vol. 29, 108-113, 2006.
doi:10.1016/j.chaos.2005.10.100

35. Varedi, S. M., M. J. Hosseini, M. Rahimi, and D. D. Ganji, "He's variational iteration method for solving a semi-linear inverse parabolic equation ," Physics Letters A, Vol. 370, 275-280, 2007.
doi:10.1016/j.physleta.2007.05.100

36. Hashemi, S. H. A., K. N. Tolou, A. Barari, and A. J. Choobbasti, "On the approximate explicit solution of linear and nonlinear non-homogeneous dissipative wave equations," Istanbul Conferences, Torque, accepted, 2008.

37. He, J. H., "Variational approach for nonlinear oscillators," Chaos, Solitons and Fractals, Vol. 34, 1430-1439, 2007.
doi:10.1016/j.chaos.2006.10.026

38. Ganji, S. S., D. D. Ganji, H. Babazadeh, and S. Karimpour, "Variational approach method for nonlinear oscillations of the motion of a rigid rod rocking back and cubic-quintic Duffing oscillators ," Progress In Electromagnetics Research M, Vol. 4, 23-32, 2008.
doi:10.2528/PIERM08061007

39. Wu, Y., "Variational approach to higher-order water-wave equations ," Chaos & Solitons and Fractals, Vol. 32, 195-203, 2007.
doi:10.1016/j.chaos.2006.05.019

40. Xu, L., "Variational approach to solitons of nonlinear dispersive equations ," Chaos, Solitons & Fractals, Vol. 37, 137-143, 2008.
doi:10.1016/j.chaos.2006.08.016

Dr. Seyedreza Ganji

Dr. Davoodi Ganji

Dr. Salim Karimpour