Vol. 5

Front:[PDF file] Back:[PDF file]
Latest Volume
All Volumes
All Issues
2008-10-27

Approximate Analytical Solutions to Nonlinear Oscillations of Non-Natural Systems Using HE's Energy Balance Method

By Davoodi Ganji, Salim Karimpour, and Seyedreza Ganji
Progress In Electromagnetics Research M, Vol. 5, 43-54, 2008
doi:10.2528/PIERM08081501

Abstract

This paper applies He's Energy balance method (EBM) to study periodic solutions of strongly nonlinear systems such as nonlinear vibrations and oscillations. The method is applied to two nonlinear differential equations. Some examples are given to illustrate the effectiveness and convenience of the method. The results are compared with exact solutions which lead us showing a good accuracy. The method can be easily extended to other nonlinear systems and can therefore be found widely applicable in engineering and other science.

Citation


Davoodi Ganji, Salim Karimpour, and Seyedreza Ganji, "Approximate Analytical Solutions to Nonlinear Oscillations of Non-Natural Systems Using HE's Energy Balance Method," Progress In Electromagnetics Research M, Vol. 5, 43-54, 2008.
doi:10.2528/PIERM08081501
http://www.jpier.org/PIERM/pier.php?paper=08081501

References


    1. Fidlin, A., Nonlinear Oscillations in Mechanical Engineering, Springer-Verlag, Berlin Heidelberg, 2006.

    2. Dimarogonas, A. D. and S. Haddad, Vibration for Engineers, Prentice-Hall, Englewood Cliffs, New Jersey, 1992.

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

    4. He, J. H., "Homotopy perturbation technique," Computer Methods in Applied Mechanics and Engineering, Vol. 178, 257-262, 1999.
    doi:10.1016/S0045-7825(99)00018-3

    5. 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

    6. Hashemi, S. H., H. R. M. Daniali, and D. D. Ganji, "Numerical simulation of the generalized Huxley equation by He's homotopy perturbation method," Applied Mathematics and Computation, Vol. 192, 157-161, 2007.
    doi:10.1016/j.amc.2007.02.128

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

    8. Nayfeh, A. H., Introduction to Perturbation Techniques, Wiley, New York, 1981.

    9. 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

    10. 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.

    11. 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

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

    13. 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

    14. 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.

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

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

    17. 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

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

    19. 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

    20. 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

    21. 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

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

    23. 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

    24. Naghipour, M., D. D. Ganji, S. H. A. Hashemi, and K. Jafari, "Analysis of non-linear oscillations systems using analytical approach," Journal of Physics, Vol. 96, 2008.

    25. 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

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

    27. Inokuti, M., et al., "General use of the Lagrange multiplier in non–linear mathematical physics," Variational Method in the Mechanics of Solids, 1978.

    28., "Generalized variational principles for ion acoustic plasma waves by He's semi-inverse method," Chaos, Solitons & Fractals, Vol. 23, No. 2, 573-576, 2005.
    doi:10.1016/j.chaos.2004.05.005

    29. He, J. H., "Variational principles for some nonlinear partial differential equations with variable coefficient," Chaos, Solitons and Fractals, Vol. 19, No. 4, 847-851, 2004.
    doi:10.1016/S0960-0779(03)00265-0

    30. Wu, B. S., C. W. Lim, and L. H. He, "A new method for approximate analytical solutions to nonlinear oscillations of nonnatural systems," Nonlinear Dynamics, Vol. 32, 1-13, 2003.
    doi:10.1023/A:1024223118496

    31. Nayfeh, A. H. and D. T. Mook, Nonlinear Oscillations, Wiley, New York, 1979.

    32. He, J. H., "Variational iteration method — Some recent results and new interpretations," Journal of Computational and Applied Mathematics, Vol. 207, 3-17, 2007.
    doi:10.1016/j.cam.2006.07.009

    33. Dehghan, M. and F. Shakeri, "Solution of an integro-differential equation arising in oscillation magnetic fields using He's homotopy perturbation method," Progress In Electromagnetics Research, Vol. 78, 361-376, 2008.
    doi:10.2528/PIER07090403

    34. 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, 2007.
    doi:10.1016/j.physleta.2007.06.042

    35. Belendez, A., C. Pascual, M. Ortuno, C. Neipp, T. Belendez, and S. Gallego, "Application of a modified He's homotopy perturbation method to obtain higher-order approximations to a nonlinear oscillator with discontinuities," Nonlinear Analysis: Real World Applications, 2007.

    36. 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

    37. Pashaei, H., D. D. Ganji, and M. Akbarzade, "Application of energy balance method for strongly nonlinear oscillators," J. Progress In Electromagnetics Research M, Vol. 2, 47-56, 2008.
    doi:10.2528/PIERM08031602

    38. Akbarzade, M., D. D. Ganji, and H. Pashaei, "Progress analysis of nonlinear oscillators with force by He's energy balance method," J.Pr ogress In Electromagnetics Research C, Vol. 3, 57-66, 2008.
    doi:10.2528/PIERC08032901

    39. Vahdati, H. and A. Abdipour, "Nonlinear stability analysis of microwave oscillators using the periodic averaging method," Progress In Electromagnetics Research, Vol. 79, 179-193, 2008.
    doi:10.2528/PIER07100101