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Mathematical solution for nonlinear vibration equations using variational approach

  • Bayat, M. (Department of Civil Engineering, Mashhad Branch,Islamic Azad University) ;
  • Pakar, I. (Young Researchers and Elite Club, Mashhad Branch, Islamic Azad University)
  • Received : 2014.02.02
  • Accepted : 2014.06.28
  • Published : 2015.05.25
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Abstract

In this paper, we have applied a new class of approximate analytical methods called Variational Approach (VA) for high nonlinear vibration equations. Three examples have been introduced and discussed. The effects of important parameters on the response of the problems have been considered. Runge-Kutta's algorithm has been used to prepare numerical solutions. The results of variational approach are compared with energy balance method and numerical and exact solutions. It has been established that the method is an easy mathematical tool for solving conservative nonlinear problems. The method doesn't need small perturbation and with only one iteration achieve us to a high accurate solution.

Keywords

References

  1. Bayat, M. and Pakar, I. (2011a), "Nonlinear free vibration analysis of tapered beams by hamiltonian approach", J. Vibroengineering, 13(4), 654-661.
  2. Bayat, M. and Pakar, I. (2011b), "Application of he's energy balance method for nonlinear vibration of thin circular sector cylinder", Int. J. Phy. Sci., 6(23), 5564-5570.
  3. Bayat, M., Pakar, I. and Shahidi, M. (2011c), "Analysis of nonlinear vibration of coupled systems with cubic nonlinearity", Mechanika, 17(6), 620-629.
  4. Bayat, M. and Pakar, I. (2012a), "Accurate analytical solution for nonlinear free vibration of beams", Struct. Eng.Mech., 43(3), 337-347. https://doi.org/10.12989/sem.2012.43.3.337
  5. Bayat, M., Pakar, I. and Domaiirry, G, (2012b), "Recent developments of some asymptotic methods and their applications for nonlinear vibration equations in engineering problems: a review", Latin American J. Solids Struct., 9(2), 145- 234 .
  6. Bayat, M., Pakar, I. and Bayat, M. (2013a), "Analytical solution for nonlinear vibration of an eccentrically reinforced cylindrical shell", Steel Compos. Struct., 14(5), 511-521. https://doi.org/10.12989/scs.2013.14.5.511
  7. Bayat, M. and Pakar, I. (2013b), "Nonlinear dynamics of two degree of freedom systems with linear and nonlinear stiffnesses", Earthq. Eng. Eng. Vib., 12(3), 411-420 . https://doi.org/10.1007/s11803-013-0182-0
  8. Bayat, M. and Pakar, I. (2013c), "On the approximate analytical solution to non-linear oscillation systems", Shock Vib., 20(1), 43-52. https://doi.org/10.1155/2013/549213
  9. Bayat, M., Pakar, I. and Cveticanin, L. (2014a), "Nonlinear free vibration of systems with inertia and static type cubic nonlinearities: an analytical approach", Mech. Machine Theory, 77, 50-58. https://doi.org/10.1016/j.mechmachtheory.2014.02.009
  10. Bayat, M., Pakar, I. and Cveticanin, L. (2014b), "Nonlinear vibration of stringer shell by means of extended Hamiltonian approach", Arch. Appl. Mech., 84(1), 43-50. https://doi.org/10.1007/s00419-013-0781-2
  11. Bayat, M., Bayat, M. and Pakar, I. (2014c), "Nonlinear vibration of an electrostatically actuated microbeam", Latin American J. Solids Struct., 11(3), 534- 544. https://doi.org/10.1590/S1679-78252014000300009
  12. Cordero, A., Hueso, J.L., Martinezand, E. and Torregros, J.R. (2010), "Iterative methods for use with nonlinear discrete algebraic models", Math. Comput. Model., 52(7-8), 1251-1257. https://doi.org/10.1016/j.mcm.2010.02.028
  13. Dehghan, M. and Tatari, M. (2008), "Identifying an unknown function in a parabolic equation with over specified data via He's variational iteration method", Chaos, Solitons Fractals, 36(1), 157-166. https://doi.org/10.1016/j.chaos.2006.06.023
  14. He, J.H (2007), "Variational approach for nonlinear oscillators", Chaos, Solitons Fractals, 34(5), 1430-1439. https://doi.org/10.1016/j.chaos.2006.10.026
  15. He J.H. (2008), "An improved amplitude-frequency formulation for nonlinear oscillators", Int. J. Nonlinear Sci. Numer. Simul., 9(2), 211-212. https://doi.org/10.1515/IJNSNS.2008.9.2.211
  16. He, J.H. (2002), "Preliminary report on the energy balance for nonlinear oscillators", Mech. Res. Commun., 29(2), 107-111. https://doi.org/10.1016/S0093-6413(02)00237-9
  17. Kuo, B.L. and Lo, C.Y. (2009), "Application of the differential transformation method to the solution of a damped system with high nonlinearity", Nonlinear Anal., 70(4), 1732-1737. https://doi.org/10.1016/j.na.2008.02.056
  18. Mehdipour, I., Ganji, D.D. and Mozaffari, M. (2010), "Application of the energy balance method to nonlinear vibrating equations", Current Appl. Phys., 10(1), 104-112. https://doi.org/10.1016/j.cap.2009.05.016
  19. Nayfeh, A.H. and Mook, D.T. (1973), Nonlinear Oscillations, Wiley, New York.
  20. Odibat, Z., Momani, S, and Suat Erturk, V. (2008), "Generalized differential transform method: application to differential equations of fractional order", Appl. Math. Comput., 197(2), 467-477. https://doi.org/10.1016/j.amc.2007.07.068
  21. Pakar, I. and Bayat, M. (2011a), "Analytical solution for strongly nonlinear oscillation systems using energy balance method", Int. J. Phy. Sci., 6(22), 5166- 5170.
  22. Pakar, I. and Bayat, M. (2012a), "On the approximate analytical solution for parametrically excited nonlinear oscillators", J. Vibroengineering, 14(1), 423-429.
  23. Pakar, I. and Bayat, M. (2012b), "Analytical study on the non-linear vibration of Euler-Bernoulli beams", J. Vibroengineering, 14(1), 216-224.
  24. Pakar, I. and Bayat, M. (2013a), "An analytical study of nonlinear vibrations of buckled Euler_Bernoulli beams", Acta Phys. Polonica A, 123(1), 48-52. https://doi.org/10.12693/APhysPolA.123.48
  25. Pakar, I. and Bayat, M. (2013b), "Vibration analysis of high nonlinear oscillators using accurate approximate methods", Struct. Eng. Mech., 46(1), 137-151. https://doi.org/10.12989/sem.2013.46.1.137
  26. Shaban, M., Ganji, D.D. and Alipour, A.A. (2010), "Nonlinear fluctuation, frequency and stability analyses in free vibration of circular sector oscillation systems", Current Appl. Phys., 10(5), 1267-1285. https://doi.org/10.1016/j.cap.2010.03.005
  27. Shen, Y.Y. and Mo, L.F. (2009), "The max-min approach to a relativistic equation", Comput. Math. Appl., 58(11), 2131-2133. https://doi.org/10.1016/j.camwa.2009.03.056
  28. Wu, G. (2011), "Adomian decomposition method for non-smooth initial value problems", Math. Comput. Model., 54(9-10), 2104-2108. https://doi.org/10.1016/j.mcm.2011.05.018
  29. Xu, N. and Zhang, A. (2009), "Variational approachnext term to analyzing catalytic reactions in short monoliths", Comput. Math. Appl., 58(11-12), 2460-2463. https://doi.org/10.1016/j.camwa.2009.03.035
  30. Xu, L. (2008), "Variational approach to solution of nonlinear dispersive K(m, n) equation", Chaos, Solitons Fractals, 37(1), 137-143. https://doi.org/10.1016/j.chaos.2006.08.016
  31. Zeng, D.Q. and Lee, Y.Y. (2009), "Analysis of strongly nonlinear oscillator using the max-min approach", Int. J. Nonlinear Sci. Numer. Simul., 10 (10), 1361-1368.

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