Steel and Composite Structures

Techno-Press (테크노프레스)
권호 표지
DOI QR코드

DOI QR Code

The analytic solution for parametrically excited oscillators of complex variable in nonlinear dynamic systems under harmonic loading

  • Bayat, Mahdi (Department of Civil Engineering, College of Engineering, Mashhad Branch, Islamic Azad University) ;
  • Bayat, Mahmoud (Department of Civil Engineering, College of Engineering, Mashhad Branch, Islamic Azad University) ;
  • Pakar, Iman (Young Researchers and Elites Club, Mashhad Branch, Islamic Azad University)
  • Received : 2014.02.20
  • Accepted : 2014.04.12
  • Published : 2014.07.25
⟨ Previous Next ⟩

Abstract

In this paper we have considered the vibration of parametrically excited oscillator with strong cubic positive nonlinearity of complex variable in nonlinear dynamic systems with forcing based on Mathieu-Duffing equation. A new analytical approach called homotopy perturbation has been utilized to obtain the analytical solution for the problem. Runge-Kutta's algorithm is also presented as our numerical solution. Some comparisons between the results obtained by the homotopy perturbation method and Runge-Kutta algorithm are shown to show the accuracy of the proposed method. In has been indicated that the homotopy perturbation shows an excellent approximations comparing the numerical one.

Keywords

References

  1. Alicia, C., Hueso, J.L., Martinez, 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
  2. Andrianov, I.V., Awrejcewicz, J. and Manevitch, L.I. (2004), Asymptotical Mechanics of Thin-Walled Structures, Springers-Verlag Berlin, Heidelberg, Germany.
  3. Agirseven, D. and Ozis, T. (2010) "An analytical study for Fisher type equations by using homotopy perturbation method", Comput. Math. Appl., 60(3), 602-609. https://doi.org/10.1016/j.camwa.2010.05.006
  4. Bayat, M. and Pakar, I. (2011a), "Nonlinear free vibration analysis of tapered beams by Hamiltonian Approach", J. Vibroeng., 13(4), 654-661.
  5. 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.
  6. Bayat, M. and Abdollahzade, G. (2011c), "Analysis of the steel braced frames equipped with ADAS devices under the far field records", Latin Am. J. Solid. Struct., 8(2), 163-181. https://doi.org/10.1590/S1679-78252011000200004
  7. Bayat, M. and Abdollahzadeh, G.R. (2011d), "On the effect of the near field records on the steel braced frames equipped with energy dissipating devices", Latin Am. J. Solid. Struct., 8(4), 429-443. https://doi.org/10.1590/S1679-78252011000400004
  8. Bayat, M. and Pakar, I. (2012), "Accurate analytical solution for nonlinear free vibration of beams", Struct. Eng. Mech., Int. J., 43(3), 337-347. https://doi.org/10.12989/sem.2012.43.3.337
  9. Bayat, M. and Pakar, I. (2013a), "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
  10. Bayat, M. and Pakar, I. (2013b), "On the approximate analytical solution to non-linear oscillation systems", Shock Vib., 20(1), 43-52. https://doi.org/10.1155/2013/549213
  11. Bayat, M., Pakar, I. and Shahidi, M. (2011), "Analysis of nonlinear vibration of coupled systems with cubic nonlinearity", Mechanika, 17(6), 620-629.
  12. Bayat, M., Pakar, I. and Domaiirry, G. (2012), "Recent developments of some asymptotic methods and their applications for nonlinear vibration equations in engineering problems: A review", Latin Am. J. Solid. Struct., 9(2), 145-234 .
  13. Bayat, M., Pakar, I. and Bayat, M. (2013), "Analytical solution for nonlinear vibration of an eccentrically reinforced cylindrical shell", Steel Compos. Struct., Int. J., 14(5), 511-521. https://doi.org/10.12989/scs.2013.14.5.511
  14. Bayat, M., Pakar, I. and Cveticanin, L. (2014a), "Nonlinear free vibration of systems with inertia and static type cubic nonlinearities : An analytical approach", Mechi. Mach. Theory, 77, 50-58. https://doi.org/10.1016/j.mechmachtheory.2014.02.009
  15. 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
  16. Bayat, M., Bayat, M. and Pakar, I. (2014c), "Nonlinear vibration of an electrostatically actuated microbeam", Latin Am. J. Solid. Struct., 11(3), 534-544. https://doi.org/10.1590/S1679-78252014000300009
  17. 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
  18. Filobello-Nino, U., Vazquez-Leal, H., Castaneda-Sheissa, R., Yildirim, A., Hernandez-Martinez, L., Pereyra-Diaz, D., Perez-Sesma, A. and Hoyos-Reyes, C. (2012), "An approximate solution of blasius equation by using HPM method", Asian J. Math. Statistics, 5(2), 50-59. https://doi.org/10.3923/ajms.2012.50.59
  19. Ganji, D.D. (2006), "The application of He's homotopy perturbation method to nonlinear equations arising in heat transfer", Phys. Letters A, 355(4-5), 337-341. https://doi.org/10.1016/j.physleta.2006.02.056
  20. He, J.H. (1999a), "Variational iteration method: A kind of nonlinear analytical technique: Some examples," Int. J. Non-Linear Mech., 34(4), 699-708. https://doi.org/10.1016/S0020-7462(98)00048-1
  21. He, J.H. (1999b), "Homotopy perturbation technique", Comput. Method. Appl. Mech. Eng., 178(3-4), 257-262. https://doi.org/10.1016/S0045-7825(99)00018-3
  22. He, J.H. (2002), "Preliminary report on the energy balance for nonlinear oscillators", Mech. Res. Communications, 29(2), 107-111. https://doi.org/10.1016/S0093-6413(02)00237-9
  23. 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
  24. He, J.H. (2008), "An improved amplitude-frequency formulation for nonlinear oscillators", Int. J. Nonlinear Sci. Numer. Simulation, 9(2), 211-212.
  25. 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
  26. 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
  27. 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
  28. Pakar, I. and Bayat, M. (2011), "Analytical solution for strongly nonlinear oscillation systems using energy balance method", Int. J. Phy. Sci., 6(22), 5166-5170.
  29. Pakar, I. and Bayat, M. (2012), "Analytical study on the non-linear vibration of Euler-Bernoulli beams", J. Vibroeng., 14(1), 216-224.
  30. Pakar, I. and Bayat, M. (2013a), "An analytical study of nonlinear vibrations of buckled Euler-Bernoulli beams", Acta Physica Polonica A, 123(1), 48-52. https://doi.org/10.12693/APhysPolA.123.48
  31. 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
  32. Pakar, I., Bayat, M. and Bayat, M. (2012), "On the approximate analytical solution for parametrically excited nonlinear oscillators", J. Vibroeng., 14(1), 423-429.
  33. 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. Physi., 10(5), 1267-1285. https://doi.org/10.1016/j.cap.2010.03.005
  34. 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
  35. 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
  36. 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
  37. 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
  38. 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.

Cited by

  1. High conservative nonlinear vibration equations by means of energy balance method vol.11, pp.1, 2016, https://doi.org/10.12989/eas.2016.11.1.129
  2. Study of complex nonlinear vibrations by means of accurate analytical approach vol.17, pp.5, 2014, https://doi.org/10.12989/scs.2014.17.5.721
  3. A novel approximate solution for nonlinear problems of vibratory systems vol.57, pp.6, 2016, https://doi.org/10.12989/sem.2016.57.6.1039
  4. Analytical study of nonlinear vibration of oscillators with damping vol.9, pp.1, 2015, https://doi.org/10.12989/eas.2015.9.1.221
  5. Supersonic nonlinear flutter of cross-ply laminated shallow shells pp.2041-3025, 2019, https://doi.org/10.1177/0954410019827461
  6. Optimum Design of Infinite Perforated Orthotropic and Isotropic Plates vol.8, pp.4, 2014, https://doi.org/10.3390/math8040569