Structural Engineering and Mechanics

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Accurate analytical solution for nonlinear free vibration of beams

  • Bayat, M. (Department of Civil Engineering, Mashhad Branch, Islamic Azad University) ;
  • Pakar, I. (Department of Civil Engineering, Mashhad Branch, Islamic Azad University)
  • Received : 2011.08.23
  • Accepted : 2012.06.26
  • Published : 2012.08.10
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Abstract

In this study, Hamiltonian Approach (HA) is applied to analysis the nonlinear free vibration of beams. Two well-known examples are illustrated to show the efficiency of this method. One of them deals with the Nonlinear vibration of an electrostatically actuated microbeam and the other is the nonlinear vibrations of tapered beams. This new approach prepares us to achieve the beam's natural frequencies and mode shapes easily and a rapidly convergent sequence is obtained during the solution. The effects of the small parameters on the frequency of the beams are discussed. Some comparisons are conducted between the results obtained by the Hamiltonian Approach (HA) and numerical solutions using to illustrate the effectiveness and convenience of the proposed methods.

Keywords

References

  1. Bayat, M., Pakar, I. and Bayat, M. (2011), "Analytical study on the vibration frequencies of tapered beams", Latin Am. J. Solids Struct., 8(2), 149-162. https://doi.org/10.1590/S1679-78252011000200003
  2. Bayat, M., Shahidi, M., Barari, A. and Domairry, G. (2011a), "Analytical evaluation of the nonlinear vibration of coupled oscillator systems", Zeitschrift fur Naturforschung Section A-A J. Phys. Sci., 66(1-2), 67-74.
  3. Bayat, M., Barari, A. and Shahidi, M. (2011b), "Dynamic response of axially loaded euler-bernoulli beams", Mechanika, 17(2), 172-177.
  4. Bayat, M. and Pakar, I. (2011c), "Application of He's energy balance method for nonlinear vibration of thin circular sector cylinder", Int. J. Phy. Sci., 6(23), 5564-5570.
  5. Bayat, M., Bayat, M. and Bayat, M. (2011d), "An analytical approach on a mass grounded by linear and nonlinear springs in series", Int. J. Phy. Sci., 6(2), 229-236.
  6. Bayat, M., Shahidi, M. and Bayat, M. (2011e), "Application of iteration perturbation method for nonlinear oscillators with discontinuities", Int. J. Phy. Sci., 6(15), 3608-3612.
  7. Bayat, M., Pakar, I. and Shahidi, M. (2011f), "Analysis of nonlinear vibration of coupled systems with cubic nonlinearity", Mechanika, 17(6), 620-629.
  8. Bayat, M. and Abdollahzadeh, G.R. (2011g), "Analysis of the steel braced frames equipped with ADAS devices under the far field records", Latin Am. J. Solids Struct., 8(2), 163-181. https://doi.org/10.1590/S1679-78252011000200004
  9. Bayat, M. and Abdollahzadeh, G.R. (2011h), "On the effect of the near field records on the steel braced frames equipped with energy dissipating devices", Latin Am. J. Solids Struct., 8(4), 429-443. https://doi.org/10.1590/S1679-78252011000400004
  10. Burgreen, D. (1951), "Free vibrations of a pin-ended column with constant distance between pin ends", J. Appl. Mech., ASME, 18, 135-139.
  11. Dumir, P. and Bhaskar, A. (1988), "Some erroneous finite element formulations of non-linear vibrations of beams and plates", J. Sound Vib., 123(3), 517-527. https://doi.org/10.1016/S0022-460X(88)80167-6
  12. Fu, Y.M., Zhang, J. and Wan, L.J. (2011), "Application of the energy balance method to a nonlinear oscillator arising in the microelectromechanical system (MEMS)", Curr Appl. Phys., 11, 482-485. https://doi.org/10.1016/j.cap.2010.08.037
  13. Ganji, D.D., Kachapi, H. and Seyed, H. (2011), "Analytical and numerical methods in engineering and applied sciences", Prog. Nonlin. Sci., 3, 1-579.
  14. Ganji, D.D., Bararnia, H., Soleimani, S. and Ghasemi, E. (2009), "Analytical solution of magneto-hydrodinamic flow over a nonlinear stretching sheet", Modern Phys. Lett. B, 23, 2541-2556. https://doi.org/10.1142/S0217984909020692
  15. Ghasemi, E., Bayat, M. and Bayat, M. (2011), "Visco-elastic MHD flow of walters liquid B fluid and heat transfer over a non-isothermal stretching sheet", Int. J. Phy. Sci., 6(21), 5022-5039.
  16. Goorman, D.J. (1975), "Free vibrations of beams and shafts", Appl. Mech., ASME, 18, 135-139.
  17. He, J.H. (2002), "Preliminary report on the energy balance for nonlinear oscillations", Mech. Res. Comm., 29, 107-111. https://doi.org/10.1016/S0093-6413(02)00237-9
  18. He, J.H. (2010), "Hamiltonian approach to nonlinear oscillators", Phys. Lett. A., 374, 2312-2314. https://doi.org/10.1016/j.physleta.2010.03.064
  19. He, J.H., Zhong, T. and Tang, L. (2010), "Hamiltonian Approach to duffing-harmonic equation", Int. J. Nonlin. Sci. Numer., 11(S), 43-46.
  20. Hoseini, S.H., Pirbodaghi, T., Ahmadian, M.T. and Farrahi, G.H. (2009), "On the large amplitude free vibrations of tapered beams: an analytical approach", Mech. Res. Comm., 36, 892-897. https://doi.org/10.1016/j.mechrescom.2009.08.003
  21. Jamshidi, N. and Ganji, D.D. (2010), "Application of energy balance method and variational iteration method to an oscillation of a mass attached to a stretched elastic wire", Current Appl. Phys., 10(2), 484-486. https://doi.org/10.1016/j.cap.2009.07.004
  22. Liu, J.F. (2009), "He's variational approach for nonlinear oscillators with high nonlinearity", Comput. Math. Appl., 58(11-12), 2423-2426. https://doi.org/10.1016/j.camwa.2009.03.074
  23. Lou, C. and Sikarskie, D. (1975), "Nonlinear vibration of beams using a form-function approximation", J. Appl. Mech., 42(1), 209-214. https://doi.org/10.1115/1.3423520
  24. 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
  25. Oztur, B. and Coskun, S.B. (2011), "The Homotopy Perturbation Method for free vibration analysis of beam on elastic foundation", Struct. Eng. Mech., 37(4), 415-425. https://doi.org/10.12989/sem.2011.37.4.415
  26. Pakar, I., Bayat, M. and Bayat, M. (2011a), "Analytical evaluation of the nonlinear vibration of a solid circular sector object", Int. J. Phy. Sci., 6(30), 6861-6866.
  27. Pakar, I. and Bayat, M. (2011b), "Analytical solution for strongly nonlinear oscillation systems using energy balance method", Int. J. Phy. Sci., 6(22), 5166-5170.
  28. Pakar, I. and Bayat, M. (2012), "Analytical study on the non-linear vibration of Euler-Bernoulli beams", J. Vibroengineering, 14(1), 216-224.
  29. Pakar, I., Bayat, M. and Bayat, M. (2012), "On the approximate analytical solution for parametrically excited nonlinear oscillators", J. Vibroengineering, 14(1), 423-429.
  30. Prathap, G. and Varadan, T. (1978), "The large amplitude vibration of tapered clamped beams", J. Sound Vib., 58(1), 87-94. https://doi.org/10.1016/S0022-460X(78)80062-5
  31. Sathyamoorthy, M. (1982), "Nonlinear analysis of beams, Part-I: A survey of recent advances", Shock Vib. Dig, 14, 19-35.
  32. Shahidi, M., Bayat, M., Pakar, I. and Abdollahzadeh, G.R. (2011), "On the solution of free non-linear vibration of beams", Int. J. Phy. Sci., 6(7), 1628-1634.
  33. Singh, G., Sharma, A. and Venkateswara Rao, G. (1990), "Large-amplitude free vibrations of beams--A discussion on various formulations and assumptions", J. Sound Vib., 142(1), 77-85. https://doi.org/10.1016/0022-460X(90)90583-L
  34. Xu, L. and He, J.H. (2010), "Determination of limit cycle by hamiltonian approach for strongly nonlinear oscillators", Int. J. Nonlin. Sci., 11(12), 1097-1101.

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