DOI QR코드

DOI QR Code

On the large amplitude free vibrations of axially loaded Euler-Bernoulli beams

  • Bayat, Mahmoud (Department of Civil Engineering, Mashhad Branch, Islamic Azad University) ;
  • Pakar, Iman (Department of Civil Engineering, Mashhad Branch, Islamic Azad University) ;
  • Bayat, Mahdi (Department of Civil Engineering, Mashhad Branch, Islamic Azad University)
  • 투고 : 2012.06.02
  • 심사 : 2012.11.28
  • 발행 : 2013.01.25

초록

In this paper Hamiltonian Approach (HA) have been used to analysis the nonlinear free vibration of Simply-Supported (S-S) and for the Clamped-Clamped (C-C) Euler-Bernoulli beams fixed at one end subjected to the axial loads. First we used Galerkin's method to obtain an ordinary differential equation from the governing nonlinear partial differential equation. The effect of different parameter such as variation of amplitude to the obtained on the non-linear frequency is considered. Comparison of HA with Runge-Kutta 4th leads to highly accurate solutions. It is predicted that Hamiltonian Approach can be applied easily for nonlinear problems in engineering.

키워드

참고문헌

  1. Azrar, L., Benamar, R. and White, R.G. (1999), "A semi-analytical approach to the non-linear dynamic response problem of S-S and C-C beams at large vibration amplitudes part I: General theory and application to the single mode approach to free and forced vibration analysis", J. Sound Vib., 224(2), 183-207. https://doi.org/10.1006/jsvi.1998.1893
  2. Bayat, M., Pakar, I. and Shahidi, M. (2011), "Analysis of nonlinear vibration of coupled systems with cubic nonlinearity", Mechanika, 17(6), 620-629.
  3. Bayat, M., Barari, A. and Shahidi, M. (2011), "Dynamic response of axially loaded Euler-Bernoulli beams", Mechanika, 17(2), 172-177.
  4. Bayat, M., Bayat, M., and Bayat, M. (2011) "An analytical approach on a mass grounded by linear and nonlinear springs in series", Int. J. Phy. Sci., 6(2), 229-236.
  5. Bayat, M., Pakar, I., and M, Bayat. (2011a) "Analytical study on the vibration frequencies of tapered beams", Latin American J. Solids Struct., 8(2), 149-162. https://doi.org/10.1590/S1679-78252011000200003
  6. Bayat, M. and Pakar, I. (2011b), "Nonlinear free vibration analysis of tapered beams by Hamiltonian Approach", J. Vibroengineering., 13(4), 654-661.
  7. 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.
  8. Pakar, I. and Bayat, M. (2011d) "Analytical solution for strongly nonlinear oscillation systems using energy balance method", Int. J. Phy. Sci., 6(22), 5166- 5170.
  9. Bayat, M. and Pakar, I. (2012a), "Analytical study on the non-linear vibration of Euler-Bernoulli beams", J. of Vibroengineering, 14(1), 216-224.
  10. Pakar, I., Bayat, M. and Bayat, M. (2012b), "On the approximate analytical solution for parametrically excited nonlinear oscillators", J. Vibroengineering, 14(1), 423-429.
  11. Bayat, M. and Pakar, I. (2012c), "On the approximate analytical solution to non-linear oscillation systems", Shock and Vibration, DOI: 10.3233/SAV-2012-0726.
  12. Bayat, M., Shahidi, M., Barari, A. and Domairry, G. (2010), "The approximate analysis of nonlinear behavior of structure under harmonic loading", Int. J. Phy. Sci., 5(7), 1074-1080.
  13. Bayat, M., Shahidi, M., Barari, A. and Domairry, G. (2011) "Analytical evaluation of the nonlinear vibration of coupled oscillator systems", Zeitschrift fur Naturforschung Section A-A Journal of Physical Sciences, 66(1-2), 67-74.
  14. Bayat, M., Shahidi, M. and Bayat, M. (2011), "Application of iteration perturbation method for nonlinear oscillators with discontinuities", Int. J. Phy. Sci., 6(15), 3608-3612.
  15. 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 American J. Solids and Struct., 9(2), 145-234.
  16. Biondi, B. and Caddemi, S. (2005), "Closed form solutions of Euler-Bernoulli beams with singularities", Inter. J. Solids and Struct., 42(9-10), 3027-3044. https://doi.org/10.1016/j.ijsolstr.2004.09.048
  17. Ganji, D.D., Rafei, M., Sadighi, A. and Ganji, Z.Z. (2009), "A comparative comparison of He's method with perturbation and numerical methods for nonlinear vibrations equations," Inter. J. Nonlinear Dyn. in Eng. Sci., 1(1), 1-20.
  18. 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.
  19. He, J.H. (2007), "Variational Approach for nonlinear oscillators", Chaos. Soliton. Fractals., 34(5), 1430-1439. https://doi.org/10.1016/j.chaos.2006.10.026
  20. He, J.H. (2010), "Hamiltonian Approach to nonlinear oscillators", Physics Letters A., 374(23), 2312-2314. https://doi.org/10.1016/j.physleta.2010.03.064
  21. He, J.H. (2002), "Preliminary report on the energy balance for nonlinear oscillations", Mech. Res. Communications., 29(2-3), 107-111. https://doi.org/10.1016/S0093-6413(02)00237-9
  22. He, J.H. (2008), "An improved amplitude-frequency formulation for nonlinear oscillators", Inter. J. of Nonlinear Sci. and Numerical Simulation., 9(2), 211-212.
  23. Lai, H.Y., Hsu, J.C. and Chen, C.K. (2008), "An innovative eigenvalue problem solver for free vibration of Euler-Bernoulli beam by using the Adomian decomposition method", Comput. Math. Appl., 56(12), 3204-3220. https://doi.org/10.1016/j.camwa.2008.07.029
  24. Xu, L. and Zhang, N. (2009), "A variational approach next 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
  25. Lewandowski, R. (1987), "Application of the Ritz method to the analysis of nonlinear free vibrations of beams", J. Sound and Vib., 114(1), 91-101. https://doi.org/10.1016/S0022-460X(87)80236-5
  26. Liu, Y. and Gurram, S.C. (2009), "The use of He's variational iteration method for obtaining the free vibration of an Euler-Bernoulli beam", Math. Comput. Modelling, 50(11-12), 1545-1552. https://doi.org/10.1016/j.mcm.2009.09.005
  27. Naguleswaran, S. (2003), "Vibration and stability of an Euler-Bernoulli beam with up to three-step changes in cross-section and in axial force", Inter. J. Mech. Sci., 45(9),1563-1579. https://doi.org/10.1016/j.ijmecsci.2003.09.001
  28. Padovan, J. (1980), "Nonlinear vibrations of general structures", J. Sound Vib., 72, 427-441. https://doi.org/10.1016/0022-460X(80)90355-7
  29. Pirbodaghi, T., Ahmadian, M.T. and Fesanghary, M. (2009), "On the homotopy analysis method for non-linear vibration of beams", Mech. Res. Communications, 36(2), 143-148. https://doi.org/10.1016/j.mechrescom.2008.08.001
  30. Sathyamoorthy, M. (1982), "Nonlinear analysis of beams, Part-I: A survey of recent advances", Shock Vib. Dig., 14, 19-35.
  31. 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.
  32. Shen, Y.Y. and Mo, L.F. (2009), "The max-min approach to a relativistic equation", Comput. Math. Appl., 58, 2131-2133. https://doi.org/10.1016/j.camwa.2009.03.056
  33. Soleimani, Kutanaei, S., Ghasemi, E. and Bayat, M. (2011), "Mesh-free modeling of two-dimensional heat conduction between eccentric circular cylinders", Int. J. Phy. Sci., 6(16), 4044-4052.
  34. Tse, F.S., Morse, I.E. and Hinkle, R.T. (1987), Mechanical Vibrations: Theory and Applications, 2nd Edition, Allyn and Bacon Inc., Bosto.

피인용 문헌

  1. Nonlinear vibration of stringer shell: An analytical approach vol.229, pp.1, 2015, https://doi.org/10.1177/0954408913509090
  2. Analytical study of nonlinear vibration of oscillators with damping vol.9, pp.1, 2015, https://doi.org/10.12989/eas.2015.9.1.221
  3. Vibration analysis of a pre-stressed laminated composite curved beam vol.19, pp.3, 2015, https://doi.org/10.12989/scs.2015.19.3.635
  4. Nonlinear dynamics of two degree of freedom systems with linear and nonlinear stiffnesses vol.12, pp.3, 2013, https://doi.org/10.1007/s11803-013-0182-0
  5. Nonlinear vibration of an electrostatically actuated microbeam vol.11, pp.3, 2014, https://doi.org/10.1590/S1679-78252014000300009
  6. Nonlinear vibration of Euler-Bernoulli beams resting on linear elastic foundation vol.15, pp.4, 2013, https://doi.org/10.12989/scs.2013.15.4.439
  7. Nonlinear Vibration Analysis of Membrane SAR Antenna Structure Adopting a Vector Form Intrinsic Finite Element vol.31, pp.03, 2015, https://doi.org/10.1017/jmech.2014.97
  8. Nonlinear free vibration of systems with inertia and static type cubic nonlinearities: An analytical approach vol.77, 2014, https://doi.org/10.1016/j.mechmachtheory.2014.02.009
  9. Analytical solution for nonlinear vibration of an eccentrically reinforced cylindrical shell vol.14, pp.5, 2013, https://doi.org/10.12989/scs.2013.14.5.511
  10. 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
  11. 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
  12. Nonlinear stability and bifurcations of an axially accelerating beam with an intermediate spring-support vol.2, pp.2, 2013, https://doi.org/10.12989/csm.2013.2.2.159
  13. Vibration of electrostatically actuated microbeam by means of homotopy perturbation method vol.48, pp.6, 2013, https://doi.org/10.12989/sem.2013.48.6.823
  14. Dynamic Response of a Damped Euler–Bernoulli Beam Having Elastically Restrained Boundary Supports pp.2250-0553, 2018, https://doi.org/10.1007/s40032-018-0485-z
  15. Vibration analysis of high nonlinear oscillators using accurate approximate methods vol.46, pp.1, 2013, https://doi.org/10.12989/sem.2013.46.1.137
  16. Accurate semi-analytical solution for nonlinear vibration of conservative mechanical problems vol.61, pp.5, 2013, https://doi.org/10.12989/sem.2017.61.5.657