Steel and Composite Structures

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Nonlinear vibration of Euler-Bernoulli beams resting on linear elastic foundation

  • Javanmard, Mehran (Department of Civil Engineering, University of Zanjan) ;
  • Bayat, Mahdi (Department of Civil Engineering, Zanjan Branch, Islamic Azad University) ;
  • Ardakani, Alireza (Faculty of Engineering and Technology, Imam Khomeini International University)
  • Received : 2013.06.18
  • Accepted : 2013.08.31
  • Published : 2013.10.25
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Abstract

In this study simply supported nonlinear Euler-Bernoulli beams resting on linear elastic foundation and subjected to the axial loads is investigated. A new kind of analytical technique for a non-linear problem called He's Energy Balance Method (EBM) is used to obtain the analytical solution for non-linear vibration behavior of the problem. Analytical expressions for geometrically non-linear vibration of Euler-Bernoulli beams resting on linear elastic foundation and subjected to the axial loads are provided. The effect of vibration amplitude on the non-linear frequency and buckling load is discussed. The variation of different parameter to the nonlinear frequency is considered completely in this study. The nonlinear vibration equation is analyzed numerically using Runge-Kutta $4^{th}$ technique. Comparison of Energy Balance Method (EBM) with Runge-Kutta $4^{th}$ leads to highly accurate solutions.

Keywords

References

  1. Al-Hosani, K., Fadhil, S. and El-Zafrany, A. (1999), "Fundamental solution and boundary element analysis of thick plates on Winkler foundation", Comput. Struct., 70(3), 325-336. https://doi.org/10.1016/S0045-7949(98)00171-0
  2. Arikoglu, A. and Ozkol, I. (2006), "Solution of differential-difference equations by using differential transform method", Appl. Math. Comput., 181(1), 153-162. https://doi.org/10.1016/j.amc.2006.01.022
  3. Auersch, L. (2008), "Dynamic interaction of various beams with the underlying soil-finite and infinite, half-space and Winkler models", Eur. J. Mech. A-Solid, 27(5), 933-958. https://doi.org/10.1016/j.euromechsol.2008.02.001
  4. Azrar, L., Benamar, R. and White, R.G. (1999), "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
  5. 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
  6. Bayat, M. and Pakar, I. (2013a), "On the approximate analytical solution to non-linear oscillation systems", Shock Vib., 20(1), 43-52. https://doi.org/10.1155/2013/549213
  7. Bayat, M. and Pakar, I. (2013b), "On the large amplitude free vibrations of axially loaded Euler-Bernoulli beams", Steel Compos. Struct. Int. J., 14(1), 73-83. https://doi.org/10.12989/scs.2013.14.1.073
  8. 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 Struct., 9(2), 145-234.
  9. 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
  10. Eisenberger, M. and Clastornik, J. (1987), "Vibrations and buckling of a beam on a variable Winkler elastic foundation", J. Sound Vib., 115(2), 233-241. https://doi.org/10.1016/0022-460X(87)90469-X
  11. Gorbunov-Posadov, M.I. (1973), The Design of Structures on an Elastic Foundation, Gosstroiizdat, Moscow, 628. [in Russian]
  12. Gupta, U., Ansari, A. and Sharma, S. (2006), "Buckling and vibration of polar orthotropic circular plate resting on Winkler foundation", J. Sound Vib., 297(3-5), 457-476. https://doi.org/10.1016/j.jsv.2006.01.073
  13. 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
  14. He, J.H. (2007), "Variational approach for nonlinear oscillators", Chaos, Soliton. Fract., 34(5), 1430-1439. https://doi.org/10.1016/j.chaos.2006.10.026
  15. He, J.H. (2008), "Max-min approach to nonlinear oscillators", Int. J. Nonlinear Sci. Numer. Simul., 9(2), 207-210.
  16. Lee, H.P. (1988), "Dynamic response of a Timoshenko beam on a Winkler foundation subjected to a moving mass", Appl. Acoust., 55(3), 203-215.
  17. Lewandowski, R. (1987), "Application of the Ritz method to the analysis of non-linear free vibrations of beams", J. Sound Vib., 114(1), 91-101. https://doi.org/10.1016/S0022-460X(87)80236-5
  18. Liu, Y. and Gurram, C.S. (2009), "The use of He's variational iteration method for obtaining the free vibration of an Euler-Bernoulli beam", Math. Comput. Model., 50(11-12), 1545-1552. https://doi.org/10.1016/j.mcm.2009.09.005
  19. Pakar, I. and Bayat, M. (2012), "Analytical study on the non-linear vibration of Euler-Bernoulli beams", J. Vibroeng., 14(1), 216-224.
  20. Pakar, I. and Bayat, M. (2013a), "An analytical study of nonlinear vibrations of buckled Euler-Bernoulli Beams", Acta Phys. Pol. A, 123(1), 48-52. https://doi.org/10.12693/APhysPolA.123.48
  21. Pakar, I. and Bayat, M. (2013b), "Vibration analysis of high nonlinear oscillators using accurate approximate methods", Struct. Eng. Mech. Int. J., 46(1), 137-151. https://doi.org/10.12989/sem.2013.46.1.137
  22. Pakar, I., Bayat, M. and Bayat, M. (2012)," On the approximate analytical solution for parametrically excited nonlinear oscillators", J. Vibroeng., 14(1), 423-429.
  23. Pirbodaghi, T., Ahmadian, M. and Fesanghary, M. (2009), "On the homotopy analysis method for non-linear vibration of beams", Mech. Res. Commun., 36(2), 143-148. https://doi.org/10.1016/j.mechrescom.2008.08.001
  24. Rao, S.S. (2007), Vibration of Continuous Systems, Wiley Online Library.
  25. Ren, Z.F. and Gui, W.K. (2011), "He's frequency formulation for nonlinear oscillators using a golden mean location", Comput. Math. Appl., 61(8), 1987-1990. https://doi.org/10.1016/j.camwa.2010.08.047
  26. Ruge, P. and Birk, C. (2007), "A comparison of infinite Timoshenko and Euler-Bernoulli beam models on Winkler foundation in the frequency-and time-domain", J. Sound Vib., 304(3-5), 932-947. https://doi.org/10.1016/j.jsv.2007.04.001
  27. Shou, D.H. (2009), "The homotopy perturbation method for nonlinear oscillators", Comput. Math. Appl., 58(11-12), 2456-2459. https://doi.org/10.1016/j.camwa.2009.03.034
  28. Soldatos, K. and Selvadurai, A. (1985), "Flexure of beams resting on hyperbolic elastic foundations", Int. J. Solids Struct., 21(4), 373-388. https://doi.org/10.1016/0020-7683(85)90062-9
  29. Tse, F., Morse, I.E. and Hinkte, R.E. (1987), Mechanical Vibrations: Theory and Applications, Cengage Learning, Independence, KY, USA.
  30. Xu, L. (2007), "He's parameter-expanding methods for strongly nonlinear oscillators", J. Comput. Appl. Math., 207(1), 148-154. https://doi.org/10.1016/j.cam.2006.07.020
  31. Zhou, D.A. (1993), "General solution to vibrations of beams on variable Winkler elastic foundation", Comput. Struct., 47(1), 83-90. https://doi.org/10.1016/0045-7949(93)90281-H

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