- Volume 15 Issue 4
DOI QR Code
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
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
elastic foundation;nonlinear vibration;analytical method;Runge-Kutta
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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.
- 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
- 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
- Gorbunov-Posadov, M.I. (1973), The Design of Structures on an Elastic Foundation, Gosstroiizdat, Moscow, 628. [in Russian]
- 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
- 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
- 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
- He, J.H. (2008), "Max-min approach to nonlinear oscillators", Int. J. Nonlinear Sci. Numer. Simul., 9(2), 207-210.
- 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.
- 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
- 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
- Pakar, I. and Bayat, M. (2012), "Analytical study on the non-linear vibration of Euler-Bernoulli beams", J. Vibroeng., 14(1), 216-224.
- 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
- 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
- Pakar, I., Bayat, M. and Bayat, M. (2012)," On the approximate analytical solution for parametrically excited nonlinear oscillators", J. Vibroeng., 14(1), 423-429.
- 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
- Rao, S.S. (2007), Vibration of Continuous Systems, Wiley Online Library.
- 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
- 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
- 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
- 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
- Tse, F., Morse, I.E. and Hinkte, R.E. (1987), Mechanical Vibrations: Theory and Applications, Cengage Learning, Independence, KY, USA.
- 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
- 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
- A new approach to modeling the dynamic response of Bernoulli-Euler beam under moving load vol.3, pp.3, 2014, https://doi.org/10.12989/csm.2014.3.3.247
- Forced nonlinear vibration by means of two approximate analytical solutions vol.50, pp.6, 2014, https://doi.org/10.12989/sem.2014.50.6.853
- Effect of soil–structure interaction on the nonlinear response of an inextensional beam on elastic foundation vol.85, pp.2, 2015, https://doi.org/10.1007/s00419-014-0918-y