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http://dx.doi.org/10.12989/scs.2013.15.4.439

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)
Publication Information
Steel and Composite Structures / v.15, no.4, 2013 , pp. 439-449 More about this Journal
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
elastic foundation; nonlinear vibration; analytical method; Runge-Kutta $4^{th}$;
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Times Cited By KSCI : 4  (Citation Analysis)
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1 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.   DOI   ScienceOn
2 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.   DOI   ScienceOn
3 Bayat, M. and Pakar, I. (2012), "Accurate analytical solution for nonlinear free vibration of beams", Struct. Eng. Mech. Int. J., 43(3), 337-347.   DOI   ScienceOn
4 Bayat, M. and Pakar, I. (2013a), "On the approximate analytical solution to non-linear oscillation systems", Shock Vib., 20(1), 43-52.   DOI
5 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.   DOI   ScienceOn
6 Arikoglu, A. and Ozkol, I. (2006), "Solution of differential-difference equations by using differential transform method", Appl. Math. Comput., 181(1), 153-162.   DOI   ScienceOn
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.   DOI   ScienceOn
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.   DOI   ScienceOn
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.   DOI   ScienceOn
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.   DOI   ScienceOn
13 He, J.H. (2002), "Preliminary report on the energy balance for nonlinear oscillations", Mech. Res. Communications, 29(2-3), 107-111.   DOI   ScienceOn
14 He, J.H. (2007), "Variational approach for nonlinear oscillators", Chaos, Soliton. Fract., 34(5), 1430-1439.   DOI   ScienceOn
15 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.   DOI   ScienceOn
16 He, J.H. (2008), "Max-min approach to nonlinear oscillators", Int. J. Nonlinear Sci. Numer. Simul., 9(2), 207-210.
17 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.
18 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.   DOI   ScienceOn
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.   DOI
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.   DOI   ScienceOn
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.   DOI   ScienceOn
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.   DOI   ScienceOn
26 Tse, F., Morse, I.E. and Hinkte, R.E. (1987), Mechanical Vibrations: Theory and Applications, Cengage Learning, Independence, KY, USA.
27 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.   DOI   ScienceOn
28 Shou, D.H. (2009), "The homotopy perturbation method for nonlinear oscillators", Comput. Math. Appl., 58(11-12), 2456-2459.   DOI   ScienceOn
29 Soldatos, K. and Selvadurai, A. (1985), "Flexure of beams resting on hyperbolic elastic foundations", Int. J. Solids Struct., 21(4), 373-388.   DOI   ScienceOn
30 Xu, L. (2007), "He's parameter-expanding methods for strongly nonlinear oscillators", J. Comput. Appl. Math., 207(1), 148-154.   DOI   ScienceOn
31 Zhou, D.A. (1993), "General solution to vibrations of beams on variable Winkler elastic foundation", Comput. Struct., 47(1), 83-90.   DOI   ScienceOn