Nonlinear Vibration Analysis of an Euler-Bernoulli Beam Resting on a Nonlinear Elastic Foundation under Compressive Axial Force

Document Type : Persian


1 Assistant Professor, Mechanical and Aerospace Engineering, Department of Mechanical Engineering, Parand Branch, Islamic Azad University, Tehran, Iran.

2 Professor, Mechanical Engineering, Department of Mechanical Engineering, Sharif university of Technology, Tehran, Iran


This paper studies the nonlinear vibration analysis of a simply supported Euler-Bernoulli beam resting on a nonlinear elastic foundation under compressive axial load using nonlinear normal modes concept in the case of three-to-one (3:1) internal resonance. The beam’s governing nonlinear PDE of motion and also its boundary conditions are derived and then solved using the method of Multiple Time Scales. Under three to-one-internal resonance condition applying nonlinear normal modes the steady state stability analysis of the beam’s vibrations is performed. Then the effect of changing the value of different parameters on the beam’s dynamic response and the steady state stability analysis is investigated.


[1] Nayfeh A.H., Lacarbonara W., Chin C.-M., Nonlinear normal modes of buckled beams: three-to-one and one-to-one internal resonances, Nonlinear Dynamics, Vol. 18, 1999, pp., 253-273.
[2] Santee D.M., Goncalves P.B., Oscillations of a beam on a non-linear elastic foundation under periodic loads, Shock and Vibration, Vol. 13, 2006, pp. 273-284.
[3] Tsiatas G.C., Nonlinear analysis of non-uniform beams on nonlinear elastic foundation, Acta Mechanical, Vol.  209, 2010, pp. 141-152.
[4] Kuo Y.H., Lee S.Y., Deflection of nonuniform beams resting on a nonlinear elastic foundation, Computers and Structures, Vol.  51, 1994, pp. 513-519.
[5] Hsu M.H., Mechanical analysis of non-uniform beams resting on nonlinear elastic foundation by the differential quadrature method, Structural Engineering and Mechanics, Vol. 22, 2006, pp. 279-292.
[6] Oz H.R., Pakdemirli M., Ozkaya E., Yilmaz M., Non-linear vibrations of a slightly curved beam resting on a non-linear elastic foundation, Journal of Sound and Vibration, Vol. 212, 1998, pp. 295-309.
[7] Pellicano F., Mastroddi F., Nonlinear dynamics of a beam on elastic foundation, Nonlinear Dynamics, Vol. 14, 1997, pp. 335-355.
[8] Balkaya M., Kaya M.O., Saglamer A., Analysis of the vibration of an elastic beam supported on elastic soil using the differential transform method, Achieve of Applied Mechanics, Vol. 79, 2009, pp. 135-146.
[9] Birman V., On the Effects of nonlinear elastic foundation on free vibration of beams, Journal of applied Mechanics, Vol. 53, 1986, pp. 471-474.
[10] King M.E., Vakakis A.F., An energy-based approach to computing resonant nonlinear normal modes, Journal of Applied Mechanics, Vol. 63, 1996, pp. 810-819.
[11] King M.E., Vakakis A.F., An energy-based formulation for computing nonlinear normal modes in undamped continuous systems, Journal of Vibration and Acoustics, Vol. 116, 1994, 332-340.
[12] Vakakis A.F., Nonlinear mode localization in systems governed by partial differential equations, Applied Mechanics Review, Vol. 49, 1996, pp. 87-99.
[13] Pellicano F., Vakakis A.F., Normal modes and boundary layers for a slender tensioned beam on a nonlinear foundation, Nonlinear Dynamics, Vol. 25, 2001, pp.79-93.
[14] Jiang D., Pierre C., Shaw S.W., The construction of non-linear normal modes for systems with internal resonance, International Journal of Non-Linear Mechanics, Vol. 40, 2005, pp. 729-746.
[15] Mazzilli C.E.N., Sanches C.T., Baracho O.G.P., Wiercigroch M., Keber M., Non-linear modal analysis for beams subjected to axial loads: analytical and finite-element solutions, International Journal of Non-Linear Mechanics, Vol. 43(6), 2008, pp. 551-561.
[16] Nayfeh A.H., Mook D.T., Nonlinear oscillations, Wiley-Interscience, New York, 1995.