Dynamic analysis of a simply supported beam resting on a nonlinear elastic foundation under compressive axial load using nonlinear normal modes techniques under threetoone internal resonance condition
 Ahmad Mamandi,
 Mohammad H. Kargarnovin,
 Salman Farsi
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Abstract
In this paper, the Nonlinear Normal Modes (NNMs) analysis for the case of threetoone (3:1) internal resonance of a slender simply supported beam in presence of compressive axial load resting on a nonlinear elastic foundation is studied. Using the Euler–Bernoulli beam model, the governing nonlinear PDE of the beam’s transverse vibration and also its associated boundary conditions are extracted. These nonlinear motion equation and boundary condition relations are solved simultaneously using four different approximateanalytical solution techniques, namely the method of Multiple Time Scales, the method of Normal Forms, the method of Shaw and Pierre, and the method of King and Vakakis. The obtained results at this stage using four different methods which are all in time–space domain are compared and it is concluded that all the methods result in a similar answer for the amplitude part of the transverse vibration. At the next step, the nonlinear normal modes are obtained. Furthermore, the effect of axial compressive force in the dynamic analysis of such a beam is studied. Finally, under threetooneinternal resonance condition the NNMs of the beam and the steadystate stability analysis are performed. Then the effect of changing the values of different parameters on the beam’s dynamic response is also considered. Moreover, 3D plots of stability analysis in the steadystate condition and the beam’s amplitude frequency response curves are presented.
 Nayfeh, A.H., Lacarbonara, W., Chin, C.M. (1999) Nonlinear normal modes of buckled beams: threetoone and onetoone internal resonances. Nonlinear Dyn. 18: pp. 253273 CrossRef
 Santee, D.M., Goncalves, P.B. (2006) Oscillations of a beam on a nonlinear elastic foundation under periodic loads. Shock Vib. 13: pp. 273284
 Tsiatas, G.C. (2010) Nonlinear analysis of nonuniform beams on nonlinear elastic foundation. Acta Mech. 209: pp. 141152 CrossRef
 Kuo, Y.H., Lee, S.Y. (1994) Deflection of nonuniform beams resting on a nonlinear elastic foundation. Comput. Struct. 51: pp. 513519 CrossRef
 Hsu, M.H. (2006) Mechanical analysis of nonuniform beams resting on nonlinear elastic foundation by the differential quadrature method. Struct. Eng. Mech. 22: pp. 279292
 Oz, H.R., Pakdemirli, M., Ozkaya, E., Yilmaz, M. (1998) Nonlinear vibrations of a slightly curved beam resting on a nonlinear elastic foundation. J. Sound Vib. 212: pp. 295309 CrossRef
 Pellicano, F., Mastroddi, F. (1997) Nonlinear dynamics of a beam on elastic foundation. Nonlinear Dyn. 14: pp. 335355 CrossRef
 Balkaya, M., Kaya, M.O., Saglamer, A. (2009) Analysis of the vibration of an elastic beam supported on elastic soil using the differential transform method. Arch. Appl. Mech. 79: pp. 135146 CrossRef
 Birman, V. (1986) On the effects of nonlinear elastic foundation on free vibration of beams. J. Appl. Mech. 53: pp. 471474 CrossRef
 King, M.E., Vakakis, A.F. (1996) An energybased approach to computing resonant nonlinear normal modes. J. Appl. Mech. 63: pp. 810819 CrossRef
 King, M.E., Vakakis, A.F. (1994) An energybased formulation for computing nonlinear normal modes in undamped continuous systems. J. Vib. Acoust. 116: pp. 332340 CrossRef
 Vakakis, A.F. (1996) Nonlinear mode localization in systems governed by partial differential equations. Appl. Mech. Rev. 49: pp. 8799 CrossRef
 Pellicano, F., Vakakis, A.F. (2001) Normal modes and boundary layers for a slender tensioned beam on a nonlinear foundation. Nonlinear Dyn. 25: pp. 7993 CrossRef
 Jiang, D., Pierre, C., Shaw, S.W. (2005) The construction of nonlinear normal modes for systems with internal resonance. Int. J. NonLinear Mech. 40: pp. 729746 CrossRef
 Pierre, C., Jiang, D., Shaw, S.W. (2006) Nonlinear normal modes and their application in structural dynamics. Math. Probl. Eng. 2006: CrossRef
 Mazzilli, C.E.N., Sanches, C.T., Baracho, O.G.P., Wiercigroch, M., Keber, M. (2008) Nonlinear modal analysis for beams subjected to axial loads: analytical and finiteelement solutions. Int. J. NonLinear Mech. 43: pp. 551561 CrossRef
 Casini, P., Giannini, O., Vestroni, F. (2011) Persistent and ghost nonlinear normal modes in the forced response of nonsmooth systems. Physica D.
 Vestroni, F., Luongo, A., Paolone, A. (2008) A perturbation method for evaluating nonlinear normal modes of a piecewise linear twodegreesoffreedom system. Nonlinear Dyn. 54: pp. 379393 CrossRef
 Casini, P., Vestroni, F. (2011) Characterization of bifurcating nonlinear normal modes in piecewise linear mechanical systems. Int. J. NonLinear Mech. 46: pp. 142150 CrossRef
 Vakakis, A.F., Manevitch, L.I., Mikhlin, Y.V., Pilipchuk, V.N., Zevin, A.A. (2001) Normal Modes and Localization in Nonlinear Systems. Kluwer Academic, Dordrecht
 Vakakis, A.F., Gendelman, O.V., Bergman, L.A., McFarland, D.M., Kerschen, G., Lee, Y.S. (2008) Nonlinear Targeted Energy Transfer in Mechanical and Structural Systems. Springer Science, Business Media, B.V., New York
 Nayfeh, A.H. (2011) The Method of Normal Forms. Wiley, New York CrossRef
 Murdock, J.A. (2003) Normal Forms and Unfoldings for Local Dynamical Systems. Springer, New York
 Nayfeh, A.H., Mook, D.T. (1995) Nonlinear Oscillations. WileyInterscience, New York CrossRef
 Title
 Dynamic analysis of a simply supported beam resting on a nonlinear elastic foundation under compressive axial load using nonlinear normal modes techniques under threetoone internal resonance condition
 Open Access
 Available under Open Access This content is freely available online to anyone, anywhere at any time.
 Journal

Nonlinear Dynamics
Volume 70, Issue 2 , pp 11471172
 Cover Date
 20121001
 DOI
 10.1007/s1107101205201
 Print ISSN
 0924090X
 Online ISSN
 1573269X
 Publisher
 Springer Netherlands
 Additional Links
 Topics
 Keywords

 Beam’s nonlinear dynamics
 Nonlinear elastic foundation
 The 3:1 internal resonance
 Steadystate stability analysis
 Beam’s frequency response
 Industry Sectors
 Authors

 Ahmad Mamandi ^{(1)}
 Mohammad H. Kargarnovin ^{(2)}
 Salman Farsi ^{(3)}
 Author Affiliations

 1. Department of Mechanical Engineering, Parand Branch, Islamic Azad University, Tehran, Iran
 2. Department of Mechanical Engineering, Sharif University of Technology, Tehran, Iran
 3. Department of Mechanical Engineering, Tarbiat Modares University, Tehran, Iran