The modal interactions and nonlinear responses of inextensional beams resting on elastic foundations with two-to-one internal resonances are investigated and the primary resonance excitations are considered. The multimode discretization and the method of multiple scales are applied to obtain the modulation equations. The equilibrium and dynamic solutions of the modulation equations are examined by the Newton–Raphson, shooting, and continuation methods. Numerical simulations are performed to investigate the chaotic dynamics of the beam. It is shown that the nonlinear responses may undergo different bifurcations and exhibit rich nonlinear phenomena. Finally, the effects of the foundation models on the nonlinear interactions of the beam are examined.
Issue Section:
Research Papers
References
1.
Cheng
, F. Y.
, and Pantelides
, C. P.
, 1988
, “Dynamic Timoshenko Beam-Columns on Elastic Media
,” J. Struct. Eng.
, 114
, pp. 1524
–1550
.10.1061/(ASCE)0733-9445(1988)114:7(1524)2.
Lai
, Y. C.
, Ting
, B. Y.
, Lee
, W. S.
, and Becker
, B. R.
, 1992
, “Dynamic Response of Beams on Elastic Foundation
,” J. Struct. Eng.
, 118
, pp. 853
–858
.10.1061/(ASCE)0733-9445(1992)118:3(853)3.
Thambiratnam
, D.
, and Zhuge
, Y.
, 1996
, “Free Vibration Analysis of Beams on Elastic Foundation
,” Comput. Struct.
, 60
, pp. 971
–980
.10.1016/0045-7949(96)00053-34.
Valsangkar
, A. J.
, and Pradhanang
, R.
, 1988
, “Vibrations of Beam-Columns on Two-Parameter Elastic Foundations
,” Earthquake Eng. Struct. Dyn.
, 16
, pp. 217
–225
.10.1002/eqe.42901602055.
Wang
, L.
, Ma
, J.
, Zhao
, Y.
, and Liu
, Q.
, 2012
, “Refined Modeling and Free Vibration of Inextensional Beams on the Elastic Foundation
,” ASME J. Appl. Mech.
(in press).10.1115/1.40230326.
Zhu
, B.
, and Leung
, A. Y. T.
, 2009
, “Linear and Nonlinear Vibration of Non-Uniform Beams on Two-Parameter Foundations Using p-Elements
,” Comput. Geotech.
, 36
, pp. 743
–750
.10.1016/j.compgeo.2008.12.0067.
Coskun
, I.
, and Engin
, H.
, 1999
, “Non-Linear Vibrations of a Beam on an Elastic Foundation
,” J. Sound Vib.
, 223
, pp. 335
–354
.10.1006/jsvi.1998.19738.
Nayfeh
, A. H.
, and Lacarbonara
, W.
, 1997
, “On the Discretization of Distributed-Parameter Systems With Quadratic and Cubic Nonlinearities
,” Nonlinear Dyn.
, 13
, pp: 203
–220
.10.1023/A:10082539012559.
Ansari
, M.
, Esmailzadeh
, E.
, and Younesian
, D.
, 2010
, “Internal-External Resonance of Beams on Non-Linear Viscoelastic Foundation Transversed by Moving Load
,” Nonlinear Dyn.
, 61
, pp. 163
–182
.10.1007/s11071-009-9639-010.
Pellicano
, F.
, and Mastroddi
, F.
, 1997
, “Nonlinear Dynamics of a Beam on Elastic Foundation
,” Nonlinear Dyn.
, 14
, pp. 335
–355
.10.1023/A:100829772125311.
Pellicano
, F.
, and Vakakis
, A. F.
, 2001
, “Normal Modes and Boundary Layers for a Slender Tensioned Beam on a Nonlinear Foundation
,” Nonlinear Dyn.
, 25
, pp. 79
–93
.10.1023/A:101298612897612.
Lacarbonara
, W.
, Rega
, G.
, and Nayfeh
, A. H.
, 2003
, “Resonant Non-Linear Normal Modes. Part I: Analytical Treatment for Structural One-Dimensional Systems
,” Int. J. Non-Linear Mech.
, 38
, pp. 851
–872
.10.1016/S0020-7462(02)00033-113.
Nayfeh
, A. H.
, 2000
, Non-Linear Interactions
, Wiley-Interscience
, New York
.14.
Feng
, Z. C.
, and Leal
, L. G.
, 1995
, “Symmetries of the Amplitude Equations of an Inextensional Beam With Internal Resonance
,” ASME J. Appl. Mech.
, 62
, pp. 235
–238
.10.1115/1.289591115.
Srinil
, N.
, and Rega
, G.
, 2007
, “The Effects of Kinematic Condensation on Internally Resonant Forced Vibrations of Shallow Horizontal Cables
,” Int. J. Non-Linear Mech.
, 42
, pp. 180
–195
.10.1016/j.ijnonlinmec.2006.09.00516.
Minitab
, 2003
, “Mett MINITAB
,” Release 14 for Windows, 1st ed, Minitab Inc., State College, PA.17.
Nayfeh
, A. H.
, and Balachandran
, B.
, 1994
, Applied Nonlinear Dynamics
, Wiley-Interscience
, New York
.18.
Wang
, L.
, Zhao
, Y.
, and Rega
, G.
, 2009
, “Multimode Dynamics and Out-of-Plane Drift in Suspended Cable Using the Kinematically Condensed Model
,” ASME J. Vibr. Acoust.
, 131
, p. 061008
.10.1115/1.4000470Copyright © 2013 by ASME
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