On the Nonlinear Dynamics of a Buckled Beam Subjected to a Primary-Resonance Excitation

We investigate the nonlinear response of a clamped-clamped buckled beamto a primary-resonance excitation of its first vibration mode. The beamis subjected to an axial force beyond the critical load of the firstbuckling mode and a transverse harmonic excitation. We solve thenonlinear buckling problem to determine the buckled configurations as afunction of the applied axial load. A Galerkin approximation is used todiscretize the nonlinear partial-differential equation governing themotion of the beam about its buckled configuration and obtain a set ofnonlinearly coupled ordinary-differential equations governing the timeevolution of the response. Single- and multi-mode Galerkinapproximations are used. We found out that using a single-modeapproximation leads to quantitative and qualitative errors in the staticand dynamic behaviors. To investigate the global dynamics, we use ashooting method to integrate the discretized equations and obtainperiodic orbits. The stability and bifurcations of the periodic orbitsare investigated using Floquet theory. The obtained theoretical resultsare in good qualitative agreement with the experimental results obtainedby Kreider and Nayfeh (Nonlinear Dynamics15, 1998, 155–177.

[1]  Ali H. Nayfeh,et al.  Investigation of Natural Frequencies and Mode Shapes of Buckled Beams , 1995 .

[2]  Sathya Hanagud,et al.  Chaotic Vibrations of Beams: Numerical Solution of Partial Differential Equations , 1993 .

[3]  A. M. Abou-Rayan,et al.  Nonlinear response of a parametrically excited buckled beam , 1993 .

[4]  F. Moon.  Experiments on Chaotic Motions of a Forced Nonlinear Oscillator: Strange Attractors , 1980 .

[5]  J. G. Eisley,et al.  Nonlinear Vibration of Buckled Beams , 1972 .

[6]  J. Eisley.  Nonlinear vibration of beams and rectangular plates , 1964 .

[7]  John Dugundji,et al.  Nonlinear Vibrations of a Buckled Beam Under Harmonic Excitation , 1971 .

[8]  J. Eisley,et al.  Stability of large amplitude forced motion of a simply supported beam , 1970 .

[9]  J. G. Eisley,et al.  Large amplitude vibration of buckled beams and rectangular plates , 1964 .

[10]  Philip Holmes,et al.  Strange Attractors and Chaos in Nonlinear Mechanics , 1983 .

[11]  Sathya Hanagud,et al.  Nonlinear vibration of buckled beams: some exact solutions , 2001 .

[12]  T. S. Sankar,et al.  Bifurcations, catastrophes and chaos in a pre-buckled beam , 1994 .

[13]  A. Nayfeh,et al.  Reduction Methods for Nonlinear Vibrations of Spatially Continuous Systems with Initial Curvature , 2000 .

[14]  Ali H. Nayfeh,et al.  Experimental Validation of Reduction Methods for Nonlinear Vibrations of Distributed-Parameter Systems: Analysis of a Buckled Beam , 1998 .

[15]  Earl H. Dowell,et al.  The role of higher modes in the chaotic motion of the buckled beam—II , 1996 .

[16]  Colin H. Hansen,et al.  NON-LINEAR RESPONSE OF A POST-BUCKLED BEAM SUBJECTED TO A HARMONIC AXIAL EXCITATION , 2000 .

[17]  Earl H. Dowell,et al.  On the Threshold Force for Chaotic Motions for a Forced Buckled Beam , 1988 .

[18]  P. Holmes,et al.  A nonlinear oscillator with a strange attractor , 1979, Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences.

[19]  Ali H. Nayfeh,et al.  Experimental Investigation of Single-Mode Responses in a Fixed-Fixed Buckled Beam , 1996 .