Nonlinear Responses of Buckled Beams to Subharmonic-Resonance Excitations

We investigated theoretically and experimentally the nonlinear responseof a clamped-clamped buckled beam to a subharmonic resonance of orderone-half of its first vibration mode. We used a multi-mode Galerkindiscretization to reduce the governing nonlinear partial-differentialequation in space and time into a set of nonlinearly coupledordinary-differential equations in time only. We solved the discretizedequations using the method of multiple scales to obtain a second-orderapproximate solution, including the modulation equations governing itsamplitude and phase, the effective nonlinearity, and the effectiveforcing. To investigate the large-amplitude dynamics, we numericallyintegrated the discretized equations using a shooting method to computeperiodic orbits and used Floquet theory to investigate their stabilityand bifurcations. We obtained interesting dynamics, such as phase-lockedand quasiperiodic motions, resulting from a Hopf bifurcation,snapthrough motions, and a sequence of period-doubling bifurcationsleading to chaos. Some of these nonlinear phenomena, such as Hopfbifurcation, cannot be predicted using a single-mode Galerkindiscretization. We carried out an experiment and obtained results ingood qualitative agreement with the theoretical results.

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

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

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

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

[5]  A. Nayfeh,et al.  Applied nonlinear dynamics : analytical, computational, and experimental methods , 1995 .

[6]  A. Nayfeh.  Introduction To Perturbation Techniques , 1981 .

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

[8]  Ali H. Nayfeh,et al.  On the Nonlinear Dynamics of a Buckled Beam Subjected to a Primary-Resonance Excitation , 2004 .

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

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

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

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

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

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

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