Nonlinear phenomena in the vibration of a slender, Euler-Bernoulli beam in compression and under periodic transverse loading is investigated. A feature of this system is the coexistence of distinct bifurcation branches which provide a rich resource for numerous solution states. An indepth study based on an energy approach is done to illustrate the presence of multiple stability resulting from the multiplicity of resonant solutions. Although the behaviors may exhibit a variety of different motions, the ultimate state is very sensitively dependent upon the initial conditions. The structures of boundary basins for the coexisting attractors are illustrated and the unpredictability of outcome is discussed in detail.
Issue Section:
Technical Papers
1.
Tseng
, W. Y.
, and Dugundji
, J.
, 1971
, “Nonlinear Vibrations of a Buckled Beam Under Harmonic Excitation
,” ASME J. Appl. Mech.
, 38
, pp. 467
–476
.2.
Holmes
, P.
, 1979
, “A Nonlinear Oscillator With a Strange Attractor
,” Philos. Trans. R. Soc. London, Ser. A
, 292
, pp. 419
–448
.3.
Holmes
, P.
, and Marsden
, J.
, 1981
, “A Partial Differential Equation With Infinitely Many Periodic Orbits: Chaotic Oscillations of a Forced Beam
,” Arch. Ration. Mech. Anal.
, 76
, pp. 135
–166
.4.
Ueda
, Y.
, 1979
, “Randomly Transitionally Phenomena in the System Governed by Duffing’s Equation
,” J. Stat. Phys.
, 20
, pp. 181
–195
.5.
Symonds
, P. S.
, and Yu
, T. X.
, 1985
, “Counterintuitive Behavior in a Problem of Elastic-Plastic Beam Dynamics
,” ASME J. Appl. Mech.
, 52
, pp. 517
–522
.6.
Leung
, A. Y. T.
, and Mao
, S. G.
, 1995
, “A Simplectic Galerkin Method for Nonlinear Vibration of Beams and Plates
,” J. Sound Vib.
, 183
, pp. 475
–491
.7.
Leung
, A. Y. T.
, and Mao
, S. G.
, 1995
, “Simplectic Integration of an Accurate Beam Finite Element in Nonlinear Vibration
,” Comput. Struct.
, 54
, pp. 1135
–1147
.8.
Ramu
, S. A.
, Sankar
, T. S.
, and Ganesan
, R.
, 1994
, “Bifurcations, Catastrophes and Chaos in a Pre-buckled Beam
,” Int. J. Non-Linear Mech.
, 29
, pp. 449
–462
.9.
Moorthy
, R. I. K.
, Kakodkar
, A.
, Srirangarajan
, H. R.
, and Suryanarayan
, S.
, 1993
, “Finite Element Simulation of Chaotic Vibrations of a Beam With Nonlinear Boundary Conditions
,” Comput. Struct.
, 49
, pp. 589
–596
.10.
Baran
, D. D.
, 1994
, “Mathematical Models Used in Studying the Chaotic Vibration of Buckled Beams
,” Mech. Res. Commun.
, 21
, pp. 189
–196
.11.
Abhyankar, N. S., 1986, “Studies in Nonlinear Structural Dynamics: Chaotic Behavior and Pointing Effect,” Ph.D. Thesis, School of Aerospace Engineering, Georgia Institute of Technology, Atlanta, GA.
12.
Soliman
, M. S.
, and Thompson
, J. M. T.
, 1992
, “Indeterminate Sub-critical Bifurcations in Parametric Resonance
,” Proc. R. Soc. London, Ser. A
, 438
, pp. 511
–518
.13.
Soliman
, M. S.
, and Thompson
, J. M. T.
, 1996
, “Indeterminate Bifurcational Phenomena in Hardening Systems
,” Proc. R. Soc. London, Ser. A
, 452
, pp. 487
–494
.14.
Soliman
, M. S.
, 1993
, “Jumps to Resonance: Long Chaotic Transients, Unpredictable Outcome, and the Probability of Restabilization
,” ASME J. Appl. Mech.
, 60
, pp. 669
–676
.15.
Stewart
, H. B.
, and Ueda
, Y.
, 1991
, “Catastrophes With Indeterminate Outcome
,” Proc. R. Soc. London, Ser. A
, 432
, pp. 113
–123
.16.
Ueda
, Y.
, Yoshida
, S.
, Stewart
, H. B.
, and Thompson
, J. M. T.
, 1990
, “Basin Explosions and Escape Phenomena in the Twin-Well Duffing Oscillator—Compound Global Bifurcations Organizing Behavior
,” Philos. Trans. R. Soc. London, Ser. A
, 332
, pp. 169
–186
.17.
Borino
, G.
, Perego
, U.
, and Symonds
, P. S.
, 1989
, “An Energy Approach to Anomalous Damped Elastic-Plastic Response to Short Pulse Loading
,” ASME J. Appl. Mech.
, 56
, pp. 430
–438
.18.
Perego
, U.
, Borino
, G.
, and Symonds
, P. S.
, 1989
, “The Role of Damping in Anomalous Response to Short Pulse Loading
,” J. Eng. Mech.
, 115
, pp. 2782
–2788
.19.
Guckenheimer, J., and Holmes, P., 1983, Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields, Springer-Verlag, New York.
Copyright © 2002
by ASME
You do not currently have access to this content.
