Abstract
The purpose of this paper is to examine a highly nonlinear model of a slender beam which yields chaotic solutions for some forcing amplitudes. The study is unique in that the governing partial differential equations are solved directly, and that the model lends itself to a more physical analysis of the beam than traditional chaotic models. In addition, the analysis will provide proof that a beam experiencing moderate deformations without stops or an initial axial force can exhibit chaotic motion. The model represents a simply-supported. Euler-Bernoulli beam subjected to a transverse load. The forcing function is sinusoidally distributed in space with an amplitude which also varies sinusoidally in time and is assumed to reach a maximum sufficient to allow nonlinearities associated with finite deformations to become important. During motion, even though displacements are large, the beam is assumed to attain only small strain levels and thus is assumed to be linearly elastic. The results indicate that for most levels of the forcing function the response of the beam is periodic. However, the steady state motion is not sinusoidal in time and in fact exhibits some bifurcated motions. At a certain level of the forcing amplitude, an asymmetry is observed and the periodicity of the motion breaks down as the beam experiences a period doubling cascade which culminates in a chaotic motion. The progression from periodic to chaotic motion is presented through a series of phase plane and Poincané plots, and physical variables such as bending moment are examined.
Similar content being viewed by others
References
Holmes, P. J., ‘A nonlinear oscillator with a strange attractor’, Philosophical Transactions of the Royal Society, London, 292, #1394, Oct. 1979, 419–448.
Shaw, S. W., ‘Chaotic dynamics of a slender beam rotating about its longitudinal axis’, J. Sound Vib. 124(2), 1988, 329–343.
Moon, F. C. and Holmes, P. J., ‘A magnetoelastic strange attractor’, J. Sound Vib. 65, 1979, 285–296.
Dowell, E. H. and Pezeshki, C., ‘On the understanding of chaos in Duffing's equation including a comparison with experiment’, J. Appl. Mech. 53, 1986, 5–9.
Abhyankar, N. S., ‘Studies in nonlinear structural dynamics: Chaotic behavior and Poynting effect’, Ph.D. Thesis, Georgia Institute of Technology, Atlanta, GA, 1987.
Moon, F. C., Chaotic Vibrations, John Wiley & Sons, New York, NY, 1987.
Pezeshki, C. and Dowell, E. H., ‘On chaos and fractal behavior in a generalized Duffing's system’, Physica D 32, 1988, 194–209.
Zavodney, L. D. and Nayfeh, A. H., ‘The non-linear response of a slender beam carrying a lumped mass to a principal parametric excitation: Theory and experiment’, Int. J. Non-linear Mechanics 24(2), 1989, 105–125.
Crespo da Silva, M. R. M., ‘Non-linear flexural-flexural-torsional-extensional dynamics of beams — I. Formulation’, Int. J. Solids Structures 24 (12), 1988, 1225–1234.
Hall II, E. K., ‘Chaotic motion of beams due to finite deformations’, Proceedings of the 31st Structures, Structural Dynamics and Materials Conference, Long Beach, CA, April 2–4, 1990.
Feigenbaum, M. J., ‘Quantitative universality for a class of nonlinear transformations’, J. Stat. Phys., 19, 1978, 25–52.
Devaney, R. L., An Introduction to Chaotic Dynamical Systems, 2nd Ed., Addison-Wesley, Redwood City, CA, 1989.
Rudnick, J., ‘Subharmonics and the transition to chaos’, Lecture Notes in Physics: 179, Dynamical Systems and Chaos, Springer-Verlag, Berlin, 1982, 115–129.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Hall, E.K., Hanagud, S.V. Chaos in a single equilibrium point system: Finite deformations. Nonlinear Dyn 2, 157–170 (1991). https://doi.org/10.1007/BF00045721
Issue Date:
DOI: https://doi.org/10.1007/BF00045721