Abstract
A nonlinear analysis of the response of a simply-supported buckled beam to a harmonic axial load is presented. The method of multiple scales is used to determine to second order the amplitude- and phase-modulation equations. Floquet theory is used to analyze the stability of periodic responses. The perturbation results are verified by integrating the governing equation using both digital and analog computers. For small excitation amplitudes, the analytical results are in good agreement with the numerical solutions. The large-amplitude responses are investigated by using a digital computer and are compared with those obtained via an analog-computer simulation. The complicated dynamic behaviors that were found include period-multiplying and period-demultiplying bifurcations, period-three and period-six motions, jump phenomena, and chaos. In some cases, multiple periodic attractors coexist, and a chaotic attractor coexists with a periodic attractor. Phase portraits, spectra of the responses, and a bifurcation set of the many solutions are presented.
Similar content being viewed by others
References
Nayfeh, A. H. and Mook, D. T., Nonlinear Oscillations, Wiley-Interscience, New York, 1979.
Tseng, W. Y. and Dugundji, J., ‘Nonlinear vibrations of a buckled beam under harmonic excitation’, Journal of Applied Mechanics 38, 1971, 467–476.
Moon, F. C., ‘Experiments on chaotic motions of a forced nonlinear oscillator: strange attractors’, Journal of Applied Mechanics 47, 1980, 638–644.
Raty, A., Isomaki, H. M., and VonBoehn, J., ‘Chaotic motion of a classical anharmonic oscillator’, Acta Polytechnica Scandinavica, Mechanical Engineering Series 85, 1984, 1–30.
Moon, F. C. and Li, G.-X., ‘Fractal basin boundaries and homoclinic orbits for periodic motion in two-well potential’, Physical Review Letters 55, 1985, 1439–1442.
Holmes, P. J. and Moon, F. C., ‘Strange attractors and chaos in nonlinear mechanics’, Journal of Applied Mechanics 50, 1983, 1021–1032.
Dowell, E. H., ‘Chaotic oscillations in mechanical systems’, Computers & Structures 30, 1988, 171–184.
Dowell, E. H. and Pezeshki, C., ‘On the understanding of chaos in Duffing equation including a comparison with experiment’, Journal of Applied Mechanics 53, 1986, 5–9.
Tang, D. M. and Dowell, E. H., ‘On the threshold force for chaotic motions for a forced buckled beam’, Journal of Applied Mechanics 55, 1988, 190–196.
Ueda, Y., ‘Randomly transitional phenomena in the system governed by Duffing's equation’, Journal of Statistical Physics 20, 1979, 181–196.
Holmes, P. J., ‘A nonlinear oscillator with a strange attractor’, Philosophical Transactions of the Royal Society of London 292, 1979, 419–447.
Nayfeh, A. H., ‘On the low-frequency drumming of bowed structures’, Journal of Sound and Vibration 94, 1984, 551–562.
Szemplinska-Stupnicka, W., Plaut, R. H., and Hsieh, J.-C., ‘Period doubling and chaos in unsymmetric structures under parametric excitation’, Journal of Applied Mechanics 56, 1989, 947–952.
Zavodney, L. D. and Nayfeh, A. H., ‘The response of a single-degree-of-freedom system with quadratic and cubic non-linearities to a fundamental parametric resonance’, Journal of Sound and Vibration 120, 1988, 63–93.
Zavodney, L. D., Nayfeh, A. H., and Sanchez, N. E., ‘The response of a single-degree-of-freedom system with quadratic and cubic non-linearities to a principal parametric resonance’, Journal of Sound and Vibration 129, 1989, 417–442.
Zavodney, L. D., Nayfeh, A. H., and Sanchez, N. E., ‘Bifurcation and chaos in parametrically excited single-degree-of-freedom systems’, Nonlinear Dynamics 1, 1990, 1–21.
Sanchez, N. E. and Nayfeh, A. H., ‘Prediction of bifurcations in a parametrically excited Duffing oscillator’, International Journal of Non-Linear Mechanics 25, 1990, 163–176.
Nayfeh, A. H., Introduction to Perturbation Techniques, Wiley-Interscience, New York, 1981.
Rahman, Z. and Burton, T. D., ‘On higher order methods of multiple scales in non-linear oscillations-periodic steady state response’, Journal of Sound and Vibration 133, 1989, 369–379.
Nayfeh, A. H., ‘Perturbation methods in nonlinear dynamics’, Nonlinear Dynamics Aspects of Particle Accelerators, Lecture Notes in Physics, Springer-Verlag, New York, 247, 1986, 238–314.
Nayfeh, M. A., Hamdan, A. M. A., and Nayfeh, A. H., ‘Chaos and instability in a power system: subharmonic-resonant case’, Nonlinear Dynamics 2, 1991, 53–72.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Abou-Rayan, A.M., Nayfeh, A.H., Mook, D.T. et al. Nonlinear response of a parametrically excited buckled beam. Nonlinear Dyn 4, 499–525 (1993). https://doi.org/10.1007/BF00053693
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/BF00053693