Abstract
The response of a structure to a simple-harmonic excitation is investigated theoretically and experimentally. The structure consists of two light-weight beams arranged in a T-shape turned on its side. Relatively heavy and concentrated weights are placed at the upper and lower free ends and at the point where the two beams are joined. The base of the ‘T’ is clamped to the head of a shaker. Because the masses of the concentrated weights are much larger than the masses of the beams, the first three natural frequencies are far below the fourth; consequently, for relatively low frequencies of the excitation, the structure has, for all practical purposes, only three degrees of freedom. The lengths and weights are chosen so that the third natural frequency is approximately equal to the sum of the two lower natural frequencies, an arrangement that produces an autoparametric (also called an internal) resonance. A linear analysis is performed to predict the natural frequencies and to aid in the design of the experiment; the predictions and observations are in close agreement. Then a nonlinear analysis of the response to a prescribed transverse motion at the base of the ‘T’ is performed. The method of multiple scales is used to obtain six first-order differential equations describing the modulations of the amplitudes and phases of the three interacting modes when the frequency of the excitation is near the third natural frequency. Some of the predicted phenomena include periodic, two-period quasiperiodic, and phase-locked (also called synchronized) motions; coexistence of multiple stable motions and the attendant jumps; and saturation. All the predictions are confirmed in the experiments, and some phenomena that are not yet explained by theory are observed.
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Nayfeh, A. H., ‘Application of the method of multiple scales to nonlinearly coupled oscillators’, in Lasers, Molecules and Methods, edited by J. O.Hirschfelder, R. E.Wyatt, and R. D.Coalson, John Wiley & Sons, New York, 1989, pp. 137–196.
Nayfeh, A. H. and Balachandran, B., ‘Modal interactions in dynamical and structural systems’, Applied Mechanics Reviews 42, 1989, 175–201.
Nayfeh, A. H. and Mook, D. T., Nonlinear Oscillations, Wiley-Interscience, New York, 1979.
Froude, W., ‘Remarks on Mr. Scott-Russell's paper on rolling’, Transactions of the Institute of Naval Architects 4, 1863, 232–275.
Nayfeh, A. H. and Mook, D. T., ‘A saturation phenomenon in the forced response of systems with quadratic nonlinearities’, Proceedings of the 8th International Conference on Nonlinear Oscillations, September 11–15, 1978, Prague, Czechoslovakia, pp. 511–516.
Ibrahim, R. A. and Barr, A. D. S., ‘Autoparametric resonance in a structure containing a liquid, Part II: Three mode interaction’, Journal of Sound and Vibration 42, 1975, 181–200.
Ibrahim, R. A., Woodall, T. D., and Heo, H., ‘Modal analysis of structural systems involving nonlinear coupling’, The Shock and Vibration Bulletin 54, 1984, 19–27.
Bux, S. L. and Roberts, J. W., ‘Non-linear vibratory interactions in systems of coupled beams’, Journal of Sound and Vibration 104, 1986, 497–520.
Nayfeh, T. A., Nayfeh, A. H., and Mook, D. T., ‘A theoretical-experimental investigation of a three-degree-of-freedom structure’, AIAA Paper No. 90-1081, 1990.
Ashworth, R. P. and Barr, A. D. S., ‘The resonances of structures with quadratic inertial non-linearity under direct and parametric harmonic excitation’, Journal of Sound and Vibration 118, 1987, 47–68.
Sridhar, S., Mook, D. T., and Nayfeh, A. H., ‘Non-linear resonances in the forced responses of plates, Part I: Symmetric responses of circular plates’, Journal of Sound and Vibration 41, 1975, 359–373.
Hadian, J. and Nayfeh, A. H., ‘Modal interaction in circular plates’, Journal of Sound and Vibration 142, 1990, 279–292.
Cartmell, M. P. and Roberts, J. W., ‘Simultaneous combination resonances in an autoparametrically resonant system’, Journal of Sound and Vibration 123, 1988, 81–101.
Nayfeh, A. H., Perturbation Methods, Wiley-Interscience, New York, 1973.
Nayfeh, A. H., Introduction to Perturbation Techniques, Wiley-Interscience, New York, 1981.
Nayfeh, A. H. and Jebril, A. E. S., ‘The response of two-degree-of-freedom systems with quadratic and cubic nonlinearities to multifrequency parametric excitations’, Journal of Sound and Vibration 115, 1987, 83–101.
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Nayfeh, T.A., Nayfeh, A.H. & Mook, D.T. A theoretical and experimental investigation of a three-degree-of-freedom structure. Nonlinear Dyn 6, 353–374 (1994). https://doi.org/10.1007/BF00053391
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DOI: https://doi.org/10.1007/BF00053391