Abstract
The nonlinear response of an initially buckled beam in the neighborhood of 1:1 internal resonance is investigated analytically, numerically, and experimentally. The method of multiple time scales is applied to derive the equations in amplitudes and phase angles. Within a small range of the internal detuning parameter, the first mode; which is externally excited, is found to transfer energy to the second mode. Outside this region, the response is governed by a unimodal response of the first mode. Stability boundaries of the unimodal response are determined in terms of the excitation level, and internal and external detuning parameters. Boundaries separating unimodal from mixed mode responses are obtained in terms of the excitation and internal detuning parameters. Stationary and non-stationary solutions are found to coexist in the case of mixed mode response. For the case of non-stationary response, the modulation of the amplitude depends on the integration increment such that the motion can be periodically or chaotically modulated for a choice of different integration increments. The results obtained by multiple time scales are qualitatively compared with those obtained by numerical simulation of the original equations of motion and by experimental measurements. Both numerical integration and experimental results reveal the occurrence of multifurcation, escaping from one well to the other in an irregular manner. and chaotic motion.
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Afaneh, A.A., Ibrahim, R.A. Nonlinear response of an initially buckled beam with 1:1 internal resonance to sinusoidal excitation. Nonlinear Dyn 4, 547–571 (1993). https://doi.org/10.1007/BF00162232
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DOI: https://doi.org/10.1007/BF00162232