Bifurcation of coupled-mode responses by modal coupling in cubic nonlinear systems

Pak, C.; Lee, Young

doi:10.1090/qam/1402

Authors: C. H. Pak and Young S. Lee
Journal: Quart. Appl. Math. 74 (2016), 1-26
MSC (2010): Primary 37G15; Secondary 37M20
DOI: https://doi.org/10.1090/qam/1402
Published electronically: December 3, 2015
MathSciNet review: 3472517
Full-text PDF Free Access

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Abstract: When a stable normal mode loses stability in nonlinear conservative 2-degree-of-freedom systems, the phenomenon of internal resonance occurs involving rigorous energy exchange between modes and generating a stable coupled mode (called a modal coupling). Based on this observation, bifurcation of the coupled-mode responses is studied when the system is weakly damped and under a small sinusoidal excitation applied to one mode. The motions are not necessarily assumed to be small throughout. To analyze the stability of the driving mode in the underlying conservative system, a procedure is formulated to construct the stability curve in a stability chart. It is found that if the driving mode loses stability, then a stable coupled-mode response is formed and can be expressed in Fourier series. Assuming that the stability curve of the driving mode enters the $p$th unstable region with $p=2,3,4,\ldots$, the coupled-mode response for $p=2,3,4,\ldots$ can be determined with two terms as the first-order approximation; i.e., each coordinate is expressed by the sum of two predominant harmonic terms. One-term approximation of coupled-mode response is plausible if $p=1$, which may result in 1:1 internal resonance. If the stability curve passes through the $p$th unstable region with $p=2,3,4,\ldots$ and if the coupled-mode responses are expressed in the first-order approximation form, then the frequency response curve of the stable coupled-mode response is overlapped with the curve of the unstable response. As the order of approximation increases, two curves are separated from each other. The proposed method is compared with other perturbation techniques in the systems that exhibit 1:1 and 3:1 internal resonances.

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