Triad mode resonant interactions in suspended cables

A triad mode resonance, or three-wave resonance, is typical of dynamical systems with quadratic nonlinearities. Suspended cables are found to be rich in triad mode resonant dynamics. In this paper, modulation equations for cable＇s triad resonance are formulated by the multiple scale method. Dynamic...

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Bibliographic Details
Published in:Science China. Physics, mechanics & astronomy mechanics & astronomy, 2016-03, Vol.59 (3), p.75-88, Article 634501
Main Authors: Guo, TieDing, Kang, HouJun, Wang, LianHua, Zhao, YueYu
Format: Article
Language:English
Subjects:
Astronomy
Bifurcations
Cables
Classical and Continuum Physics
Dynamical systems
Mathematical analysis
Modulation
Nonlinearity
Observations and Techniques
Physics
Physics and Astronomy
Resonance
Resonant interactions
三波共振
倍周期分岔
共振相互作用
分岔特性
和弦
悬索
非线性动力学系统
非线性行为
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Summary:A triad mode resonance, or three-wave resonance, is typical of dynamical systems with quadratic nonlinearities. Suspended cables are found to be rich in triad mode resonant dynamics. In this paper, modulation equations for cable＇s triad resonance are formulated by the multiple scale method. Dynamic conservative quantities, i.e., mode energy and Manley-Rowe relations, are then constructed. Equilibrium/dynamic solutions of the modulation equations are obtained, and full investigations into their stability and bifurcation characteristics are presented. Various bifurcation behaviors are detected in cable＇s triad resonant re- sponses, such as saddle-node, Hopf, pitchfork and period-doubling bifurcations. Nonlinear behaviors, like jump and saturation phenomena, are also found in cable＇s responses. Based upon the bifurcation analysis, two interesting properties associated with activation of cable＇s triad resonance are also proposed, i.e., energy barrier and directional dependence. The first gives the criti- cal amplitude of high-frequency mode to activate cable＇s triad resonance, and the second characterizes the degree of difficulty for activating cable＇ s triad resonance in two opposite directions, i.e., with positive or negative internal detuning parameter.
ISSN:1674-7348
1869-1927
DOI:10.1007/s11433-015-5766-4