Chaos in an embedded single-walled carbon nanotube
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Considering the geometrical nonlinearity of an embedded single-walled carbon nanotube, the analytical condition and the numerical results of chaotic vibration of the carbon nanotube are presented in this paper. Firstly, based on the Galerkin approximation method, a Duffing-type model is derived from the equation of motion that describes the oscillation of the embedded single-walled carbon nanotube clamped at both ends under a transverse load. And then, the Melnikov function of the Duffing-type model is derived. From the Melnikov function, the analytical condition of the chaos in the nanotube is obtained. Finally, a structure-preserving difference scheme for the original oscillating model is constructed based on the generalized multi-symplectic framework and the chaotic vibration of the nanotube is reproduced to verify the accuracy and the validity of the analytical condition. The analytical condition obtained in this paper gives some guidance on the property studying and the structure designing of some carbon nanotube devices.
KeywordsEmbedded single-walled carbon nanotube Chaos Generalized multi-symplectic Galerkin approximation Melnikov function Structure-preserving
The authors wish to thank Professor Thomas J. Bridges of Surrey University for giving us several good suggestions. The research is supported by the National Natural Science Foundation of China (11002115, 10972182, 11172239), the Science Foundation of Aviation of China (2010ZB53021), 111 project (B07050) to the Northwestern Polytechnical University, the NPU Foundation for Fundamental Research (JC20110259), the Doctoral Program Foundation of Education Ministry of China (20126102110023), the Open Foundation of State Key Laboratory of Mechanical System & Vibration (MSV-2011-21), and the Open Foundation of State Key Laboratory of Structural Analysis of Industrial Equipment (GZ0802).
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