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Research Papers

Vibrations of Double-Walled Carbon Nanotubes With Different Boundary Conditions Between Inner and Outer Tubes

[+] Author and Article Information
Kai-Yu Xu

Department of Mechanics, Shanghai Institute of Applied Mathematics and Mechanics, Shanghai University, 99 Shangda Road, Shanghai 200444, China; College of Engineering, Michigan Technological University, 1400 Townsend Drive, Houghton, MI 49931kyxu@staff.shu.edu.cn.

Elias C. Aifantis

College of Engineering, Michigan Technological University, 1400 Townsend Drive, Houghton, MI 49931

Yong-Hua Yan

Department of Mechanics, Shanghai Institute of Applied Mathematics and Mechanics, Shanghai University, 99 Shangda Road, Shanghai 200444, China

J. Appl. Mech 75(2), 021013 (Feb 26, 2008) (9 pages) doi:10.1115/1.2793133 History: Received August 03, 2006; Revised May 31, 2007; Published February 26, 2008

Free vibrations of a double-walled carbon nanotube (DWNT) are studied. The inner and outer carbon nanotubes are modeled as two individual elastic beams interacting each other by van der Waals forces. An original method is proposed to calculate the first seven order resonant frequencies and relative vibrational modes. Detailed results are demonstrated for DWNTs according to the different boundary conditions between inner and outer tubes, such as fixed-free, cantilever-free, fixed-simple and fixed-fixed (reduced form) supported ends. Our results indicate that there is a special invariable frequency for a DWNT that is not affected by different combinations of boundary conditions. All vibrational modes of the DWNT must be coaxial when the resonant frequency is smaller than this frequency. Some noncoaxial vibrations will occur when their resonant frequencies exceed the frequency. Especially, the first noncoaxial resonant frequency is still invariable for all different boundary conditions. A change of resonant frequency for various lengths of DWNTs is discussed in detail. In addition, our model predicts a new coaxial-noncoaxial vibrational mode in fixed-simple supports for inner and outer tubes of a DWNT.

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Copyright © 2008 by American Society of Mechanical Engineers
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Figures

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Figure 1

The first seven vibrational modes of a DWNT with a free inner tube and a fixed outer tube

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Figure 2

The first seven vibrational modes of a DWNT with a free inner tube and a cantilever outer tube

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Figure 3

The first seven vibrational modes of a DWNT with a simple inner tube and a fixed outer tube

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Figure 4

The first six vibrational modes of a DWNT with both fixed inner and outer tubes

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Figure 5

Resonant frequency attenuation for the increasing length of a DWNT with a free inner tube and a fixed outer tube with an inner diameter of 0.7nm and an outer diameter of 1.4nm

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