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Frontiers of Structural and Civil Engineering

Front. Struct. Civ. Eng.    2014, Vol. 8 Issue (2) : 151-159
Vibration analysis of multi-walled carbon nanotubes embedded in elastic medium
Pattabhi R. BUDARAPU1,*(),Sudhir Sastry YB2,Brahmanandam JAVVAJI3,D. Roy MAHAPATRA3
1. Institute of Structural Mechanics, Bauhaus University of Weimar, Weimar 99423, Germany
2. Deptarment of Aeronautical Engineering, Institute of Aeronautical Engineering, Hyderabad 500043, India
3. Department of Aerospace Engineering, Indian Institute of Science, Bangalore 560012, India
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We propose a method to estimate the natural frequencies of the multi-walled carbon nanotubes (MWCNTs) embedded in an elastic medium. Each of the nested tubes is treated as an individual bar interacting with the adjacent nanotubes through the inter-tube Van der Waals forces. The effect of the elastic medium is introduced through an elastic model. The mathematical model is finally reduced to an eigen value problem and the eigen value problem is solved to arrive at the inter-tube resonances of the MWCNTs. Variation of the natural frequencies with different parameters are studied. The estimated results from the present method are compared with the literature and results are observed to be in close agreement.

Keywords natural frequencies      multi-walled carbon nanotubes (MWCNTS)      elastic medium     
Corresponding Authors: Pattabhi R. BUDARAPU   
Issue Date: 19 May 2014
 Cite this article:   
Pattabhi R. BUDARAPU,Sudhir Sastry YB,Brahmanandam JAVVAJI, et al. Vibration analysis of multi-walled carbon nanotubes embedded in elastic medium[J]. Front. Struct. Civ. Eng., 2014, 8(2): 151-159.
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Pattabhi R. BUDARAPU
Sudhir Sastry YB
Brahmanandam JAVVAJI
Fig.1  Transverse vibrations of a bar. (a) Deformed configuration of a bar subjected to flexural vibration, indicating the material properties and the external forces on the bar; (b) Free body diagram of an element of length dx, of the bar in (a)
Fig.2  Schematic of MWCNTs embedded in elastic medium
Fig.3  Variation of the stiffness constant with the parameter nπd/2L
Fig.4  Variation of the first three natural frequencies of the SWCNTs (a) with the length, for a constant inner diameter of 0.7 nm and (b) with the diameter, for constant length of 20 nm
Fig.5  Variation of the first natural frequency of the SWCNTs (a) with the length for various inner diameters and with the diameter for various lengths (b) with the Young’s modulus of the medium
Fig.6  Natural frequencies of the DWCNTs. (a) Comparison of the first mode natural frequencies of the DWCNTs without the elastic medium, from the present method with Ref. [32]; (b) Effect of the elastic medium on the natural frequencies of the DWCNTs. Variation of the natural frequencies with L/d ratio are plotted with and without the elastic medium
Fig.7  Variation of the first and the second natural frequencies of the first (ω11, ω21) and the second nanotubes (ω12, ω22) of the DWCNTs with length, for diameters equal to 1, 1.5 and 2 nm
Fig.8  Variation of the first five natural frequencies of the DWCNTs (a) with the length for a constant inner diameter of the first tube equal to 0.7 nm and (b) with the diameter for a constant length of 20 nm
Fig.9  Variation of the first five natural frequencies of a 5 wall CNTs (a) with the modulus of the elastic medium and (b) with the L/d ratio
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