Short carbon nanotubes of smaller aspect ratio (say, between 10 and 50) are finding significant application in nanotechnology. This paper studies vibration of such short carbon nanotubes whose higher-order resonant frequencies fall within terahertz range. Because rotary inertia and shear deformation are significant for higher-order modes of shorter elastic beams, the carbon nanotubes studied here are modeled as Timoshenko beams instead of classical Euler beams. Detailed results are demonstrated for double-wall carbon nanotubes of aspect ratio 10, 20, or 50 based on the Timoshenko-beam model and the Euler-beam model, respectively. Comparisons between different single-beam or double-beam models indicate that rotary inertia and shear deformation, accounted for by the Timoshenko-beam model, have a substantial effect on higher-order resonant frequencies and modes of double-wall carbon nanotubes of small aspect ratio (between 10 and 20). In particular, Timoshenoko-beam effects are significant for both large-diameter and small-diameter double-wall carbon nanotubes, while double-beam effects characterized by noncoaxial deflections of the inner and outer tubes are more significant for small-diameter than large-diameter double-wall carbon nanotubes. This suggests that the Timoshenko-beam model, rather than the Euler-beam model, is relevant for terahertz vibration of short carbon nanotubes.

1.
Rueckers
,
T.
,
Kim
,
K.
,
Joselevich
,
E.
,
Tseng
,
G. T.
,
Cheung
,
C. L.
, and
Lieber
,
C. M.
,
2000
, “
Carbon Nanotube-Based Nonvolatile Random Access Memory for Molecular Computing
,”
Science
,
289
,
94
97
.
2.
Postma
,
H. W. C.
,
Teepen
,
T.
,
Yao
,
Z.
,
Grifoni
,
M.
, and
Dekker
,
C.
,
2000
, “
Carbon Nanotube Single-Electron Transistors at Room Temperature
,”
Science
,
293
,
76
79
.
3.
Roschier
,
L.
,
Tarkiainen
,
R.
,
Ahlskog
,
M.
,
Paalanen
,
M.
, and
Hakonen
,
P.
,
2001
, “
Multiwalled Carbon Nanotubes as Ultrasensitive Electrometers
,”
Appl. Phys. Lett.
,
78
,
3295
3297
.
4.
Ahlskog
,
M.
,
Hakonen
,
P.
,
Paalanen
,
M.
,
Roschier
,
L.
, and
Tarkiainen
,
R.
,
2001
, “
Multiwalled Carbon Nanotubes as Building Blocks in Nanoelectronics
,”
J. Low Temp. Phys.
,
124
,
335
352
.
5.
Dai
,
H.
,
Hafner
,
J. H.
,
Rinzler
,
A. G.
,
Colbert
,
D. T.
, and
Smalley
,
R. E.
,
1996
, “
Nanotubes as Nanoprobes in Scanning Probe Microscopy
,”
Nature (London)
,
384
,
147
150
.
6.
Kim
,
P.
, and
Lieber
,
C. M.
,
1999
, “
Nanotube Nanotweezers
,”
Science
,
286
,
2148
50
.
7.
Cumings
,
J.
, and
Zettel
,
A.
,
2000
, “
Low-Friction Nanoscale Linear Bearing Realized From Multiwall Carbon Nanotubes
,”
Science
,
289
,
602
604
.
8.
Thostenson
,
E. T.
,
Ren
,
Z.
, and
Chou
,
T. W.
,
2001
, “
Advances in the Science and Technology of Carbon Nanotubes and Their Composites: A Review
,”
Compos. Sci. Technol.
,
61
,
1899
1912
.
9.
Qian
,
D.
,
Wagner
,
G. J.
,
Liu
,
W. K.
,
Yu
,
M. F.
, and
Ruoff
,
R. S.
,
2002
, “
Mechanics of Carbon Nanotubes
,”
Appl. Mech. Rev.
,
55
,
495
533
.
10.
Ru, C. Q., 2004, “Elastic Models For Carbon Nanotubes,” Encyclopedia of Nanoscience and Nanotechnology, Vol. 2, H. S. Nalwa, ed., American Scientific Publishers, Stevenson Ranch, CA, pp. 731–744.
11.
Wong
,
E. W.
,
Sheehan
,
P. E.
, and
Lieber
,
C. M.
,
1997
, “
Nanobeam Mechanics: Elasticity, Strength, and Toughness of Nanorods and Nanotubes
,”
Science
,
277
,
1971
75
.
12.
Treacy
,
M. M. J.
,
Ebbesen
,
T. W.
, and
Gibson
,
J. M.
,
1996
, “
Exceptonally High Young’s Modulus Observed for Individual Carbon Nanotubes
,”
Nature (London)
,
381
,
678
680
.
13.
Poncharal
,
P.
,
Wang
,
Z. L.
,
Ugarte
,
D.
, and
de Heer
,
W. A.
,
1999
, “
Electrostatic Deflections and Electromechanical Resonances of Carbon Nanotubes
,”
Science
,
283
,
1513
16
.
14.
Harik
,
V. M.
,
2001
, “
Ranges of Applicability for the Continuum Beam Model in the Mechanics of Carbon Nanotubes and Nanorods
,”
Solid State Commun.
,
120
,
331
335
.
15.
Ru
,
C. Q.
,
2000
, “
Column Buckling of Multiwalled Carbon Nanotubes With Interlayer Radial Displacements
,”
Phys. Rev. B
,
62
,
16962
67
.
16.
Yoon
,
J.
,
Ru
,
C. Q.
, and
Mioduchowski
,
A.
,
2002
, “
Non-Coaxial Resonance of an Isolated Multiwall Carbon Nanotube
,”
Phys. Rev. B
,
66
,
233
402
.
17.
Yoon
,
J.
,
Ru
,
C. Q.
, and
Mioduchowski
,
A.
,
2003
, “
Vibration of Embedded Multiwall Carbon Nanotubes
,”
Compos. Sci. Technol.
,
63
,
1533
1542
.
18.
Dequesnes
,
M.
,
Rotkin
,
S. V.
, and
Aluru
,
N. R.
,
2002
, “
Calculation of Pull-In Voltages for Carbon-Nanotube-Based Nanoelectromechanical Switches
,”
Nanotechnology
,
13
,
120
131
.
19.
Snow
,
E. S.
,
Campbell
,
P. M.
, and
Novak
,
J. P.
,
2002
, “
Singlewall Carbon Nanotube Atomic Force Microscope Probes
,”
Appl. Phys. Lett.
,
80
,
2002
4
.
20.
Ishikawa
,
M.
,
Yoshimura
,
M.
, and
Ueda
,
K.
,
2002
, “
A Study of Friction by Carbon Nanotube Tip
,”
Appl. Surf. Sci.
,
188
,
456
459
.
21.
Zhao
,
Y.
,
Ma
,
C. C.
,
Chen
,
G.
, and
Jiang
,
Q.
,
2003
, “
Energy Dissipation Mechanisms in Carbon Nanotube Oscillators
,”
Phys. Rev. Lett.
,
91
,
175
504
.
22.
Li
,
C.
, and
Chou
,
T. W.
,
2004
, “
Vibrational Behaviors of Multiwalled-Carbon-Nanotube-Based Nanomechanical Resonators
,”
Appl. Phys. Lett.
,
84
,
121
123
.
23.
Timoshenko, S., 1974, Vibration Problems in Engineering, Wiley, New York.
24.
Rao
,
S. S.
,
1974
, “
Natural Vibrations of Systems of Elastically Connected Timoshenko Beams
,”
J. Acoust. Soc. Am.
,
55
,
1232
1237
.
25.
Cowper
,
G. R.
,
1996
, “
The Shear Coefficient in Timoshenko’s Beam Theory
,”
ASME J. Appl. Mech.
,
33
,
335
340
.
26.
Hutchinson
,
J. R.
,
2001
, “
Shear Coefficients for Timoshenko Beam Theory
,”
ASME J. Appl. Mech.
,
68
,
87
92
.
27.
Smith
,
B. W.
, and
Luzzi
,
D. E.
,
2000
, “
Formation Mechanism of Fullerene Peapods and Coaxial Tubes: A Path for Large Scale Synthesis
,”
Chem. Phys. Lett.
,
321
,
169
174
.
28.
Saito
,
R.
,
Matsuo
,
R.
,
Kimura
,
T.
,
Dresselhaus
,
G.
, and
Dresselhaus
,
M. S.
,
2001
, “
Anomalous Potential Barrier of Double-Wall Carbon Nanotube
,”
Chem. Phys. Lett.
,
348
,
187
193
.
29.
Bandow
,
S.
,
Takizawa
,
M.
,
Hirahara
,
K.
,
Yudasaka
,
M.
, and
Iijima
,
S.
,
2001
, “
Raman Scattering Study of Double-Wall Carbon Nanotubes Derived From the Chains of Fullerenes in Single-Wall Carbon Nanotubes
,”
Chem. Phys. Lett.
,
337
,
48
54
.
30.
Dresselhaus
,
M. S.
, and
Eklund
,
P. C.
,
2000
. “
Phonons in Carbon Nanotubes
,”
Adv. Phys.
,
49
,
705
814
.
You do not currently have access to this content.