DOI : 10.17577/IJERTV1IS10339
- Open Access
- Total Downloads : 545
- Authors : S Sanyasi Naidu, Ch Varun, G Satyanarayana
- Paper ID : IJERTV1IS10339
- Volume & Issue : Volume 01, Issue 10 (December 2012)
- Published (First Online): 28-12-2012
- ISSN (Online) : 2278-0181
- Publisher Name : IJERT
- License:
This work is licensed under a Creative Commons Attribution 4.0 International License
Fundamental Natural Frequencies of Double-Walled Carbon with Different Boundary Conditions
S Sanyasi Naidu1 Ch Varun2 G Satyanarayana3
Abstract
In the present work fundamental natural frequencies of double walled carbon nanotubes (DWCNTs) are studied using Bubnov-Galerkin method. The main objective of this method is for quick and effective evaluation of fundamental frequencies. The inner and outer carbon nanotubes are modeled as two individual Euler-Bernoullis elastic beams interacting each other by Vander Waals force. The fundamental natural frequencies of double walled carbon nanotubes (DWCNTs) are studied for three cases, i.e. (i) inner and outer carbon nanotubes (CNTs) with same boundary conditions, such as simply supported- simply supported and clamped-clamped (ii) inner and outer carbon nanotubes (CNTs) with different boundary conditions, such as simply supported- clamped, cantilever-clamped, and cantilever- simply supported and (iii) left and right ends of carbon nanotubes (CNTs) with different boundary conditions, such as simply supported- clamped, simply supported- free and clamped-free.
The fundamental natural frequencies are validating with those available in literature and observed a good agreement between them. The Effect of aspect ratio is studied for different boundary conditions. Out of all these boundary conditions clamped clamped boundary condition has the highest natural frequencies. These observations may be useful for the designer to estimate the fundamental natural frequencies in each two series.
-
Introduction:
Afterthediscoveryofmulti-walledcarbonnanotubes (MWCNTs) in1991 by Iijima, hasstimulatedever-broaderresearchactivities in science and engineering devoted entirely to carbon nanostructures and their applications. This is due in large part to the combination of their expected structural perfection, small size, low density, high stiffness, high strength (the strength of the outer most shell of MWCNT is approximately 100 times greater than that of aluminum), and excellent electronic properties. As a result, carbon nanotube (CNT) may find use in a wide range of application in material reinforcement, field emission pane display, chemical senthesing, drug delivery, Nano electronicsetc
Carbon nanotubes (CNTs) have the most promising materials for nanoelectronics, nanodevices, and nanocomposites because of their unusual electronic properties and superior mechanical strength [1]. Many proposed applications and designs of CNTs are involved with aspect ratio about 10. Such examples include suspended crossing CNTs with spans about 20 nm, CNTs as single -electron transistors of length down to 20, MWNTs of aspect ratio around 20 as electrometers or building blocks in nanoelectronics, CNT-nanomechanical switches of aspect ratio around 10 and CNTs of aspect ratio about 10-25 as atomic force microscope (AFM).Owing to the hollow structure of CNTs, short CNTs are preferred in many cases to prevent undesirable kinking and buckling. Therefore, the vibrational behavior of short CNTs say, of aspect ratio between 10 and 30, is of practical significance.
Most CNTs to date have been synthesized with closed ends[1]. For applications of MWCNTs, both its ends can be restricted only on the outer tube. For example, in a nanoelectricalmechanicalsystem (NEMS), the small size and unique properties of CNTs suggest that they can be used in sensor devices with unprecedented sensitivity[2]. Other relevant issues to be clarified are the effects of differential boundary supports between the inner and outer tubes on the vibration of MWCNTs and boundary effects on transverse vibration devices composed of rods in microelectromechanicalsystems (MEMSs)[3]. It is
expected that the differences of boundary condition, which are ignored in existing beam model, would play an important role in the vibration of a DWCNT when the vibrational modes at a resonant frequency between the two tubes are considered. Especially for short DWCNTs, some changes of boundary conditions may effect the vibrational modes more sensitively. For this reason, the relevance of the existing model, in which both tubes have the same boundary conditions for the vibration of DWCNTs, is questionable. To clarify this issue, free vibrations of DWCNTs with differential boundary supports between inner and outer tubes are studied in this work.
-
Analysis:
The governing differential equations for free vibration of the DWCNTs are
c1 ( w
2 – w
1 )= EI1
4
w
1
x 4
4 w
+ A 1
2
w
1
t 2
2 w
– c1 ( w
2 – w
1 )= EI 2
2
x 4
+ A 2
2
t 2
(1)
Where x is the axial coordinate, t is the time, w j ( x ,t) the transverse displacement, I j
the moment of
j
inertia and A the cross-sectional area of the j th nanotube; the indexes j 1,2 denote the inner and outer nanotube, respectively.
The exact solution for various boundary conditions were considered by Xu et at.[ 4,5,6]. Their derivation necessitates numerical evaluation of 8×8 determinant and attendant cumbersome numerical analysis. Therefore the expressions for natural frequencies are obtained in this work by approximate method areexplained in the following sections.
-
Case: 1 both inner and outer nanotubes are with same boundary condition
-
Simply Supported DWCNTs: Polynomial Approximate Solution Here transverse displacement is consider as w D( )sin(t) (2)
Where, ( ) is a coordinate function and x / L is a non-dimensional axial coordinate.
The coordinate function depends upon the boundary conditions of the carbon nanotubes (CNTs). For
2 w
simply supported: at left end 1, deflection and bending moment are zero (i.e. w 0 ,
2 w
2
0 )
and at right end 1,
deflection and bending moment are zero (i.e. w 0 ,
2
0 ). We have to select
the degree of coordinate function equal to number of independent boundary conditions plus one. So the coordinate function for this boundary condition is 5 a 4 b 3 c 2 d . The boundary
conditions are applied to the coordinate function and found that a 0 , b 10 , c 0 , d 7
. Then
the coordinate function becomes 3 5 10 3 7
3 3
(3)
Now the displacements consider as follows:
w1 D1 sin(t) , w2 D2 sin(t) (4)
Substitute the expressions (4) into governing differential eq. (1), and multiplying the result of the substitution by and integrating over the length of the beam. The following two equations are obtained
in D1 and D2 as
(L4 A 2 L4c 99EI )D (L4c )D 0
1 1 1 1 1 2
(L4c )D (L4 A 2 L4c 99EI )D 0
(5)
1 1 2 1 2 2
We demand the determinant
(L4 A 2 L4 c
-
99EI
-
L4 c
1 1
-
L4 c
1 1
L4 A 2 L4 c 99EI
(6)
1 2 1 2
to vanish . This leads to the frequency equation as given below
L8 2 A A 4 (99L4 A EI L8 A c
-
-
99EI
L4 A L8c A )2 99L4c EI
-
-
99EI L4c
1 2 1 2 1 1
1 2 1 2
1 2 1 1
1 2
9801E 2 I I 0 (7)
With roots 2
[ L4 A c L4c A 99A EI
99EI A (L8 A2c2 2L8 A c2 A
198L4 A2c EI
1,1
1 1 1 2 1 2
1 2 1 1
1 1 2
1 1 2
198L4 A c EI A L8c2 A2 198L4c A A EI
198L4c A2 EI 9801A2 E 2 I 2
1 1 1 2 1 2
1 2 1 2
1 2 1 1 2
19602A E 2 I I A 9801E 2 A2 I 2 )1/ 2 ]/ 2L4 A A
1 2 1 2 2 1 1 2
2 [ L4 A c
-
L4c A
-
99A EI
-
99EI A
(L8 A2c2 2L8 A c2 A
198L4 A2c EI
2,1
1 1 1 2 1 2
1 2 1 1
1 1 2
1 1 2
198L4 A c EI A L8c2 A2 198L4c A A EI
198L4c A2 EI 9801A2 E 2 I 2
1 1 1 2 1 2
1 2 1 2
1 2 1 1 2
19602A E 2 I I A 9801E 2 A2 I 2 )1/ 2 ]/ 2L4 A A
1 2 1 2 2 1
1 2 (8)
-
-
Clamped DWCNTs:
The above procedure is repeated with 1 cos(2 ) , which is satisfying the clamped boundary condition. The frequency equation as given below
9L8 2 A A 4 ( 23378 L4 A EI
-
9L8 A c
23378 EI
L4 A
9L8c A )2
1 2 5 1 2 1 1 5 1 2 1 2
23278 L4 c EI
23278 EI
L4 c
2429100E 2 I I 0
5 1 2 5 1 1 1 2
With roots
2 [
1 L4 A c
-
1 L4 c A
8572 A EI
8572 EI A 1 8
2 2 1 L8 A c2 A
1,1
2 1 1 2
1 2 33 1
2 33
1 2 ( 4 L A1 c1 2
1 1 2
8572 L4 A2 c EI
-
85733 L4 A c EI A
1 L8c2 A2 8572 L4 c A A EI
33 1 1
2 33
1 1 1 2 4
1 2 33
1 2 1 2
8572 L4 c A2 EI 67474A2 E 2 I 2 134950A E 2 I I A
67474E 2 A2 I 2 )1/ 2 ]/ L4 A A
33 1 2 1 1 2
1 2 1 2
2 1 1 2
2 [
1 L4 A c
-
-
1 L4 c A
8572 A EI
8572 EI A 1 8
2 2 1 L8 A c2 A
2,1
2 1 1 2
1 2 33 1
2 33
1 2 ( 4 L A1 c1 2
1 1 2
8572 L4 A2 c EI
-
85733 L4 A c EI A
1 L8c2 A2 8572 L4 c A A EI
33 1 1
2 33
1 1 1 2 4
1 2 33
1 2 1 2
8572 L4 c A2 EI 67474A2 E 2 I 2 134950A E 2 I I A
67474E 2 A2 I 2 )1/ 2 ]/ L4 A A
33 1 2 1 1 2
1 2 1 2
2 1 1 2
-
Case 2: Both inner and outer nanotubes with different boundary conditions
-
Simply supported-clamped DWCNT: Approximate Solution
Here transverse displacement as
w D( )sin(t) (9)
Where, ( ) is a coordinate function.
The coordinate function is depends upon the boundary conditions of the carbon nanotubes (CNTs). For simply supported-clamped, at left end 1 transverse displacement, slope and bending moment are
zero (i.e. w 0,
w
2 w
0 , 2
0 ) and at right end 1,transverse displacement, slope and bending
moment are zero (i.e. w 0,
w
2 w
0 , 2
0 ).We have to select the degree of coordinate function as
equal to number of independent boundary conditions plus one. So the coordinate functionfor this boundary condition 7 a 6 b 5 c 4 d 3 e 2 f . The boundary conditions are applied
to the coordinate function and find that a 0 , b 3, c 0 , d 3 , e 0, f 1 .Then the coordinate
function is 7 3 5 3 3 1
(10)
We now the displacements as follows:
w1 D1 sin(t) , w2 D2 sin(t) (11)
We substitute the expressions (11) into governing differential eq. (1), and multiplying the result of the substitution by and integrating over the length of the beam, the following two equations are obtain in
D and D : (L4 A 2 L4c 241EI )D (L4c )D 0
1 2 1 1 1 1 1 2
(L4c )D (L4 A
2 L4c 241EI )D 0
1 1 2
1 2 2
(12)
We demand the determinant
(L4 A 2 L4 c
241EI
-
-
-
L4 c
1 1
-
L4 c
1 1
L4 A 2 L4 c 241EI
(13)
1 2 1 2
to vanish . This leads to the frequency equation
L8 2 A A 4 (241L4 A EI L8 A c 241EI L4 A L8c A )2 241L4c EI
1 2 1 2 1 1 1 2 1 2 1 2
1
241EI
L4c
58081E 2 I I 0
(14)
1
1
2
With roots
2 [ L4 A c
-
-
-
L4c A
-
241A EI
-
241EI A
(L8 A2c2 2L8 A c2 A
-
482L4 A2c EI
1,1
1 1 1 2 1 2
1 2 1 1
1 1 2
1 1 2
-
482L4 A c EI A L8c2 A2 482L4c A A EI
482L4c A2 EI 58081A2 E 2 I 2
1 1 1 2 1 2
1 2 1 2
1 2 1 1 2
116162A E 2 I I A 58081E 2 A2 I 2 )1/ 2 ]/ 2L4 A A
1 2 1 2 2 1 1 2
2 [ L4 A c
-
-
L4c A
-
241A EI
-
241EI A
(L8 A2c2 2L8 A c2 A
-
482L4 A2c EI
2,1
1 1 1 2 1 2
1 2 1 1
1 1 2
1 1 2
-
482L4 A c EI A L8c2 A2 482L4c A A EI
482L4c A2 EI 58081A2 E 2 I 2
1 1 1 2 1 2
1 2 1 2
1 2 1 1 2
116162A E 2 I I A 58081E 2 A2 I 2 )1/ 2 ]/ 2L4 A A
1 2 1 2
2 1 1 2
(15)
-
Cantilever-Clamped DWCNTs
The above procedure is repeated with
7 2 6 5 4 4 3 2 2
, which is satisfies the
clamped boundary condition. The frequency equation as given below
L8 2 A A 4 (448L4 A EI L8 A c 448EI L4 A L8c A )2 448L4c EI
1 2 1 2 1 1 1 2 1 2 1 2
1
448EI
L4 c
200704E 2 I I 0
(16)
1
1
2
With roots
2 [ L4 A c
-
-
-
L4c A
-
448A EI
-
448EI A
(L8 A2c2 2L8 A c2 A
-
896L4 A2c EI
1,1
1 1 1 2 1 2
1 2 1 1
1 1 2
1 1 2
896L4 A c EI A L8c2 A2 896L4c A A EI
896L4c A2 EI 200704A2 E 2 I 2
1 1 1 2 1 2
1 2 1 2
1 2 1 1 2
401408A E 2 I I A 200704E 2 A2 I 2 )1/ 2 ]/ 2L4 A A
1 2 1 2 2 1 1 2
2 [ L4 A c
-
L4c A
-
448A EI
-
448EI A
(L8 A2c2 2L8 A c2 A
-
896L4 A2c EI
2,1
1 1 1 2 1 2
1 2 1 1
1 1 2
1 1 2
896L4 A c EI A L8c2 A2 896L4c A A EI
896L4c A2 EI 200704A2 E 2 I 2
1 1 1 2 1 2
1 2 1 2
1 2 1 1 2
401408A E 2 I I A 200704E 2 A2 I 2 )1/ 2 ]/ 2L4 A A
(17)
1 2 1 2 2 1 1 2
-
Cantilever-Simply Supported DWCNTs
The above procedure is repeated with
8 7 5 6 27 5 14 4 34 3 9 2 15
, which is
satisfies the clamped boundary condition. The frequency equation as given below
L8 2 A A 4 (146L4 A EI L8 A c 146EI L4 A L8c A )2 146L4c EI
1 2 1 2 1 1 1 2 1 2 1 2
1
146EI
L4c
21316E 2 I I 0
(18)
1
1
2
With roots
2 [ L4 A c
-
-
L4c A
146A EI
146EI A
(L8 A2c2 2L8 A c2 A
-
292L4 A2c EI
1,1
1 1 1 2 1 2
1 2 1 1
1 1 2
1 1 2
-
292L4 A c EI A L8c2 A2 292L4c A A EI
292L4c A2 EI 21316A2 E 2 I 2
1 1 1 2 1 2
1 2 1 2
1 2 1 1 2
42632A E 2 I I A 21316E 2 A2 I 2 )1/ 2 ]/ 2L4 A A
1 2 1 2 2 1 1 2
2 [ L4 A c
-
-
L4c A
146A EI
146EI A
(L8 A2c2 2L8 A c2 A
-
292L4 A2c EI
2,1
1 1 1 2 1 2
1 2 1 1
1 1 2
1 1 2
-
292L4 A c EI A L8c2 A2 292L4c A A EI
292L4c A2 EI 21316A2 E 2 I 2
1 1 1 2 1 2
1 2 1 2
1 2 1 1 2
42632A E 2 I I A 21316E 2 A2 I 2 )1/ 2 ]/ 2L4 A A
(19)
1 2 1 2 2 1 1 2
-
Case:3 Both left and right ends of nanotubes with different boundary condition
-
Simply supported-Clamped DWCNT: Approximate Solution
Here consider the transverse displacement is considered as w D( )sin(t)
Where, ( ) is a coordinate fuction.
(20)
The coordinate function depending upon the boundary conditions of the carbon nanotubes (CNTs). For simply supported-clamped boundary condition at left end 1transverse displacement and bending
2 w
moment are zero (i.e. w 0 ,
2
0 ) and at right end 1 transverse displacement and deflection are
zero (i.e. w 0 , w 0 ).We have to select the degree of coordinate function is equal to number of
independent boundary conditions plus one. So the coordinate function for this boundary condition is
5 a 4 b 3 c 2 d . The boundary conditions are applied to the coordinate function and
found that a 1 , b 5 , c 1 , d 3 . Then the coordinate function becomes
2 2 2 2
2 5 4 5 3 2 3
(21)
Now the displacements considered are as follows w1 D1 sin(t) ,, w2 D2 sin(t) (22)
We substitute the expressions (18) into governing differential Eq. (1), and multiplying the result of the
substitution by and integrating over the length of the beam, the following two equations for D1 and
D : (L4 A 2 L4c 145EI
)D (L4c )D 0
2 1 1 1 1 1 2
(L4c )D (L4 A 2 L4c 145EI )D 0(23)
1 1 2 1 2 2
We demand the determinant
(L4 A 2 L4c
145EI
-
-
-
L4c
1 1
-
L4 c
1 1
L4 A 2 L4 c 145EI
(24)
1 2 1 2
to vanish . This leads to the frequency equation
L8 2 A A 4 (145L4 A EI L8 A c 145EI L4 A L8c A )2 145L4c EI
1 2 1 2 1 1 1 2 1 2 1 2
1
145EI
L4 c
21025E 2 I I
0 (25)
1
1
2
With roots
2 [ L4 A c
-
-
-
L4c A
145A EI
145EI A
(L8 A2c2 2L8 A c2 A
-
290L4 A2c EI
1,1
1 1 1 2 1 2
1 2 1 1
1 1 2
1 1 2
-
290L4 A c EI A L8c2 A2 290L4c A A EI
290L4c A2 EI 21025A2 E 2 I 2
1 1 1 2 1 2
1 2 1 2
1 2 1 1 2
42050A E 2 I I A 21025E 2 A2 I 2 )1/ 2 ]/ 2L4 A A
1 2 1 2 2 1 1 2
2 [ L4 A c
-
-
L4c A
145A EI
145EI A
(L8 A2c2 2L8 A c2 A
-
290L4 A2c EI
2,1
1 1 1 2 1 2
1 2 1 1
1 1 2
1 1 2
-
290L4 A c EI A L8c2 A2 290L4c A A EI
290L4c A2 EI 21025A2 E 2 I 2
1 1 1 2 1 2
1 2 1 2
1 2 1 1 2
42050A E 2 I I A 21025E 2 A2 I 2 )1/ 2 ]/ 2L4 A A
1 2 1 2
2 1 1 2
(26)
-
Clamped-Free DWCNTs:
The above procedure is repeated with
17 5 36 4 26 3 124 2 97
, which is satisfies the
Clamped-Free boundary condition. The frequency equation as given below
L8 2 A A 4 (7L4 A EI L8 A c
-
-
7EI
L4 A L8c A )2 7L4c EI
-
-
7EI L4c
1 2 1 2 1 1
1 2 1 2
1 2 1 1
1 2
49E 2 I I 0
(27)
With roots
2 [ L4 A c
-
L4c A
-
7A EI
-
7EI A
(L8 A2c2 2L8 A c2 A
14L4 A2c EI
1,1
1 1 1 2 1 2 1 2 1 1
1 1 2
1 1 2
14L4 A c EI A L8c2 A2 14L4c A A EI
14L4c A2 EI 49A2 E 2 I 2
1 1 1 2 1 2
1 2 1 2
1 2 1 1 2
98A E 2 I I A 49E 2 A2 I 2 )1/ 2 ]/ 2L4 A A
1 2 1 2 2 1 1 2
2 [ L4 A c
-
L4c A
-
7A EI
-
7EI A
(L8 A2c2 2L8 A c2 A
14L4 A2c EI
2,1
1 1 1 2 1 2 1 2 1 1
1 1 2
1 1 2
14L4 A c EI A L8c2 A2 14L4c A A EI
14L4c A2 EI 49A2 E 2 I 2
1 1 1 2 1 2
1 2 1 2
1 2 1 1 2
98A E 2 I I A 49E 2 A2 I 2 )1/ 2 ]/ 2L4 A A
(28)
1 2 1 2 2 1 1 2
-
Simply Supported-Free DWCNTs:
The above procedure is repeated with
3 5 5 4 10 3 30 2 32
, which is satisfies the
Simply Supported-Free boundary condition. The frequency equation as given below
L8 2 A A 4 (5L4 A EI L8 A c 5EI L4 A L8c A )2 5L4c EI
1 2 1 2 1 1 1 2 1 2 1 2
1
5EI
L4 c
25E 2 I I 0
(29)
1
1
2
With roots
2 [ L4 A c
-
-
L4c A
-
5A EI
-
5EI A
(L8 A2c2 2L8 A c2 A
10L4 A2c EI
1,1
1 1 1 2 1 2 1 2 1 1
1 1 2
1 1 2
10L4 A c EI A L8c2 A2 10L4c A A EI
10L4c A2 EI 25A2 E 2 I 2
1 1 1 2 1 2
1 2 1 2
1 2 1 1 2
50A E 2 I I A 25E 2 A2 I 2 )1/ 2 ]/ 2L4 A A
1 2 1 2 2 1 1 2
2 [ L4 A c
-
L4c A
-
5A EI
-
5EI A
(L8 A2c2 2L8 A c2 A
10L4 A2c EI
2,1
1 1 1 2 1 2 1 2 1 1
1 1 2
1 1 2
10L4 A c EI A L8c2 A2 10L4c A A EI
10L4c A2 EI 25A2 E 2 I 2
1 1 1 2 1 2
1 2 1 2
1 2 1 1 2
50A E 2 I I A 25E 2 A2 I 2 )1/ 2 ]/ 2L4 A A
(30)
1 2 1 2 2 1 1 2
-
For numerical analysis the following data is taken for DWCNTs
Youngs modulus (E) = 1 TPa Mass density ( ) = 2.3 g/ cm3
Vander Waals interlayer interaction coefficient ( c 1 ) = 71.11GPa Inner radius ( R1 ) = 0.35 nm
Outer radius ( R2 ) = 0.70 nm
Wall thickness each nanotube = 0.34 nm
-
Results and Discussions:
Table:1 First natural frequenciesof DWCNTs with same boundary conditions:
S. No.
Aspect ratio
S-S
C-C
1,1
(THz)
2,1
(THz)
1,1
(THz)
2,1
(THz)
1
10
0.4794
7.7609
1.0919
7.7692
2
11
0.3963
7.7578
0.9029
7.7529
3
12
0.3330
7.7558
0.7590
7.7426
4
13
0.2838
7.7545
0.6469
7.7358
5
14
0.2447
7.7536
0.5579
7.7312
6
15
0.2131
7.7530
0.4860
7.7280
7
16
0.1873
7.7526
0.4272
7.7257
8
17
0.1659
7.7522
0.3785
7.7240
9
18
0.1480
7.7520
0.3376
7.7228
10
19
0.1329
7.7518
0.3030
7.7218
11
20
0.1199
7.7517
0.2735
7.7211
1.2
1
Frequency (THz)
0.8
0.6
0.4
0.2
7.78
7.77
Frequency (THz)
7.76
w 21 C-C
w 21 S-S
w 11 C-C
w 11 S-S
7.75
7.74
7.73
7.72
0
0 5 10 15 20 25
Aspect Ratio (L/D)
7.71
0 5 10 15 20 25
Aspect Ratio (L/D)
-
(b)
-
-
-
Fig.1 Variation of Co-axial and Non Co-axial Frequencies with aspect ratio with same boundary conditions
Table: 2 First natural frequencies of DWCNTs with different boundary condition:
|
S. No. |
Aspect ratio |
S-C |
Cant.-C |
Cant.-S |
|||
|
1,1 (THz) |
2,1 (THz) |
1,1 (THz) |
2,1 (THz) |
1,1 (THz) |
2,1 (THz) |
||
|
1 |
10 |
0.7424 |
7.7413 |
1.0192 |
7.7956 |
0.5821 |
7.7655 |
|
2 |
11 |
0.6133 |
7.7339 |
0.8427 |
7.7815 |
0.4812 |
7.7609 |
|
3 |
12 |
0.5153 |
7.7292 |
0.7084 |
7.7725 |
0.4044 |
7.7580 |
|
4 |
13 |
0.4390 |
7.7261 |
0.6037 |
7.7666 |
0.3446 |
7.7561 |
|
5 |
14 |
0.3785 |
7.7240 |
0.5206 |
7.7626 |
0.2971 |
7.7548 |
|
6 |
15 |
0.3297 |
7.7225 |
0.4536 |
7.7598 |
0.2588 |
7.7539 |
|
7 |
16 |
0.2897 |
7.7215 |
0.3987 |
7.7578 |
0.2275 |
7.7533 |
|
8 |
17 |
0.2566 |
7.7207 |
0.3532 |
7.7564 |
0.2015 |
7.7528 |
|
9 |
18 |
0.2289 |
7.7201 |
0.3150 |
7.7553 |
0.1798 |
7.7524 |
|
10 |
19 |
0.2054 |
7.7197 |
0.2827 |
7.7545 |
0.1613 |
7.7522 |
|
11 |
20 |
0.1854 |
7.7193 |
0.2552 |
7.7538 |
0.1456 |
7.7520 |
1.2
1
Frequency (THz)
0.8
0.6
0.4
0.2
0
0 5 10 15 20 25
Aspect Ratio (L/D)
7.8
7.79
7.78
w 11 S-C
w 11 Cant-C
w 11 Cant-S
Frequency (THz)
7.77
7.76
7.75
7.74
7.73
7.72
7.71
w 21 S-C
w 21 Cant-C
w 21 Cant-S
0 5 10 15 20 25
Aspect Ratio (L/D)
(a) (b)
Fig.2 Variation of Co-axial and Non Co-axial Frequencies with aspect ratio with different boundary conditions
Table:3First natural frequencies of DWCNTs with different boundary conditions at left and right ends:
|
S. No. |
Aspect ratio |
S-C |
S-F |
C-F |
|||
|
1,1 (THz) |
2,1 (THz) |
1,1 (THz) |
2,1 (THz) |
1,1 (THz) |
2,1 (THz) |
||
|
1 |
10 |
0.5801 |
7.7654 |
0.1078 |
7.7516 |
0.1708 |
7.7518 |
|
2 |
11 |
0.4795 |
7.7609 |
0.0891 |
7.7514 |
0.1412 |
7.7515 |
|
3 |
12 |
0.4030 |
7.7580 |
0.0748 |
7.7513 |
0.1187 |
7.7514 |
|
4 |
13 |
0.3434 |
7.7561 |
0.0638 |
7.7512 |
0.1012 |
7.7513 |
|
5 |
14 |
0.2961 |
7.7548 |
0.0550 |
7.7512 |
0.0872 |
7.7512 |
|
6 |
15 |
0.2579 |
7.7539 |
0.0439 |
7.7512 |
0.0759 |
7.7512 |
|
7 |
16 |
0.2267 |
7.7533 |
0.0421 |
7.7511 |
0.0667 |
7.7512 |
|
8 |
17 |
0.2008 |
7.7528 |
0.0373 |
7.7511 |
0.0591 |
7.7511 |
|
9 |
18 |
0.1791 |
7.7524 |
0.0333 |
7.7511 |
0.0527 |
7.7511 |
|
10 |
19 |
0.1608 |
7.7522 |
0.0299 |
7.7511 |
0.0473 |
7.7511 |
|
11 |
20 |
0.1451 |
7.7520 |
0.0269 |
7.7511 |
0.0427 |
7.7511 |
0.7
0.6
frequency (THz)
0.5
0.4
0.3
0.2
0.1
0
0 5 10 15 20 25
Aspect Ratio (L/D)
7.768
7.766
Frequency (THz)
7.764
w 21 S-C
w 21 C-F
w 21 S-F
w 11 S-C
w 11 C-F
w 11 S-F
7.762
7.76
7.758
7.756
7.754
7.752
7.75
0 5 10 15 20 25
Aspect Ratio (L/D)
Fig.3 Variation of Co-axial and Non Co-axial Frequencies with aspect ratio with left and right ends of CNTs with different boundary conditions
-
Comparison with K.Y. xuet.al[5]:
K.Y. xu et.al[5] studied vibration of a double-walled carbon nanotube aroused by nonlinear interlayer van der Waals _vdW_forces with different boundary conditions for aspect ratio 10 and 20. The natural frequencies obtained by present method is compare with the with the natural frequencies K.Y. xu et.al[5]. The frequencies are tabulated below. It is observed that a good agreement between them. This shows the accuracy of the present method.
S.No
Aspect ratio (L/D)
Boundary condition
Present
K.Y. Xu et.al.
1,1
(THz)
2,1
(THz)
1,1
(THz)
2,1
(THz)
1
10
S-S
0.4794
7.7609
0.4
7.71
2
C-C
1.0919
7.7692
1.06
7.75
3
C-F
0.1708
7.7518
0.17
7.7
4
20
S-S
0.1199
7.7517
0.11
7.7
5
C-C
0.2735
7.7211
0.26
7.7
6
C-F
0.0427
7.7511
0.04
7.7
-
Comparison with Natsuki et.al.[7]:
Most recently, Natusuki et al. [7] analyzed free vibration characteristics of DWCNT. He calculated the natural frequencies of DWCNT; both ends simply supported .Specifically Natusuki et al. [7] adopted the following formula for the Vander Waals interaction coefficient c1 :
R R 6 1001 4 11120 6
c 1 2 H 13 H 7
(31)
3
9
1 4
/ 2
1 2
1
(1 K cos 2 )m / 2
Where
H m (R
-
R )m
d , ( m 7,13 ) (32)
0
And
K 4R1 R2
(33)
2
(R1 R2 )
For evaluation of
c1 , Natsuki used the following data
0.34 nm, 2.967 mev, 0.142 nm, din 4.8 nm, and dout 5.5 nm, Yield c1 1.474825922044788×1011 whereasNatuski [1] informs that his value is 1.50×10 11 , showing an excellent comparison.
According to Natsuki [7] the first natural frequency for L 10nm equals to 4.04 THz. The numerical data which is used by Natsuki is applied to our exact and approximate methods for simply supported boundary conditions of DWCNT, which yields 4.0339 THz and 4.0453 THz respectively, which shows the close agreement with Natsuki results.
S. No.
1,1 (THz)
% Error
1
Natsuki et.al
4.04
_
2
Trigonometric Solution
4.0339
0.15099
3
Polynomial Solution
4.0453
0.13118
-
-
Conclusions:
This paper studies free vibration analysis of DWNTs modeled as elastic beams for different boundary conditions between inner and outer tubes. The results obtained are compared with those available in literature and some discussions are summarizedas follows.
-
This method estimates the natural frequencies with minimum error.
-
Increasing the Aspect ratio decreasing the natural frequencies.
-
The effect of Aspect ratio for second series is very less when compare to the first series.
-
The natural frequencies of DWCNTs in Clamped boundary conditions showshighest than other.
Nomenclature:
1
A Cross-sectional area of inner nanotube (nm 2 )
2
A Cross-sectional area of outer nanotube (nm 2 )
c1 Vander Waals interlayer interaction coefficient (GPa)
D Diameter of outer nanotube (nm)
E Youngs modulus (TPa)
I1 Moment of inertia of inner nanotube (nm 4 ) I 2 Moment of inertia of outer nanotube (nm 4 ) L Length of the nanotube (nm)
m Mode number ( m 1, 2, 3 ..)
t Time
x Axial coordinate
w1 Transverse displacement of inner nanotube
w2 Transverse displacement of outer nanotube
Mass density (g/cm 3 )
Non-dimensional axial coordinate
1,1 Fundamental natural frequency of first series (THz)
2,1 Fundamental natural frequency of second series (THz)
Vander Waals radius (nm)
The well depth of the Lennard-Jones potential (mev)
The carbon-carbon bond length (nm)
References:
-
D. Qian, G.J. Wagner, W.K.Liu, Mechanics of carbon nanotubes, Applied Mechanics Review 55 (2002) 495-533.
-
Maiti,A., 2001 Application of carbon nanotubes as Electromechanical Sensors: results from first- principles simulationsPhys. Status Solidi B, 226, pp 121-123.
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W.K. Liu, E.G. Karpov, H.S. Park, Nano Mechanics and Materials: Theory, Multiscale Methods and Applications, John Wiley &Sons Ltd., West Sussex, England, 2006.
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Zhang, Y.Q., Liu, G.R., and Xie, X.Y., 2005, Free transverse vibrations of Double-walled Carbon nanotubes Using a Theory of nonlocal Elasticity, Phys. Rev. B, 71, p. 195404.
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