nonlinearfreevibrationofsingle-walledcarbonnanotubesusingnonlocal timoshenkobeamtheory
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Nonlinear free vibration of single-walled carbon nanotubes using nonlocalTimoshenko beam theory
J. Yang a,n, L.L. Ke b,c, S. Kitipornchai b
a School of Aerospace, Mechanical and Manufacturing Engineering, RMIT University, PO Box 71, Bundoora, Victoria 3083, Australiab Department of Building and Construction, City University of Hong Kong, Hong Kong, PR Chinac Institute of Engineering Mechanics, Beijing Jiaotong University, 100044 Beijing, PR China
a r t i c l e i n f o
Article history:Received 24 November 2009
Received in revised form
19 January 2010
Accepted 19 January 2010Available online 25 January 2010
Keywords:
SWCNTs
Nonlinear vibration
Timoshenko beam theory
Nonlocal elasticity
DQ method
a b s t r a c t
Nonlinear free vibration of single-walled carbon nanotubes (SWCNTs) is studied in this paper based onvon Karman geometric nonlinearity and Eringens nonlocal elasticity theory. The SWCNTs are modeled
as nanobeams where the effects of transverse shear deformation and rotary inertia are considered
within the framework of Timoshenko beam theory. The governing equations and boundary conditions
are derived by using the Hamiltons principle. The differential quadrature (DQ) method is employed to
discretize the nonlinear governing equations which are then solved by a direct iterative method to
obtain the nonlinear vibration frequencies of SWCNTs with different boundary conditions. Zigzag (5, 0),
(8, 0), (9, 0) and (11, 0) SWCNTs are considered in numerical calculations and the elastic modulus is
obtained through molecular mechanics (MM) simulation. A detailed parametric study is conducted to
study the influences of nonlocal parameter, length and radius of the SWCNTs and end supports on the
nonlinear free vibration characteristics of SWCNTs.
& 2010 Elsevier B.V. All rights reserved.
1. Introduction
Nonlocal elasticity theory was proposed by Eringen [13] to
consider the scale effect in elasticity by assuming the stress at a
reference point to be a function of strain field at every point in the
body. It has been extensively applied to analyze the bending,
buckling, vibration and wave propagation of beam-like elements
in micro- or nanoelectromechanical devices [413]. Sudak [14]
studied infinitesimal column buckling of carbon nanotubes
(CNTs), incorporating the van der Waals (vdW) forces and small
scale effect, and showed that the critical axial strain decreases
compared with the results of classical beams. Wang [15]
discussed the molecular dispersion relationships for CNTs by
taking into account the small scale effect. Wang and Hu [16]
studied flexural wave propagation in a SWCNT by using thecontinuum mechanics and dynamic simulation. Lu et al. [17]
investigated the wave propagation and vibration properties of
single- or multi-walled CNTs based on nonlocal beam model.
Wang et al. [18] presented analytical solutions for the free
vibration of nonlocal Timoshenko beams. Reddy[19] developed
nonlocal theories for EulerBernoulli, Timoshenko, Reddy, and
Levinson beams. Analytical bending, vibration and buckling
solutions are obtained which bring out the nonlocal effect on
bending deformation, buckling load, and natural frequencies.More recently, Tounsi and his co-workers [2023] investigated
the sound wave propagation in single- and double-walled CNTs
taking into account the nonlocal effect, temperature and initial
axial stress. Furthermore, they [24,25] derived the consistent
governing equation of motion for the free vibration of fluid-
conveying CNTs with nonlocal effect, which is an important
application of nonlocal elastic theory in CNTs. Yang et al. [26]
investigated the pull-in instability of nano-switches subjected to
combined electrostatic and intermolecular forces within the
framework of nonlocal elasticity theory. Aydogdu[27]presented
a generalized nonlocal beam theory to study bending, buckling
and free vibration of nanobeams.
Previous theoretic and experimental investigations [28,29]
showed that the deformation of nanostructures, such as CNTs, isnonlinear in nature when subjected to large external loads.
Fu et al. [28] investigated the nonlinear free vibration of
embedded multiwall CNTs considering inter-tube radial displace-
ment and the related internal degrees of freedom. Shen and Zhang
[30,31] considered the buckling and postbuckling behavior of
single- and double-walled CNTs in thermal environments. Yan
et al. [32]analyzed the nonlinear vibration characteristics of the
fluid-filled DWNTs. To the best of authors knowledge, however,
no previous work has been done concerning the small scale effect
on the nonlinear vibration behavior of nanostructures.
This paper makes the first attempt to study the nonlinear
free vibration of SWCNTs based on von Karman geometric
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Contents lists available atScienceDirect
journal homepage: www .elsevier.com/locate/physe
Physica E
1386-9477/$- see front matter& 2010 Elsevier B.V. All rights reserved.
doi:10.1016/j.physe.2010.01.035
n Corresponding author. Tel.: +61 3 99256169; fax: +61 3 99256108.
E-mail address: [email protected] (J. Yang).
Physica E 42 (2010) 17271735
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nonlinearity, Timoshenko beam theory and Eringens nonlocal
elasticity theory. The Hamiltons principle is employed to derive
the governing equations and boundary conditions which are
solved by using the differential quadrature (DQ) method. A direct
iterative technique is then used to obtain the nonlinear vibration
frequencies of nonlocal SWCNTs with different end supports. In
numerical calculations, zigzag (5, 0), (8, 0), (9, 0) and (11, 0)
SWCNTs are considered and the elastic modulus is obtained by
using MM simulation. The influences of nonlocal parameter,length and radius of the SWCNTs and end supports on the
nonlinear free vibration characteristics of the SWCNTs are
discussed in detail.
2. Nonlocal nanobeam model
Unlike the constitutive equation in classical elasticity,
Eringens nonlocal elasticity theory [13] states that the stress
at a pointx in a body depends not only on the strain at that point
but also on those at all points of the body. This observation is in
accordance with atomic theory of the lattice dynamics and
experimental observation of the phonon dispersion [2]. Thus,
the nonlocal stress tensor rat point x is expressed as
r
ZV
a9x0x9; tTx0 dx0; 1
where T(x) is the classical, macroscopic stress tensor at point x.
a(9x0x9,t) is the nonlocal modulus or attenuation functionintroducing into the constitutive equation the nonlocal effect at
the reference pointx produced by local strain at the source x0.9x0-x9 is the Euclidean distance, and t e0a=l is defined as the scalecoefficient that incorporates the small scale factor, where e0 is a
material constant determined experimentally or approximated by
matching the dispersion curves of plane waves with those of
atomic lattice dynamics, and a andl are the internal and external
characteristic lengths (e.g. crack length, wavelength), respectively.
The stress tensorT(x) at pointx in a Hookean solid is related to
the strain tensor e(x) at the point by the generalized Hookes law[19]
Tx Cx : ex; 2
whereC(x) is the fourth-order elasticity tensor and : denotes the
double-dot product.
From Eqs. (1) and (2), the integral nonlocal constitutive
relations can be represented in an equivalent differential form
as[2]
1tl2r2r T: 3
3. Nonlinear vibration analysis of nonlocal SWCNTs
Fig. 1 shows a SWCNT modeled as a Timoshenko nanobeamwith length L, radius r, and effective tube thickness h. I t is
assumed that the SWCNTs vibrate only in the xzplane. Based on
Timoshenko beam theory, the displacements of an arbitrary point
in the beam along the x- and z-axes, denoted by ~Ux;z; t and~Wx;z; t, respectively, are
~Ux;z; t Ux; t zCx; t; ~Wx;z; t Wx; t; 4
where U(x,t) and W(x,t) are displacement components in the
midplane,C is the rotation of beam cross-section and tis time.
The von Karman type nonlinear strain-displacement relations give
exx@U
@x
1
2
@W
@x
2z
@C
@x ; gxz
@W
@x C; 5
where,exx is the axial strain, and gxz is the shear strain.
The strain energy Vis given by
V1
2
Z L0
ZA
sxxexxsxzgxzdAdx; 6
where A is the cross-sectional area of the beam, sxx and txz arenormal and shear stresses, respectively. By submitting Eq. (5) into
Eq. (6), the strain energy Vcan be represented as
V1
2
Z L0
ZA
sxx@U
@x
1
2
@W
@x
2" #sxxz
@C
@x sxz
@W
@x C
( )dAdx
1
2
Z L0
Nx@U
@x
1
2
@W
@x
2" # Mx
@C
@x Qx
@W
@x C
( )dx;
7
where the normal resultant force Nx, bending moment Mx, and
transverse shear force Qxare calculated from
Nx
ZA
sxx dA; Mx
ZA
sxxzdA; Qx
ZA
sxzdA: 8
The kinetic energy Kcan be calculated from
K12
Z L
0rA
@U
@t
2
rA @W
@t
2
rI @C
@t
2
" #dx; 9
whereIis the second moment of area and r is the mass density ofbeam material.
For a beam type structure, the thickness and width are much
smaller than its length. Therefore, for beams with transverse
motion in thexzplane, the nonlocal constitutive relations (3) can
be approximated to one-dimensional form as[19]
sxxe0a2d
2sxxdx2
Eexx; sxze0a2d
2sxzdx2
Ggxz; 10
where E and G are Youngs modulus and shear modulus,
respectively. The constitutive relations in classical elasticity
theories can be recovered by setting the nonlocal parametere0a =0.
Using the Hamiltons principleZ t0
dKdVdt 0; 11
substituting Eqs. (7) and (9) into Eq. (11), integrating by parts and
setting the coefficients of dU, dW and dC to zero leads to the
equations of motion as [3335]
@Nx@x
rA@2U
@t2 ; 12a
@Qx
@x
@
@x
Nx@W
@x rA @
2W
@t2 ; 12b
r
L
x
z
h
Fig. 1. A single wall carbon nanotube (SWCNT) modeled as a nonlocal Timoshenko
nanobeam.
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4. Solution method
The differential quadrature (DQ) method [37,38] is used to
solve nonlinear Eq. (20) and the associated boundary conditions
to determine the nonlinear free vibration frequencies of nonlocal
SWCNTs. The fundamental idea of the DQ method is to
approximate the derivative of a function at a sample point as a
linear weighted sum of the function values at all of the sample
points in the problem domain. Hence, u, w and c and their kthderivatives with respect to x can be approximated by
fu; w;cg XN
m 1
lmx umxm; t; wmxm; t;cmxm; t
; 25
@k
@xk u; w;c
9x xi XN
m 1
Ck im um xm; t ; wm xm; t ; cm xm; t
;
26
whereNis the total number of nodes distributed along the x-axis,
lm(x) is the Lagrange interpolation polynomials, and Cim(k) is the
weighting coefficients whose recursive formula can be found in
[3740]. The cosine pattern is used to generate the DQ pointsystem
xi1
2 1cos
pi1N1
; i 1; 2;. . .N: 27
Applying Eqs (25) and (26) to Eq. (20), one obtains a set of
nonlinear ordinary differential equations
a11XN
m 1
C2im
um1
Z
XNm 1
C1im
wmXN
m 1
C2im
wm
! I1 uim
2XN
m 1
C2im
um
!;
28a
a55 XN
m 1
C2im wmZXN
m 1
C1imcm !S1im2S2i I1 wim
2XN
m 1
C2im wmm2XN
m 1
C2im umm4XN
m 1
C4im um
!;
28b
d11XN
m 1
C2imcma55Z
XNm 1
C1im
wmZci
! I3
cim2XN
m 1
C2im
cm
" #;
28c
where the dot represents the derivative with respect to the
dimensionless timei,
S1ia11
ZX
N
m 1
C2
im
um XN
m 1
C1
im
wm 3
2ZX
N
m 1
C1
im
wm !2
24XN
m 1
C2im wmXN
m 1
C1im umXN
m 1
C2im wm
#; 29
S2ia11Z2
3XN
m 1
C2im wm
!39
XNm 1
C1im wmXN
m 1
C2im wmXN
m 1
C3im wm
24
3
2
XNm 1
C1im wm
!2 XNm 1
C4im wm
35a11
Z
XNm 1
C4im umXN
m 1
C1im wm
"
3 XN
m 1
C3im um XN
m 1
C2im wm3 XN
m 1
C2im um XN
m 1
C3im wm
XN
m 1
C1im umXN
m 1
C4im wm
#: 30
The associated boundary conditions can be handled in the
same way. For example, the boundary conditions of a hinged
hinged SWCNT are written as
d11XN
m 1
C11mcma11Z
XNm 1
C21mumXN
m 1
C11mwm
"
3
2Z
XNm 1
C11mwm
!2 XNm 1
C21mwmXN
m 1
C11mumXN
m 1
C21mwm
35
m2 I3XN
m 1C21m cmI1Z w1m2I1Z
XNm 1
C21mum !
0;
u1 w1 0; at z 0; 31a
d11XN
m 1
C1Nmcma11Z
XNm 1
C2NmumXN
m 1
C1Nmwm
"
3
2Z
XNm 1
C1Nmwm
!2 XNm 1
C2NmwmXN
m 1
C1NmumXN
m 1
C1Nmwm
35
m2 I3XN
m 1
C2NmcmI1Z wNm
2I1ZXN
m 1
C2Nmum
! 0;
uN wN 0; at z 1: 31b
Denoting the unknown dynamic displacement vector
d uif gT; wif g
T; ci Tn oT
; i 1; 2;. . .N; 32Table 2
Dimensionless linear and nonlinear fundamental frequencies of nonlocal (8, 0)
SWCNTs: results with varying total number of nodes N.
N HH CH CC
ol onl ol onl ol onl
5 0.45690 0.66723 0.81629
6 0.41943 0.43755 0.60107 0.60978 0.81256
7 0.42263 0.44016 0.60342 0.61924 0.80536 0.83042
8 0.42333 0.44055 0.60523 0.61963 0.80550 0.82071
10 0.42333 0.44055 0.60526 0.61975 0.80551 0.81888
16 0.42333 0.44055 0.60526 0.61975 0.80551 0.81888
20 0.42333 0.44055 0.60526 0.61975 0.80551 0.81888
Table 3
Dimensionless linear fundamental frequency of a hingedhinged nonlocal
Timoshenko nanobeam.
L/h (e0a)2=0.5 (e0a)
2=1.5 (e0a)2=2.5
Ref.[19] Present Ref.[19] Present Ref .[19] Present
10 9.6335 9.6331 9.2101 9.2097 8.8380 8.8377
20 9.6040 9.5942 9.1819 9.1726 8.8110 8.8020
100 9.5135 9.4765 9.0953 9.0600 8.7279 8.6940
Table 1
The elastic modulus of zigzag (5, 0), (8, 0), (9, 0) and (11, 0) SWCNTs obtained by
using molecular mechanics simulation.
(n, m) Number of
atoms
Length
(nm)
Diameter
(nm)
Elastic modulus
(TPa)
(5, 0) 240 4.7971 0.391 1.1468
(8, 0) 384 4.8659 0.626 1.1556
(9, 0) 432 4.8749 0.705 1.1572
(11, 0) 528 4.8857 0.861 1.1621
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Eqs. (28) and (31) can be expressed in matrix form as
KL KNLd M d 0; 33
where M is the mass matrix, KL is the linear stiffness matrix and
KNL is nonlinear stiffness matrix that is the functions in d. M, KLandKNLare 3N 3Nmatrices.
Expanding the dynamic displacement vector d in the form of
d deioi; 34
whereo OLffiffiffiffiffiffiffiffiffir=E
p represents the dimensionless frequency,O is
the nonlinear vibration frequency of the SWCNT,
d ui T
; wi T
; ci Tn oT
is the vibration mode shape
vector. Substituting Eq. (34) into Eq. (33) yields the nonlinear
eigenvalue equations as follows:
KL KNLd
o2Md 0; 35
This nonlinear equation can be solved through a direct
iterative process below
Step 1: By neglecting the nonlinear matrix KNL, a linear
eigenvalue (ol) and the associated eigenvector are obtained fromEq. (35). The eigenvector is then appropriately scaled up such that
the maximum transverse displacement is equal to a given
vibration amplitude wmax. Note that wmax=w(0.5)for clamped
clamped and hingedhinged SWCNTs while wmax=w(0.57) for a
clampedhinged SWCNT.Step 2: Using the eigenvector to calculate KNL, a new
eigenvalue and eigenvector are obtained from the updated
eigensystem (35).
Step 3: The eigenvector is scaled up again and step 2 is
repeated until the relative error between the given vibration
amplitude and the maximum transverse displacement wmax iswithin 0.1%.
5. Molecular mechanics simulation for the elastic modulus of
SWCNTs
It was assumed in many previous studies [9,11,13,17,28]
dealing with vibration behavior of SWCNTs that the elastic
modulus of the SWCNT is about 1 TPa. In fact, the elastic modulus
is different for different diameters and chirality of the SWCNT and
can be determined through molecular mechanics (MM) simula-
tion. The present paper employs the MM simulation to obtain the
elastic modulus of zigzag (5, 0), (8, 0), (9, 0) and (11, 0) SWCNTs
shown inFig. 2. The interatomic interactions in the SWCNTs are
described by the COMPASS force field (condensed-phased
optimized molecular potential for atomistic simulation studies)
[41]. This is the first and only ab initio force field to enable
accurate simulation and simultaneous prediction of structural,
conformational, vibrational, and thermophysical properties for a
broad range of molecules both in isolation and in the condensed-
phase. The MM simulations are carried out at a temperature of 0 K
to avoid the thermal effect [42]. The thickness of the SWCNT is
selected as 0.34 nm. It is assumed that the two ends of the SWCNT
are fixed boundaries. The simulations of the SWCNT under
compression can be identified through a minimizer processor,
which enables the atoms in CNTs to rotate and move relative to
each other following a certain minimization algorithm tominimize the strain energy so that an equilibrium state can be
identified. In the present analysis, energy minimization is
conducted using the smart minimizer that switches from the
steepest-descent to conjugated gradient and then to the Newton
method. The strain energy of the SWCNT is collected at every
compression deformation with the incremental displacement step
of 0.01 nm. Once the strain energy at every compression step is
available, the second derivative of the strain energy with respect
to the compression can easily be obtained through a simple finite
difference method. The modulus and radius of the zigzag (5, 0), (8,
0), (9, 0) and (11, 0) SWCNTs are listed inTable 1and will be used
in the next section for nonlinear vibration analysis of SWCNTs.
These values agree well with the results obtained from the
molecular dynamic simulation [43], molecular structural
mechanics [44]and experiment[45].
It should be pointed out that for CNTs modeled as a 1-D
isotropic solid such as the Timoshenko beam model used in the
present paper, elastic modulus is the most important elastic
parameter that influences the vibration frequencies of the
SWCNTs. As in many previous studies, see, for example,
Refs. [9,18], our focus is placed on the effect of diameter and
chirality on the elastic modulus of the SWCNTs by using MM
simulation while the shear modulus is approximately determined
fromG =0.5E/(1+n). The Poissons ratio is taken as the value of thegraphite, i.e. v=0.19.
6. Numerical results
Table 2 lists the dimensionless fundamental frequencies
(oOLffiffiffiffiffiffiffiffiffir=E
p ) of (8, 0) SWNTs (m=0.15) with varying total
numbers of nodes N. o l and onl denote the linear and nonlinearfrequencies (wmax=0.4), respectively. The SWNTs are modeled as
nonlocal Timoshenko nanobeams with hingedhinged (HH),
clampedhinged (CH) and clampedclamped (CC) boundary
conditions. Consider the SWNTs with radius r=0.313 nm, length
L=5 nm, Youngs modulus E=1.1556 TPa, Poissons ratio v=0.19,
effective tube thickness h=0.34 nm and shear correction factor
Ks=0.563 [18]. It is seen that the accuracy of the results is
improved with an increasing number of nodes Nand convergent
results are obtained whenNZ10. Hence,N=10 is used in all of the
following numerical calculations.
Table 4
Dimensionless linear fundamental frequency of SWCNTs with different boundary conditions.
Boundary condition m=0.1 m=0.3 m=0.5
Ref.[18] Present Ref. [18] Present Ref.[18] Present
HH 3.0243 3.0210(0.11%) 2.6538 2.6385(0.58%) 2.2867 2.2665(0.89%)
CH 3.6939 3.6849(0.24%) 3.2115 3.1724(1.23%) 2.7471 2.6982(1.81%)
CC 4.3471 4.3269(0.47%) 3.7895 3.7032(2.33%) 3.2420 3.1372(3.34%)
Table 5
Comparisons of nonlinear frequency ratio onl/ol for isotropic homogeneoushingedhinged beam withL/h=100, h =0.3 in.
Wmax=Y Present FEM[46]
1.0 1.11920 1.1181
2.0 1.41801 1.4178
3.0 1.80919 1.8094
4.0 2.24511 2.2455
5.0 2.70429 2.7052
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Table 3 gives the dimensionless linear fundamental
frequencies (oOL2ffiffiffiffiffiffiffiffiffiffiffiffiffirA=EI
p ) of a hingedhinged nonlocal
Timoshenko nanobeam with various slenderness ratio L/h. The
parameters used in this example are[19]:L =10,E=30 106,r=1,
v=0.3, Ks=5/6. Our results are in good agreement with the
analytical results given by Reddy[19]using nonlocal Timoshenko
beam theory as well.
Table 4 presents the dimensionless linear fundamental
frequencies (oOL2 ffiffiffiffiffiffiffiffiffiffiffiffiffirA=EIp ) of the SWNTs based on nonlocalTimoshenko beam model. The analytical solutions given by
0.0 0.2 0.4 0.6 0.81.00
1.05
1.10
1.15
1.20
wmax
H-H:
= 0.00 (l= 0.4680)
= 0.10 (l= 0.4465)
= 0.15 (l= 0.4233)
= 0.20 (l= 0.3963)
0.0 0.2 0.4 0.6 0.81.00
1.02
1.04
1.06
1.08
1.10
C-H:
= 0.00 (l= 0.6765)
= 0.10 (l= 0.6420)
= 0.15 (l= 0.6053)
= 0.20 (l= 0.5628)
wmax
0.01.00
1.02
1.04
1.06
1.08
wmax
nl/
l
nl/
l
nl/
l
C-C:
= 0.00 (l= 0.9036)
= 0.10 (l= 0.8560)
= 0.15 (l= 0.8055)
= 0.20 (l= 0.7473)
0.2 0.4 0.6 0.8
Fig. 3. The effect of nonlocal parameter m on nonlinear frequency ratio versusamplitude curves of (8, 0) SWCNTs with L =5 nm: (a) hingedhinged; (b) clamped
hinged; and (c) clampedclamped.
0.01.00
1.04
1.08
1.12
1.16
wmax
nl/
l
nl/
l
nl/
l
H-H:
L = 5 nm (l= 0.4233)
L = 8 nm (l= 0.2742)
L = 12 nm (l= 0.1853)
L = 16 nm (l= 0.1397)
1.00
1.02
1.04
1.06
1.08
1.10
C-H:
L = 5 nm (l= 0.6053)L = 8 nm (
l= 0.4085)
L = 12 nm (l= 0.2808)
L = 16 nm (l= 0.2130)
wmax
1.00
1.01
1.02
1.03
1.04
1.05
1.06
C-C:
L = 5 nm (l= 0.8055)
L = 8 nm (l= 0.5666)
L = 12 nm (l= 0.3972)
L = 16 nm (l= 0.3036)
wmax
0.2 0.4 0.6 0.8
0.0 0.2 0.4 0.6 0.8
0.0 0.2 0.4 0.6 0.8
Fig. 4. The effect of lengthLon nonlinear frequency ratio versus amplitude curves
of (8, 0) SWCNTs withm=0.15: (a) hingedhinged; (b) clampedhinged; and (c)clampedclamped.
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Wang et al. [18] are also provided for a direct comparison.
The parameters used in this example are taken as [18]:
radius r=0.339 nm, Youngs modulus E=5.5 TPa, Poissons ratio
v=0.19, effective tube thicknessh =0.066 nm and shear correction
factor Ks=0.563. The figures in the brackets are the relative
errors between the present and analytical solutions.
The difference is very small at m=0.1 but tends to increaseas m increases. This is because the nonlocal effect is notincluded in the shear constitutive relationship in their
work[18].
Table 5 gives nonlinear frequency ratio onl/ol at different
maximum vibration amplitudes Wmax=Y (=1.0, 2.0, 3.0, 4.0, 5.0)
0.01.00
1.02
1.04
1.06
1.08C-C:
(5,0): r = 0.1955 nm (l= 0.6402)
(8,0): r = 0.3130 nm (l= 0.8055)
(9,0): r = 0.3525 nm (l= 0.8551)
(11,0): r = 0.4305 nm (l= 0.9415)
wmax
nl/
l
1.00
1.02
1.04
1.06
1.08
1.10C-H:
(5,0): r = 0.1955 nm (l= 0.4666)
(8,0): r = 0.3130 nm (l= 0.6053)(9,0): r = 0.3525 nm (
l= 0.6496)
(11,0): r = 0.4305 nm (l= 0.7304)
wmax
nl/
l
1.00
1.04
1.08
1.12
1.16
1.20H-H:
(5,0): r = 0.1955 nm (l= 0.3164)
(8,0): r = 0.3130 nm (l= 0.4233)
(9,0): r = 0.3525 nm (l= 0.4598)
(11,0): r = 0.4305 nm (l= 0.5302)
wmax
nl/
l
0.2 0.4 0.6 0.8
0.0 0.2 0.4 0.6 0.8
0.0 0.2 0.4 0.6 0.8
Fig.5. The effect of radius ron nonlinear frequency ratio versus amplitude curves
of the SWCNTs with m=0.15 and L=5 nm: (a) hingedhinged; (b) clampedhinged; and (c) clampedclamped.
0.0 0.2 0.4 0.6 0.8 1.00.0
0.1
0.2
0.3
0.4
C-C:
= 0.00
= 0.10
= 0.15
= 0.20
x/L
w
0.0 0.2 0.4 0.6 0.8 1.00.0
0.1
0.2
0.3
0.4
C-H:
= 0.00
= 0.10
= 0.15
= 0.20
x/L
w
0.0 0.2 0.4 0.6 0.8 1.00.0
0.1
0.2
0.3
0.4
H-H:= 0.00
= 0.10
= 0.15
= 0.20
x/L
w
Fig. 6. The effect of nonlocal parameter on the nonlinear mode shapes (deflection
w) of (8, 0) SWCNTs with wmax=0.4 andL =5 nm: (a) hingedhinged; (b) clamped
hinged; and (c) clampedclamped.
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for isotropic homogeneous hingedhinged withL/h=100,h =0.3 in.
Here, Yffiffiffiffiffiffiffi
I=Ap
is the radius of the gyration of the beam withIand
A as the cross-section area and area moment of inertia, onlandolare the dimensionless nonlinear and linear frequencies,
respectively. The results obtained by the present direct iterative
method and finite element method[46]are listed inTable 5. Good
agreement was observed between the results obtained by the
direct iterative method and finite element method.
We now investigate the nonlinear free vibration of hinged
hinged (HH), clampedhinged (CH) and clampedclamped (C
C) nonlocal SWCNTs. Zigzag (5, 0), (8, 0), (9, 0) and (11, 0) SWCNTs
are considered and their elastic modulus and radius are listed in
Table 1. The frequency is normalized aso OL ffiffiffiffiffiffiffiffiffir=Ep inFigs. 37.InFigs. 35, the linear fundamental frequenciesolare also given.Unless otherwise stated, it is assumed that the length of theSWCNTs L =5 nm, Poissons ratio v=0.19, effective tube thickness
h =0.34 nm and shear correction factor Ks=0.563[18].
Fig. 3shows the nonlocal effect on the nonlinear frequency ratioonl/olversus amplitude curves for the (8, 0) SWCNTs withL =5 nm.Note that the nonlocal parameter m=0 corresponds to classicalSWCNTs without nonlocal effect. The SWCNTs exhibit a typical hard-
spring behavior, i.e., the nonlinear frequency ratio increases as the
vibration amplitude is increased for all boundary conditions. The
nonlocal parameter has a significant effect on the nonlinear vibration
behavior. At a given vibration amplitude, an increase in the nonlocal
parameter leads to both smaller linear and nonlinear frequencies but
a higher nonlinear frequency ratio. The clampedclamped SWCNT has
the highest while the hingedhinged one has the lowest linear
frequency, nonlinear frequency and nonlinear frequency ratio since
the end support is the strongest for the clampedclamped SWCNT
and the weakest for the hingedhinged SWCNT.
Fig. 4 shows the effect of beam length L on the nonlinear
frequency ratio versus amplitude curves for (8, 0) SWCNTs with
m=0.15. Both linear frequency and nonlinear frequency ratiodecrease as the lengthL increases. As L changes from 5 to 16 nm,
the linear frequency drops remarkably while the nonlinear
frequency ratio decreases slightly. The effect of beam length L
on the nonlinear frequency ratio is seen to be very small and is
negligible for long SWCNT (LZ16 nm).
Fig. 5 shows the effect of the radius r on the nonlinear
frequency ratio versus amplitude curves for SWCNTs with m=0.15andL =5 nm. Again, zigzag (5, 0), (8, 0), (9, 0) and (11, 0) SWCNTs
are considered. The radius changes from 0.1955 nm of the (5, 0)SWCNT to 0.4305 nm of the (11, 0) SWCNT. Results show that an
increase in the radius significantly raises the linear fundamental
frequency but slightly increases the nonlinear frequency ratio.
The nonlinear fundamental mode shapes for the displacement w
and rotation c of (8, 0) SWCNTs are plotted in Figs. 6 and 7with
various nonlocal parameter (m=0.0, 0.1, 0.15, 0.2) at wmax=0.4 andL=5 nm. The maximum displacementw occurs at the midpoint of
the hingedhinged and clampedclamped SWCNTs buts slightly
deviates from the center of the clampedhinged SWCNT. The
nonlocal parameter nearly has no effect on the nonlinear mode
shape (w andc) for the hingedhinged SWCNT, but it is relatively
large for the clampedhinged and clampedclamped SWCNTs. The
similar phenomenon is also found by Wang et al. [18] for linear
vibration modes of the nonlocal Timoshenko beams. Wang et al. [18]proved that the linear vibration modes of the hingedhinged beam
do not include any nonlocal parameter, which is included in the
linear vibration modes of the clampedhinged and clamped
clamped beams. It is should be pointed out that though the nonlocal
parameter has not effect on both the linear and nonlinear modes of
the hingedhinged SWCNT, it has significant effect to both the linear
and nonlinear frequencies of the hingedhinged SWCNT, as can be
seen from all of the results inTables 3, 4andFig. 3.
7. Conclusions
This paper investigates the nonlinear free vibration of SWCNTs
based on von Karman geometric nonlinearity, Timoshenko beam
0.0 0.2 0.4 0.6 0.8 1.0-0.08
-0.06
-0.04
-0.02
0.00
0.02
0.04
0.06
0.08
H-H:
= 0.00
= 0.10
= 0.15
= 0.20
x/L
0.0 0.2 0.4 0.6 0.8 1.0-0.06
-0.04
-0.02
0.00
0.02
0.04
0.06
0.08
0.10
C-H:
= 0.00
= 0.10
= 0.15
= 0.20
x/L
0.0 0.2 0.4 0.6 0.8 1.0-0.06
-0.04
-0.02
0.00
0.02
0.04
0.06
C-C:
= 0.00
= 0.10
= 0.15
= 0.20
x/L
Fig. 7. The effect of nonlocal parameter on the nonlinear mode shapes (rotation
c) of (8, 0) SWCNTs withwmax=0.4 andL =5 nm: (a) hingedhinged; (b) clamped
hinged; and (c) clampedclamped.
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theory and Eringens nonlocal elasticity theory. Theoretical
formulations include the small scale effect and the influences of
transverse shear deformation and rotary inertia. The differential
quadrature (DQ) method and a direct iterative approach are
employed to obtain the nonlinear vibration frequencies and mode
shapes of nonlocal nanobeams with different end supports. Zigzag
(5, 0), (8, 0), (9, 0) and (11, 0) SWCNTs are considered in numerical
calculation and their elastic modulus is obtained by using MM
simulation. Numerical results show that: (1) at a given vibrationamplitude, an increase in nonlocal parameter leads to smaller
linear and nonlinear frequencies but a higher nonlinear frequency
ratio; (2) both linear frequency and nonlinear frequency ratio
become lower as the length of SWCNT increases and the radius
decreases; (3) the nonlocal parameter has an insignificant effect
on the nonlinear mode shape but can considerably change the
linear and nonlinear frequencies.
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