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7/21/2019 Nonlinear Vibrations of a Single-walled Carbon Nanotube for Delivering of Nanoparticles
1/19
Nonlinear Dyn
DOI 10.1007/s11071-014-1255-y
ORIGINAL PAPER
Nonlinear vibrations of a single-walled carbon nanotube
for delivering of nanoparticles
Keivan Kiani
Received: 9 April 2013 / Accepted: 13 January 2014
Springer Science+Business Media Dordrecht 2014
Abstract Thecapability of carbonnanotubes(CNTs)
in efficient transporting of drug molecules into the bio-
logical cells has been the focus of attention of vari-
ous scientific disciplines during the past decade. From
applied mechanics points of view, translocation of a
nanoparticle inside the pore of a CNT would result
in vibrations. The true understanding of the interac-
tive forces between the moving nanoparticle and the
inner surface of the CNT is a vital step in factual
realization of such vibrations. Herein, by employing
the nonlocal Rayleigh beam theory, nonlinear vibra-tions of single-walled carbon nanotubes (SWCNTs)
as nanoparticle delivery nanodevices are studied. The
existing van der Waals interactional forces between
the constitutive atoms of the nanoparticle and those
of the SWCNT, frictional force, and both longitudinal
and transverse inertial effects of the moving nanoparti-
cle are taken into account in the proposed model. The
nonlinear-nonlocal governing equations are explicitly
obtained and then numerically solved using Galerkin
method and a finite difference scheme in the space
and time domains, respectively. The roles of the veloc-
ity and mass weight of the nanoparticle, small-scale
effect, slenderness ratio, and vdW force on the maxi-
mumlongitudinal and transverse displacements as well
as the maximum nonlocal axial force and bending
K. Kiani (B)
Department of Civil Engineering, K.N. Toosi University
of Technology, Valiasr Ave., P.O. Box 19967-15433,
Tehran, Iran
e-mail: k_kiani@kntu.ac.ir; keivankiani@yahoo.com
moment within the SWCNT are examined. In general,
the obtained results reveal that the nonlinear analysis
should be performed when the nanotube structure is
traversed by a moving nanoparticle with high levels of
the mass weight and velocity.
Keywords Single-walled carbon nanotube
(SWCNT) Nanoparticle delivery NonlinearvibrationNonlocal Rayleigh beam theory
1 Introduction
Due to the brilliant mechanical properties of car-
bon nanotubes (CNTs) [15] as well as frictionless
nature of their inner surface for conveying fluids flow
[68], they are recognized as superior nanodevices for
nanoparticle delivery [913]. Among various forms
of CNTs, single-walled carbon nanotubes (SWCNTs)
have been broadly investigated because direct compar-
ison between thepredictedproperties by the theoretical
worksandthoseof experimentallyobserveddata would
be possible[14,15]. In order to control vibrations of
SWCNTs for transporting of nanoparticles, their vibra-
tion behaviors dueto anindividualmovingnanoparticle
should be rationally investigated.
Atomic simulationsof nanostructuresgenerally take
alotoftimeandlaborcosts.Ontheotherhand,ifatleast
onedimension of thenanostructure under study is large
enough in compare to other ones, such costs consider-
ably increase. As a result, exploiting alternative effi-
cient techniques in analyzing of such nanostructures
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K. Kiani
has been the focus of attentions of the nanotechnol-
ogy community during the past two decades. Nonlocal
continuum field theory of Eringen [16,17] is among
the successful ones, which has been frequently used
for modeling of SWCNTs. To this end, the equivalent
continuum structure (ECS) pertinent to the SWCNT
is considered. The ECS for a SWCNT is an isotropichollow cylinder solid whose most of its frequencies
are identical to those of the SWCNT under study. In
nonlocal modeling of the nanostructure, by using a so-
called small-scale effect parameter, the existing inter-
atomic bonds between the constitutive atoms of CNTs
are appropriately incorporated into the equations of
motion. For each problem, the small-scale effect para-
meter is commonly adjusted by comparing the pre-
dicted results by the nonlocal model with those of an
atomic model[1821]. In this research work,sinceonly
the longitudinal and transverse vibrations of SWCNTsdue to translocation of nanoparticles are of concern, a
nonlocal beam model is employed. Certainly, if cap-
turing the propagated circumferential waves within the
SWCNT due to the passage of a moving nanoparticle
would be also of interest, nonlocal shell models should
be replaced and then appropriately analyzed.
In the context of nonlocal continuum theory, the
investigations on the effects of small-scale as well as
mass weight and velocity of the moving nanoparticle
on the linear transverse vibrations of SWCNTs were
initiated by Kiani and Mehri [22]. Such studies werealso carried out for double-walled CNTs, and analyt-
ical expressions of elastic deformation fields for the
innermost and outermost tubes were obtained[23,24].
In other complementary works, through using nonlo-
cal beam theories, the inertial effects of the moving
nanoparticle were also taken into account in the mod-
eling of the problem[25,26]. Such studies explained
that under what situations the effects of inertial terms
of the moving nanoparticle due to the vibrations of the
hosted nanotube are not negligible at all, and should be
appropriately included in the modeling of the problem.
Simsek[27] examined transversely forced vibrations
of a SWCNT subjected to a moving harmonic force
in the context of nonlocal Euler-Bernoulli beam the-
ory for small deflections. In another work, Simsek [28]
studied laterally small vibrations of microbeams under
action of a moving microparticle on the basis of mod-
ified couple stress theory. In the latter two works, the
inertial effects of the moving micro-/nanoparticle were
not considered in the proposed models. There are also
some works on the influence of nanoparticle transloca-
tion on thesmall in-planeandout-of-planevibrations of
nanoplates [2932]. As it is seen, the undertaken works
for the problems of moving nanoparticle-SWCNT
interaction were restricted to small deflections. In some
cases, as itwas exploredin some details formacro-scale
structures subjected to a moving mass [33], the hypoth-esis of small displacements may be not reasonable. In
following up this matter, this work is mainly devoted
to answer this question that under what circumstances
the linear analysis (LA) of the problem would be no
longer satisfactory.
In the present scrutinization, nonlinear vibrations
of a SWCNT for transporting a nanoparticle with a
constant velocity are investigated in the framework of
nonlocal continuum theory of Eringen. The interac-
tional forces between the moving nanoparticle and the
vibrating SWCNT are taken into account. To this end,the mass weight of the nanoparticle, the vdW forces
between the atoms of the moving nanoparticle and
those of the SWCNT, and both the longitudinal and
transverse inertial effects of the moving nanoparticle
are incorporated into the above-mentioned interactive
forces. In the context of large displacements, the equa-
tions of motion of the SWCNT are obtained on the
basis of the nonlocal Rayleigh beam theory. Due to
the appearance of the inertial effects in the governing
equations, finding an analytical solution is a very prob-
lematic task. Thereby, the Galerkin method and a finitedifference scheme are implemented for discretization
of the nonlinear-nonlocal governing equations in the
space and time domains, respectively. The dynamic
axial andtransverse displacements as well as thenonlo-
cal axialandbending momentwithintheSWCNTacted
upon by a moving nanoparticle are numerically calcu-
lated.Theeffects of thecrucial factors on themaximum
values of the elastic field of the SWCNT are inspected
in some detail.
2 Definition and assumptions of the physical
problem
An ECS with simply supported ends acted upon by a
moving nanoparticle is considered as shown in Fig.1.
The ECS is a hollow isotropic elastic solid of length lb,
inner/outer radiusri/ro, elasticity modulusEb, cross-
sectional area Ab, and moment inertia Ib. A mov-
ing mass with mass weight mg and constant veloc-
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Nonlinear vibrations of SWCNTs for nanoparticle delivery
Fig. 1 aA SWCNT for
delivering of a nanoparticle.
bA simply supported ECS
pertinent to the SWCNT
acted upon by a moving
nanoparticle
(a)
(b)
Mg v
xM
lb
xM
lb
Directi
onofm
ovemen
t
ofthen
anopar
ticle
ityv enters the hollow space of the SWCNT from the
left-hand side. The location of the moving nanopar-
ticle on the inner surface of the ECS is denoted by
(xM= vt,zM= ri )wheretis the time parameter(see Fig.1). Due to the fairly strong attraction forces
between the constitutive atoms of the moving nanopar-
ticle and those of the SWCNT, in which thez direc-
tion component of the resultant force is represented
by FvdW, the moving nanoparticle would be in con-tact with the inner surface of the SWCNT through the
course of vibration (i.e., 0t lbv). The longitudinal
friction force between the outer surface of the mov-
ing nanoparticle and the inner surface of the SWCNT,
Ff, is assumed to obey the hypothesis of the Coulomb
friction theory. Thereby,
Ff=kMg+FvdWMD
2uzDt2
(xM,zM), (1)
where k is the kinetic friction coefficient, D2
Dt2 is
the second material derivative with respect to time,
uz= uz(x, t)is the transverse displacement field ofthe ECS, and its longitudinal one is represented by
ux= ux(x, t). By taking into account of both lon-gitudinal and transverse inertial effects of the moving
nanoparticle, the longitudinal and transverse interac-
tional forces at the contact point, which are, respec-
tively, denoted byFcxandFcz , are stated by
Fcx=
FfMD2ux
Dt2
(xM,zM)
=M
k
D2uzDt2 D2uxDt2
(xM,zM)
H(lbxM), (2a)
Fcz= M
g D2uz
Dt2
(xM,zM)
H(lbxM), (2b)
where= g+ FvdWM
, andHis the Heaviside func-
tion. Based on theRayleigh beam theory, the longitudi-
nal and the transverse components of the displacement
field would beux(x,z, t)=u(x, t)zw,x(x, t)anduz(x,z, t)=w(x, t)whereu(x, t)andw(x, t)denotethe longitudinal and the transverse dynamic displace-
ments of the neutral axisof the ECS,and [.],xrepresentsthe first derivative of[.]with respect tox. By introduc-ing such displacements to Eqs. (2a) and (2b), one can
arrive at
Fcx=Mk D2wDt2
D2u
Dt2riD
2w,x
Dt2
(xxM)H(lbxM), (3a)
Fcz=M
g D2w
Dt2
(xxM)H(lbxM), (3b)
whererepresents the Dirac delta function. There are
some notes on using Eqs. (3a) and(3b) that should be
paid attention to by the interested readers:
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K. Kiani
1. The moving nanoparticle is traveling on a straight
line. Therefore, only the longitudinal and the trans-
verse components of displacements of the SWCNT
are taken into account. For an arbitrary path of
motion of the nanoparticle on the inner surface of
the SWCNT, special treatments should be consid-
ered in both modeling and analyzing of the problemunder study.
2. Duringthecourse ofexcitation,the moving nanopar-
ticle would be entirely in contact with the SWCNT.
Therefore, the full inertial effects of the moving
nanoparticle due to the motion of the SWCNT have
been considered. Now, if someone is interested in
studying the separation of the moving nanoparticle
from the inner surface of the SWCNT, the initi-
ation of such a phenomenon could be monitored
by checking the sign of the contact force in the
z direction. When such a positive contact forcebecomes negative at a special time, the separation
definitely occurs. During the course of separation,
the particle moves within the gaseous continuum
of the pore of the SWCNT, and we have Fcz= 0.At such a time interval, the motion of the mov-
ing nanoparticle could be readily investigated via
Newtons second law. Due to the vdW interactional
forces as well as the gravitational force, surely, the
moving nanoparticle would touch again the inner
surface of the SWCNT (i.e., reattachment). In this
study, the possibility of separation of the movingnanoparticle from the inner surface of the SWCNT
will be studied. Furthermore, the role of nonlin-
earity of the strains, for the case of large deflec-
tion of the SWCNT, on such an interesting phe-
nomenon is addressed. However, the phenomenon
of attachmentreattachment is not captured by the
proposed model, since it is assumed that the fairly
strong attraction forces between the constitutive
atoms of the nanoparticle and those of the SWCNT
would prevent the moving nanoparticle from sepa-
ration.
3. The cause of motion of the nanoparticle has not
been considered, since only the effects of the parti-
cle translocation on the vibrations of the nanostruc-
ture are of particular interest. Surely, any cause of
movement of the nanoparticle and its interactional
effects with the dynamic displacement field of the
SWCNT would result in more complicated govern-
ing equations as well as more difficulties in solving
the governing equations of the problem.
In thefollowing part, thederivation of thenonlinear-
nonlocal governing equations for slender SWCNTs
subjected to a moving nanoparticle will be explained.
For the sake of generality in studying the problem, the
equations of motion are presented in the dimensionless
form. The initial and boundary conditions are imposed
to the equations of motion. For solving the resultingboundary value problem, the Galerkin method is pro-
posed in the continuing. By application of such a pow-
erful method and using appropriate mode shapes, the
nonlinear governing equations are deduced to the non-
linear ordinary differential equations (ODEs). Using a
finite difference scheme, the resulting ODEs are solved
in the time domain, and the generated dynamic dis-
placements and forces of the SWCNT due to a moving
nanoparticle are determined.
3 Nonlocal modeling of the problem under study
According to the von-Karman beam theory for the
SWCNT which has been modeled based on the NRBT,
the nonlinear axial strain and stress accounting for
largedeflections-small rotationsof theECSare approx-
imated as
xx=ux,x+1
2
u2x,x+w2x,x
u,x
+1
2 u2,x+w2,xzw,xx, (4a)xx= EbxxEb
u,x+
1
2
u2,x+w2,x
zw,xx
.
(4b)
In the context of nonlocal continuum theory of Erin-
gen, the only nonlocal stress component of the present
model is expressed by [17,3436]
nlxx(e0a)2nlxx,xx= xx, (5)
wheree0ais called small-scale parameter. The mag-nitude of this parameter could be determined by com-
paring the predicted dispersion curves by the proposed
nonlocalmodelwith thoseofanatomistic-basedmodel.
By multiplying both sides of Eq.(5) byzandz2, and
then integrating the resulting statements over the cross-
sectional area of the ECS, the nonlocal axial force,
Nnlb , and the bending moment, Mnlb , within the ECS
are related to their classical (i.e., local) counterparts as
follows:
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Nonlinear vibrations of SWCNTs for nanoparticle delivery
Nnlb (e0a)2Nnlb,xx=Nb=
Ab
xxdA=EbAb
u,x+1
2
u2,x+w2,x
,
(6a)
Mnlb (e0a)2Mnlb,xx=Mb=
Ab
zxxdA= EbIbw,xx. (6b)
In order to derive the governing equations of the ECS
fordelivering a nanoparticleon thebasisof thenonlocal
continuum theory, the kinetic energy of the ECS,T(t),
its elastic strain energy,U(t), and the work done by
the exerted forces of the moving nanoparticle on the
ECS,W(t), should be appropriately evaluated. These
parameters are stated as
T(t)=12
lb0
b Ab u2 + w2+Ibw2,xdx, (7a)U(t)=1
2
lb0
u,x+
1
2
u2,x+w2,x
Nnlb
w,xxMnlb
dx, (7b)
W(t)=
(Fcxux+Fczuz)(xxM)
(zzM) d H(lbxM), (7c)
where the over-dot sign denotes the derivative with
respect to the time, and represents the inner surface
region of the ECS. In order to derive the equations of
motion for the problem at hand, the Hamiltons princi-
ple is exploited:t2
t1(TU+W) dt=0,in which
t1andt2are two arbitrary times, andis the variation
symbol. Thereby, the nonlocal governing equations are
obtained as
bAbuNnlb,x=Fcx(xxM)H(lbxM), (8a)bAbwIbw,xx
Nnlb w,x,x
Mnlb,xx= Fczri Fcx,x (xxM)H(lbxM), (8b)
by combining Eqs.(6a) and (6b) with Eqs. (8a) and
(8b), the nonlocal axial force and bending moment
within the ECS in terms of ECSs displacements are
derived as,
Nnlb =Nb+(e0a)2 [bAbuFcx(xxM)H(lbxM)],x, (9a)
Mnlb=Mb+(e0a)2
bAbwIbw,xx
Nnlb w,x,x
Fcz
ri Fcx,x(xxM)H(lbxM)
,
(9b)
by substituting Eqs. (9a) and (9b) into Eqs. (8a) and
(8b), one can arrive at
bAbu(e0a)2u,xx
Nb,x= Fcx(xxM)(e0a)2 (Fcx(xxM)),xx
H(lbxM), (10a)
bAb
w(e0a)2w,xxbIb w,xx(e0a)2w,xx xx
Nnlb w,x
,x
(e0a)2Nnlb w,x
,xx x
Mb,xx
= Fczri Fcx,x (xxM)(e0a)2 ((Fczri Fcx,x (xxM),xxH(lbxM). (10b)Inorder toexpress Eqs. (10a)and(10b) intermsof only
displacements, from Eqs. (6a) and (6b),NbandMbas
a function of displacement components are substituted
into these equations. On the other hand, in the context
of small rotation,Nnlb w,x
,x
(e0a)2Nnlb w,x
,xxx
EbAb
u2,x+ 12
u2,x+w2,x
w,x
,xAs a result, from
Eqs. (10a) and (10b), the nonlocal equations of motion
of aSWCNTtransportingan individualmovingnanopar-
ticle in terms of displacements accounting for largedeflections are obtained as
bAbu(e0a)2u,xx
EbAb u2,x+12 u2,x+w2,x
,x
= Fcx(xxM)(e0a)2 (Fcx(xxM)),x xH(lbxM),(11a)
bAb
w(e0a)2w,x xbIb w,x x(e0a)2w,x xxx
EbAb
u2,x+
1
2
u2,x+w2,x
w,x
,x
+EbIbw,x xxx=
Fczri Fcx,x
(xxM) (e0a)2 FczriFcx,x (xxM),x xH(lbxM).
(11b)
Since only the influence of the exerted forces by the
moving nanoparticle on the deformation field of the
SWCNT is of interest, the initial deflection of the
SWCNT due to its own weight is neglected. Hence,
the following initial conditions are considered,
u(x, t=0)=0, w(x, t=0)=0. (12)
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K. Kiani
In the case of simply supported SWCNT with fixed
movable ends, the following conditions should be
satisfied:
u(x=0, t)=0, w(x=0, t)=w(x=lb, t)=0,Nnlb (x=lb, t)=0, Mnlb (x=0, t)
=Mnlb (x=lb, t)=0. (13)Formoregenerality, the followingdimensionless quan-
tities are introduced:
= xlb
, M=xMlb , u =ulb
, w = wlb
, = 1l2b
EbIbbAb
t,
=e0alb
, =lbrb
, M= MbAblb
, = vCL
, =
glbCL
,
= g
, =lbri
,[.]=[.], +2[.],+()2[.], ,(14)
whereis a dimensionless operator,rbis the gyra-tion radius of theECSs cross-section (i.e.,rb
= IbAb ),and CLis the speed of the longitudinal wave within theECS (i.e.,CL =
Ebb
). By introducing the dimen-
sionless parameters in Eq. (14) to Eqs. (11), (12),
and (13), the dimensionless nonlinear governing equa-
tions of a SWCNT subjected to a moving nanoparticle
based on the nonlocal continuum theory of Eringen are
expressed by:
u,
2 u2,+12 u2,+w2,
,
=M k ()2 w
u1
w
(xxM)
H(1M), (15a)
w, 2w,
2
u2,+1
2
u2,+w2,
w,
,
+w,
=M
()2 w1
k
()2 w
u1
w
,
(M)H(1M),(15b)
with the following initial and boundary conditions,
u( , =0)=0, w(, =0)=0,u(=0, )=0, w(=0, )=w(=1, )=0,N
nl
b (=1, )=0, Mnl
b (=0, )=Mnl
b (=1, )=0,(16)
where the dimensionless operator is defined by
[.] = [.] 2[.], . Using Eqs. (9a), (9b), and (14),the dimensionless nonlocal axial force and bending
moment within the SWCNT subjected to a moving
nanoparticle are calculated in terms of dimensionless
displacements as follows:
Nnl
b =2
u,+1
2
u2,+w2,
+2
u, M
k
()2 wu1
w,
(M)H(1M)
,
,
(17a)
Mnl
b = w, +2
w, 2w,
Nnl
b w,, M()2
w
1
k ()2 w
u1
w,
,
(M)H(1M)
,
(17b)
where Nnl
b = Nnlb l
2b
EbIband M
nl
b = Mnlb lb
EbIb. Due to the
appearance of both longitudinal and transverse iner-
tial effects of the moving nanoparticle as well as the
existence of the nonlinear terms in the formulations of
the nonlocal equations of motion, seeking an analyti-cal solution to Eqs. (17a) and(17b) is a very problem-
atic task. Thereby, suggestion of an efficient numerical
scheme in solving such equations would be of great
beneficial. In the following part, Galerkin method plus
to a special finite difference scheme is employed for
fulfilling such a crucial job.
4 Solving the coupled nonlinear partial differential
equations of motion of a SWCNT
for nanoparticle delivery
The only unknown dimensionless displacements of the
ECSare discretized in terms of mode shapesas follows:
u( , )=N Mi=1
ui ( )ui (),
w(,)=N Mi=1
wi ()wi (), (18)
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Nonlinear vibrations of SWCNTs for nanoparticle delivery
whereui ( )andwi ( )are thei th mode shapes asso-
ciated with the longitudinal and transverse displace-
ments of the SWCNT, respectively.ui ( )andwi ( )in
order are the unknown parameters pertinent to thei th
mode shapes of longitudinal and transverse displace-
ments that should be determined at the required times,
N Mis the number of vibrational modes which is con-sidered in the analysis of the problem under study. For
the considered boundary conditions of the SWCNT,
the following mode shape functions are taken into
account [25,26]:
ui ( )=sin ((i0.5) ) , wi ( )=sin (i ) .(19)
Now both sides of Eqs. (15a) and (15b) are, respec-
tively, multiplied by uand w where denotes thevariational sign. Through integrating of the resulting
relations over the dimensionless space interval [0,1],
and taking the necessary integration by parts, one can
arrive at the following set of ODEs:
Mbx, = fb, (20)
or
M
uu
b Muw
b
M
wu
b M
ww
b u,
w, = f
u
b
f
w
b , (21)where
u=T, (22a)w=< w1, w2, . . . , wN M >T, (22b)
Muu
b
i j
=1
0
ui
uj+2ui,uj,
d
+M
ui (M)2ui,(M)
uj (M)H(1M),(22c)
Muwb i j =Mui (M)2ui,(M) sgn w
()2
k
wj (M)
1
wj,(M)
H(1M),
(22d)M
ww
b
i j
=1
0
wi
wj +2wi,wj,
+ 2
wi,wj,+2wi,wj,
d
+M
wi (M)2wi,(M)
wj (M)
1
sgn
w
()2
k
wj,(M)
1
wj,(M)
H(1M), (22e)
Mwu
b
i j
= M
wi (M)2wi,(M)
uj,H(1M),
(22f)
f
u
b
i=
10
2
u,+12
u2,+w2,
1+u,
ui,d
+M
ui (M)2ui,(M)
1sgn
w
()2
2w,(M, )
+ ()2 w,(M, )
2u, (M, )+()2 u,(M, )H(1M),
(22g)f
w
b
i=
10
2w,wi,
u,+
1
2
u2,
+ w2,+w,wi, d+Mwi (M)2wi,(M)
()2 + 2w, (M, )+()2 w,(M, ),
1 1
2+ k sgn
w()2
+ 1
2u, (M, )+()2 u,(M, )
,
H(1M). (22h)
In order to evaluate the unknown parameters of
Eq. (21)at each time, let y=x,. Therefore, Eq.(20)could be rewritten as,
M z,=f,
z=
y
x
, M=
Mb 0
0 I
,f=
fby
, (23)
whereIis the identity matrix. For discretizing Eq.(23)
in the time domain, the finite difference method is
exploited. For this purpose, z, is approximated by
z,= zi+1zi , where= i+1 i , zi = z(i ),andzi+1=z(i+1). By substituting such a discretizedform ofz,into Eq.(23),
Mzi+1Mzi f= 0,M=(1 )Mi+Mi+1 ,f= (1 )fi+fi+1 , (24)whereis the weight parameter of time, and its value
is commonly considered in the range of 0 1. In all
calculations, the value of this parameter is set equal
to 0.7. By employing Newtons method for calculating
zi+1in Eq. (24)at each time step,
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K. Kiani
Kzi+1=f,K=M f,zz=zoldi+1 ,f= M zoldi+1+M zi+ f,zi+1=z newi+1 zoldi+1 ,
(25)
wherez
old
i+1is the previous value ofzi+1through theiteration process. The elements off,zhave been givenin Appendix. The unknown parameters in Eq. (25)
are determined by performing iteration process until
achievingthe accurateresults for znewi+1at each time step.
5 Results and discussion
To show the capabilities of the proposed model in
predicting dynamic response of SWCNTs due to the
nanoparticle delivery, a fairly comprehensive paramet-ric study is carried out in accordance with the NA
explained in Sect.4.The limitations of the LA in pre-
dicting the elastic field of the SWCNT due to a moving
nanoparticle are also of particular interest. To this end,
consider a SWCNT acted upon by a moving nanopar-
ticle with the following data[37]:ri = 1 nm, ro =1.34 nm, b = 2500 kg/m3,Eb = 1 TPa, =2, andk=0.3. In the following parts, the influencesof mass weight and velocity of the moving nanoparti-
cle, vdW interactional force, and small-scale parame-
ter on the maximum axial displacement and nonlocalaxial force as well as the maximum deflection and non-
local bending moment of the SWCNT subjected to a
moving nanoparticle are examined in some details. In
order to study the problem in a more reasonable frame-
work, the following normalized fields are introduced:
uN= uumax,st , wN= w
wmax,st, NnlbN=
Nnlb
Nmax,st,
andMnlbN= M
nlb
Mmax,st. In these relations,umax,stand
Nmax,stdenote themaximum dimensionless values of
axial displacement and local axial force due to the sta-
tically applied frictional force at the midspan point of
the SWCNT, respectively. Additionally, wmax,stand
Mmax,strepresentthemaximum dimensionless valuesof deflection and local bending moment due to the stat-
icallyapplied weight of thenanoparticle at themidspan
point of the SWCNT, respectively. Since the effects of
theinterested parameters on themaximum elastic fields
of the SWCNT are of particular concern, the maximum
values of the above-mentioned normalized fields are,
respectively, denoted byuN,max,wN,max,NnlbN,max,
andMnlbN,max.
To show the efficiency of the proposed method-
ology, a convergence study is performed. For this
purpose, the relative errors of the maximum normal-
ized transverse displacement and normalized nonlocal
bending moment are defined by erel,w =wN,maxwN,max(N M=15)wN,max(N M=15) and erel,M =MnlbN,maxMnlbN,max(N M=15)MnlbN,max(N M=15)
. In Fig. 2, the plottedresults oferel,w anderel,Mas a function ofN Mare
provided. As it is seen in Fig.2,the predicted relative
errors of both maximum transverse displacement and
nonlocal bending moment of the SWCNT acted upon
bya movingnanoparticlewoulddecreaseas thenumberof mode shapes increases. Furthermore, for fairly high
levels ofN M, the effect of the vibration mode number
on the above-mentioned relative errors would reduce.
Such a scrutiny reveals that the proposed numerical
scheme is an effective one in nonlinear dynamic analy-
sisofnanotubestructures subjectedto moving nanopar-
ticles.
Fig. 2 Convergence check
of the proposed numerical
model; (e0a=1 nm, =20,M=0.1, VN=0.3)
3 5 7 10 150
0.01
0.02
0.03
0.04
0.05
0.06
NM
erel,w
3 5 7 10 150
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
NM
erel,M
1 3
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Nonlinear vibrations of SWCNTs for nanoparticle delivery
Fig. 3 Comparison of the
predicted normalized
transverse displacement
under the moving object by
the proposed model with
those of Lee [38]:
aVN=0.11,bVN
=0.5; (M
=0.2,
rb= 0.03lb , ( ) LA,() NA, () Lee[38])
0 0.25 0.5 0.75 10
0.3
0.6
0.9
1.2
wN
(M
,)
0 0.25 0.5 0.75 10
0.45
0.9
1.35
1.8
wN
(M
,)
M
(a)
(b)
For the sake of verification, the predicted results
by the proposed model are compared with those of
Lee [38] in some special cases. Lee [38] studied lin-
early dynamical responses of simply supported Timo-
shenko beams which are acted upon by a moving mass.
In order to compare the predicted results by the NRBT
with those of the Timoshenko model, a slender beam
with lbrb
= 1003 is considered. The predicted normal-
ized deflections of the beam under the moving object
versus its dimensionless position have been plotted in
Fig.3a, b for two levels of the velocity of the movingobject. The predicted results by the proposed model
based on the LA and nonlinear analysis (NA) as well
as those of Lee [38] have been provided in these fig-
ures. Asit can beseenin Fig. 3a, b, there is a reasonably
good agreement between the linearly predicted results
of the present model and those of Lees model[38] for
most of the positions of the moving object. A more
detailed scrutiny of the plotted results reveals that the
maximum relative error between the predicted normal-
ized deflection by the LA and those of the model of
Lee [38] are limited to 1.5 and 3 percent forVN=0.11and 0.5, respectively. For the lower level of thevelocity
of the moving body, the predicted normalized deflec-
tions under the moving body on the basis of the NA
are in line and very close to the predicted values by the
LA. However, for the greater levels of the velocity, the
discrepancies between the results of the LA and those
of the NA are more obvious. As it is seen in Fig. 3b,
the linear model generally overestimates the results of
thenonlinear model. In the following, a comprehensive
parametric studywill bepresentedto determinethelim-
itations of the linear model in predicting the maximum
elastic fields of the SWCNTs subjected to a moving
nano-object.
The time history plots of the displacements and non-
local forces of the midspan point of the SWCNT used
for a nanoparticle delivery are demonstrated in Figs.4
and 5 for different levels of the velocity of the nanopar-
ticle. The predicted results have been provided on the
basis of both LAand NA.In these figures, frepresents
the dimensionless time of leaving the SWCNT by themoving nanoparticle. For a low level of the velocity of
the moving nanoparticle (i.e.,VN=0.1), the predictedresults by the LA and those of the NA are coincident
with a good accuracy (see Fig.4a,5a). For a fairly low
level of the velocity (i.e.,VN=0.2), the predicted lon-gitudinal displacement and nonlocal axial force of the
midspan point of the SWCNT by the LA and those of
the NA are fairly coincident. The predicted deflection
and nonlocal bending moment of the midspan point of
the SWCNT based on the LA and those of the NA are
roughly close to each other. Thediscrepancies betweenthe predicted results by the LA and those of the NA are
more obvious at the locally minimum and maximum
points of the plotted results. As the velocity of the mov-
ing nanoparticle increases, the discrepancies between
the predicted results by the LA and those of the NA
magnify. For the normalized axial displacement and
nonlocalaxial force,thismatteris more apparentduring
the course of free vibration as well as at the end of the
course of forced vibration.However, for thenormalized
1 3
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K. Kiani
0 1 25
0
5
uN
(0.5,
)
0 1 25
0
5
uN
(0.5,)
0 1 25
0
5
uN
(0.5,
)
0 1 210
0
10
uN
(0.5,
)
0 1 215
0
15
uN
(0.5,
)
/f
0 1 25
0
5
Nbn
l (0.5,
)
0 1 25
0
5
Nbn
l (0.5,
)
0 1 25
0
5
Nbn
l (0.5,
)
0 1 210
0
10
Nbn
l (0.5
,)
0 1 215
0
15
Nbn
l (0.5,
)
/f
(a)
(b)
(c)
(d)
(e)
Fig. 4 Time history plots of the normalized axial displacement
and nonlocal axial force of the midspan point of the SWCNTtraversed by a moving nanoparticle for different levels of the
velocity:a VN
=0.1, b VN
=0.3,c VN
=0.5, d VN
=0.7,
eVN= 0.9; (= 50,M= 0.3,e0a= 1 nm; ( ) LA,() NA)
deflectionandnonlocal bending moment, such discrep-
ancies are more obvious in both courses of forced and
free vibrations. As it is observed in Figs. 4and5, the
peak points of both dynamic displacements and nonlo-
cal forces of the SWCNT move from the first phase to
the second oneas the velocity of the moving nanoparti-
cle grows. In the following parts, the influences of both
the velocity and mass weight of the moving nanoparti-
cle on the maximum values of both displacements and
nonlocal forces within the SWCNT acted upon by a
moving nanoparticle are explained.
An interesting study has been conducted to exam-
ine the influence of the existing interactional vdW
forces between the constitutive atoms of the nanopar-
ticle and those of the SWCNT on the nonlinear
dynamic response of the SWCNT subjected to a mov-
ing nanoparticle. For this purpose, the plots of the pre-
dicted both linear and nonlinear results of the displace-
ments as well as nonlocal forces of the SWCNT are
provided in Fig.6ac for different levels of the mov-
ing nanoparticle velocity. For low levels of the moving
nanoparticle velocity (i.e.,VN
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Nonlinear vibrations of SWCNTs for nanoparticle delivery
0 1 22
0
2
wN
(0.5,
)
0 1 22
0
2
wN
(0.5,
)
0 1 22
0
2
wN
(0.5,
)
0 1 23
0
3
wN
(0.5,
)
0 1 22
0
2
wN
(0.5,
)
/f
0 1 21
0
1
Mbn
l (0.5,
)
0 1 22
0
2
Mbn
l (0.5,
)
0 1 22
0
2
Mbn
l (0.5,
)
0 1 22
0
2
Mbn
l (0.5,
)
0 1 22
0
2
Mbn
l (0.5,
)
/f
(a)
(b)
(c)
(d)
(e)
Fig. 5 Time history plots of the normalized deflection and non-
local bending moment of the midspan point of the SWCNT tra-
versed by a moving nanoparticle for different levels of the veloc-
ity: a VN= 0.1, bVN= 0.3, c VN= 0.5, dVN= 0.7, eVN= 0.9; (= 50, M= 0.3,e0a= 1 nm; ( ) LA,() NA)
moment by the LA and those of the NA would increase
as the magnitude of the vdW force increases. For mod-
erate levels of the moving nanoparticle velocity (i.e.,
VN=0.3), the predicted values of both axial displace-ment and nonlocal axial force within the SWCNT by
both the LA and NA linearly magnify with the vdWforce; however, the predicted values of both the deflec-
tion and nonlocal bending moment of the SWCNT
slightly vary with the vdW interactional force. For
>3, the LA can predict both the predicted axial dis-
placements and nonlocal axial forces of the SWCNT
by the NA with relativeerror lower than 3 %. Neverthe-
less, the differences between the predicted deflections
as well as nonlocal bending moments by the LA and
those of the NA are in the range of 78.5 %. For a fairly
high level of the moving nanoparticle velocity (i.e.,
VN= 0.5), the discrepancies between the predictedresults by the LA and those of the NA would com-
monly increase with the magnitude of the vdW force.
Almost for all values of , the NA underestimates the
predicted results by the LA. Based on the NA, the pre-dicted normalized maximum deflections and nonlocal
bending moment reduce with the magnitude of vdW
force.
The effect of mass weight of the nanoparticle on the
vibrations of SWCNTs for nanoparticle delivery is of
interest in this part. The plots of normalized maximum
displacements and nonlocal forces in terms of dimen-
sionless mass of the nanoparticle for different levels
of the nanoparticles velocity have been provided in
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K. Kiani
1 3 51
6
11
uN,max
1 3 51
1.05
1.1
1.15
wN,max
1 3 51
5
9
NbN,max
nl
1 3 50.9
1
MbN,max
nl
1 3 51
6
11
uN,max
1 3 51.35
1.45
1.55
wN,max
1 3 51
5
9
NbN,max
nl
1 3 51.25
1.4
MbN,max
nl
1 3 51
6
11
uN,max
1 3 51.6
1.8
2
wN,max
1 3 51
5
9
NbN,max
nl
1 3 51.4
1.7
MbN,max
nl
(a)
(b)
(c)
Fig. 6 Effect of the vdW interactional force on the maximum
values of normalized displacements and nonlocal forces of the
SWCNT for different levels of the nanoparticles velocity: a
VN=0.1, b VN=0.3,c VN=0.5; [( ) LA, () NA;=50,M=0.3,e0a=1 nm]
Figs. 7ae. Regarding the maximum values of dynami-
calaxial displacementas well as nonlocal axial force of
the SWCNT in which acted upon by a moving nanopar-
ticle, for VN0.3 and all considered levels of the massweight of the nanoparticle, the discrepancies between
the predicted results by the LA and those of the NA
are lesser than 5 %. In the case ofVN= 0.5, suchdiscrepancies are lower than 5 % for M 0.25. ForM >0.25, the discrepancies between the results of the
LA and those of the NA increase with the mass weight
of themoving nanoparticle. ForM=0.5, the NA over-estimates the result of the LA with relative error about
25 %. Commonly, the discrepancies between the pre-
dicted both axial displacement and nonlocal force by
the NA and those of the LA magnify with the velocityof the moving nanoparticle. Concerning lateral vibra-
tion of theexploited SWCNT for nanoparticledelivery,
fora low level of themovingnanoparticlevelocity (i.e.,
VN=0.1) and the considered range ofM, the discrep-ancies between both the deflection and nonlocal bend-
ing moment of the SWCNT by the NA and those of
the LA are lower than 3 %. In such a circumstance, the
predicted results by the LA would be trustable with a
good accuracy. Generally, such discrepancies increase
with the velocity of the nanoparticle, particularly for
those nanoparticles with high values of mass weight
(see Fig.7d, e). According to the LA, for velocity of
the moving nanoparticle up toVN=0.5, there exists aroughly linear relationship between mass weight of the
moving nanoparticle and both values of the normalized
maximum deflection and nonlocal bending moment
within the SWCNT (see Fig. 7be). On the basis of
the NA, the nonlinear variations of such parameters as
a function of the mass weight of the moving nanopar-
ticle are so obvious forVN0.5.Another instructive study is carried out to determine
theinfluenceof themoving nanoparticlevelocityon the
vibrational behavior of carryingnanoparticle SWCNTs
based on both LA and NA. In the case ofM=0.3 ande0a= 1 nm, the predicted displacements and nonlo-cal forces within the SWCNT in terms of the normal-
ized velocity of the moving nanoparticle are demon-
strated in Fig.8.Irrespective of the initial fluctuations
of the normalized maximum deflection and nonlocal
bending moment (forVN
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Nonlinear vibrations of SWCNTs for nanoparticle delivery
0 0.25 0.53.6
3.8
4
uN,max
0 0.25 0.51
1.1
1.2
wN,max
0 0.25 0.52.2
2.4
2.6
NbN,max
nl
0 0.25 0.50.9
0.95
1
MbN,max
nl
0 0.25 0.53
3.5
4
uN,max
0 0.25 0.51.2
1.4
1.6
wN,max
0 0.25 0.52
2.5
3
NbN,max
nl
0 0.25 0.51.2
1.4
1.6
MbN,max
nl
0 0.25 0.50
5
10
uN,max
0 0.25 0.51.5
2
2.5
wN,max
0 0.25 0.52
4
6
NbN,max
nl
0 0.25 0.51
1.5
2
MbN,max
nl
0 0.25 0.50
20
40
uN,max
0 0.25 0.51.5
2
2.5
wN,max
0 0.25 0.50
10
20
NbN,max
nl
0 0.25 0.51
2
3
MbN
,max
nl
0 0.25 0.50
50
M
uN,max
0 0.25 0.51.5
2
2.5
M
wN,max
0 0.25 0.50
20
40
M
NbN,max
nl
0 0.25 0.50
2
4
M
MbN,max
nl
(a)
(b)
(c)
(d)
(e)
Fig. 7 Effect of the nanoparticles weight on the maximum
values of normalized displacements and nonlocal forces of the
SWCNT for different levels of the nanoparticles velocity: a
VN = 0.1, b VN = 0.3, c VN = 0.5, d VN = 0.7, eVN=0.9; [( ) LA, () NA; =60,e0a=1 nm]
tically lessen with the velocity of the moving nanopar-
ticle up toVN= 1. A close scrutiny shows that theLA overestimates the predicted wN,maxby the NA
with relative error lower than 10 %. ForVN 0.68, the LA underesti-
mates the predictedMnlbN,maxby the NA. Excluding the
initial fluctuations of the normalized maximum axial
displacement and nonlocal force forVN
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K. Kiani
0 0.5 12
6
10
14
18
22
VN
uN,max
0 0.5 10.8
1
1.2
1.4
1.6
1.8
2
2.2
VN
wN,max
0 0.5 12
4
6
8
10
12
14
VN
NbN,max
nl
0 0.5 10.8
1.2
1.6
2
2.4
VN
MbN,max
nl
Fig. 8 Effect of the nanoparticle velocity on the maximum values of normalized displacements and nonlocal forces of the SWCNT;
[( ) LA, () NA;=50,M=0.3,e0a=1 nm]
in capturing the predicted results of interest by the NA,
theLAcould reproduce thenormalized maximum elas-
tic fields of the SWCNT on the basis of the NA with
relative error lower than 2 and 10 % forVN=0.1 and0.3, respectively. It is worth mentioning that no regu-
lar pattern is observed for discrepancies of the results
of the LA and those of the NA in terms of the veloc-
ity of the moving nanoparticle. As the magnitude of
the small-scale parameter increases, the discrepancies
between the predictednormalizedmaximum deflection
by the LA and that of the NA would generally mag-
nify. Except the caseVN=0.9, this fact is also true forthe maximum normalized values of the nonlocal bend-
ing moment. ForVN=0.9, the discrepancies betweenthe predicted normalized maximum nonlocal axial and
bending moment of the SWCNT by the LA and those
of the NA would lessen with the small-scale parameter.
Generally, no regular pattern for the discrepancies of
the results of the LA and those of the NA as a function
of the small-scale parameter is detectable for various
levels of the velocity of the moving nanoparticle.
The influence of the slenderness ratio on the gener-
ated displacements and nonlocal forces within the car-
rying nanoparticle SWCNTs is of interest. The plots of
the longitudinal and transverse displacements as well
as nonlocal axial force and bending moment in terms
of the slenderness ratio are provided in Fig. 10ac.
Such plots are provided for three levels of the velocity
of the moving nanoparticle (i.e., VN= 0.1, 0.3, and0.5). In the case ofVN=0.1 (see Fig.10a), the nor-malized displacements and nonlocal forces decrease
as the slenderness ratio of the nanostructure increases.
Further investigations display that the discrepancies
between the predicted results by the NA and those of
the LA would generally lessen with the slenderness
ratio. Regarding the caseVN=0.3 (see Fig.10b), thepredicted normalized axial force and bending moment
would reduce as the slenderness ratio of the SWCNT
increases. However, such a fact is not generally true for
the longitudinaland transverse displacements. Interest-
ingly, the NA predicts that the maximum normalized
transverse displacement would slightly magnify as the
slenderness ratio increases. Nevertheless, the obtained
results based on the LA showthat this parameter would
lessen with the slenderness ratio. Such a fact is more
obvious forVN= 0.5 (see Fig.10c). Excluding thenormalized maximum longitudinal displacement, the
discrepancies between the predicted results by the LA
and those of the NA would generally reduce as the
slenderness ratio increases. In the case ofVN= 0.5(see Fig.10c), except the predicted normalized maxi-
mumtransverse displacement, both LAandNA display
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Nonlinear vibrations of SWCNTs for nanoparticle delivery
0 1 23.7
3.75
3.8
uN,max
0 1 21
1.1
1.2
wN,max
0 1 22.2
2.4
2.6
NbN,max
nl
0 1 20.9
0.95
1
MbN,max
nl
0 1 23
3.5
4
uN,max
0 1 21.2
1.4
1.6
wN,max
0 1 22.4
2.6
2.8
NbN,ma
x
nl
0 1 2
1.4
1.6
MbN,max
nl
0 1 25
5.5
6
uN,max
0 1 21.6
1.8
2
wN,max
0 1 23.4
3.6
3.8
NbN,max
nl
0 1 21.5
1.6
1.7
MbN,max
nl
0 1 25
10
15
uN,max
0 1 21.9
2
2.1
wN,max
0 1 24
6
8
N
bN,max
nl
0 1 21.9
2.1
2.3
M
bN,max
nl
0 1 210
20
30
e0a(nm)
uN,max
0 1 21.5
2
2.5
e0a(nm)
wN,max
0 1 25
10
15
e0a(nm)
NbN,max
nl
0 1 21.8
2
2.2
e0a(nm)
MbN,max
nl
(a)
(b)
(c)
(d)
(e)
Fig. 9 Effect of the small-scale parameter on the maximum
values of normalized displacements and nonlocal forces of the
SWCNT for different levels of the nanoparticles velocity: a
VN = 0.1, b VN = 0.3, c VN = 0.5, d VN = 0.7, eVN=0.9; [( ) LA, () NA; =50,M=0.3]
that thenormalized maximum dynamic responsewould
decrease with the slenderness ratio. Further, the dis-
crepancies between the results of the NA and those of
the LA would reduce as the slenderness ratio increases.
Among all studied cases, in the case ofVN=0.5, vari-ation of the slenderness ratio has the most influence on
the variation of the maximum dynamic response.
Realizing the accuracy levels of the linear analysis
for the problem under study would be of great impor-
tance in practical applications, for instance, SWCNTs
for drug delivery. To this end, a relative error para-
meter is defined by erel = |[.]L A [.]N A|/|[.]L A|,where[.] is the parameter under study (i.e., [.] =uN,max, orwN,max, orN
nlbN,max, orM
nlbN,max). In
the plane ofVN M, the contour plots pertinent totheerel =0.05, 0.1, 0.2, and 0.30 are presented forthe normalized maximum displacements and nonlocal
forces in Figs.11 and12.The demonstrated results
have been given for three levels of the small-scale para-
meter (i.e.,e0a= 0, 1, and 2 nm). According to theplotted results in these figures, higher values of both
mass weight and velocity of the moving nanoparticle
would result in more discrepancies between the pre-
dictedresults by the LAand those ofthe NA.The region
between two arbitrary contours specifies the zone in
which the LA could estimate the predicted results by
the NA with the relative error in the range of those spe-
cific values pertinent to the aforementioned contours.
As it is seen in Figs.11and12,for nearly a half area
of the consideredVN Mplane, the LA could pre-dict the results of the NA with accuracy lower than 5
percent. The contour lines of deflections and nonlo-
cal bending moments of the SWCNT subjected to a
moving nanoparticle for various small-scale parame-
ters generally pursue the same trend (see Fig.12a, b).
However, this matter is not exactly true for axial dis-
placement and nonlocal axial force (see Fig.11a, b).
Higher density of the contour lines implies more sen-
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K. Kiani
20 40 603.5
4
4.5
uN,max
20 40 601
1.05
1.1
1.15
wN,max
20 40 602
3
4
5
NbN,max
nl
20 40 601
1.2
1.4
MbN,max
nl
20 40 603
4
5
6
uN,max
20 40 601.4
1.6
1.8
wN,max
20 40 602
4
6
8
NbN,max
nl
20 40 601
1.5
2
MbN,max
nl
20 40 600
5
10
15
uN,max
20 40 601.5
2
2.5
wN,max
20 40 600
10
20
30
NbN,max
nl
20 40 601.5
2
2.5
MbN,max
nl
(a)
(c)
(b)
Fig. 10 Effect of the slenderness ratio on the maximum values of normalized displacements and nonlocal forces of the SWCNT for
different levels of the nanoparticles velocity: aVN=0.1,bVN=0.3,cVN=0.5; [( ) LA, () NA;M=0.3,e0a=1 nm]
Fig. 11 Contour plots of
erelfor:aNormalizedmaximum longitudinal
displacement,bNormalized
maximum nonlocal axial
force; ((...)e0a=0,( )e0a=1 nm, ()e0a=2 nm;=50)
0 0.25 0.50
0.25
0.5
0.75
1
M
VN
0.05
0.05
0.05
0.05
0.05
0.1
0.1
0.2
0.2
0.3
0.3
0 0.25 0.50
0.25
0.5
0.75
1
M
VN
0.05
0.05
0.05
0.05
0.1
0.1
0.2
0.2
(a) (b)0.1
0.3
0.3
sitivity of the accuracy level to the variation of both
mass weight and velocity of the moving nanoparticle.
According to Fig.11a, b, for a moving nanoparticle
with M= 0.5, the variation of the velocity of the
moving nanoparticle in the range of 0.50.6 has the
most influence on the variation of the normalized max-
imum axial displacement as well as the nonlocal axial
force.
1 3
7/21/2019 Nonlinear Vibrations of a Single-walled Carbon Nanotube for Delivering of Nanoparticles
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Nonlinear vibrations of SWCNTs for nanoparticle delivery
Fig. 12 Contour plots of
erelfor:aNormalized
maximum deflection,
bNormalized maximum
nonlocal bending
moment; ((...)e0a=0,( )e0a=1 nm, ()e0a
=2 nm;
=50)
0 0.25 0.50
0.25
0.5
0.75
1
M
VN
0.05
0.0
5
0.05
0.05
0.1
0 0.25 0.50
0.25
0.5
0.75
1
M
VN
0.05
0.
05
0.05
0.1
0.1
0.2
0.3
0.1 0.05
(a) (b)
6 Conclusions
Nonlinear longitudinal and transverse vibrations of
SWCNTs fornanoparticledelivery areexploredvia the
nonlocal Rayleigh beam theory. Without considering
the cause of motion of the nanoparticle, it is assumed
that the nanoparticle slips on a straightpath on the inner
surface of the SWCNT. By considering the interac-
tionalvdWforcesbetween theconstitutive atoms of the
nanoparticle and those of the SWCNT, a simple fric-
tional model is employed. Both longitudinal and trans-
verse inertial effects of the moving nanoparticle are
incorporated into the interactional forces. By making
some reasonable assumptions, the nonlocal governing
equations of the model are constructed. The resulting
nonlinear-coupled equations of motion are solved via
Galerkin approach. The influences of the velocity and
the mass weight of the nanoparticle, the vdW interac-
tional force, the small-scale parameter, and the slen-
derness ratio on the maximum elasto-dynamic fields of
the SWCNT are addressed in some detail. The major
obtained results are as
1. As the velocity and the mass weight of the moving
nanoparticle magnify, the discrepancies between
the predicted results by the LA and those of the
NA would increase.
2. The maximum longitudinal displacement as well
as nonlocal axial force within the SWCNT would
increase as the vdW force between the mov-
ing nanoparticle and the nanotube intensifies. For
lower levels of the velocity, both maximum trans-
verse displacement and nonlocal bending moment
would increase as the magnitude of the vdW force
increases. However, for higher velocities, these
parameters would decrease with the vdW force.
Further, the discrepancies between the results of
the LA and those of the NA would magnify with
the vdW force.
3. Generally, the discrepancies between the results of
the LA and those of the NA would decrease as the
slenderness ratio of the SWCNT increases.
4. For moderate levels of the velocity of the moving
nanoparticle, the maximum elasto-dynamic fields
of the SWCNT would generally increase with the
small-scale parameter. However, for high levels of
the moving nanoparticle velocity, variation of the
small-scale parameter has a trivial influence on the
variation of the elasto-dynamic fields of the carrier
SWCNT.
Appendix
Theconstitutive submatrices of thematrixf,zwith theirelements are as,
f,z=
fy
fx
I 0
; f
y=
fu
b
u,
fu
b
w,
fwb
u,
fwb
w,
,f
x= fubu fubw
fwb
u
fwb
w
, (26)1 3
7/21/2019 Nonlinear Vibrations of a Single-walled Carbon Nanotube for Delivering of Nanoparticles
18/19
K. Kiani
wheredenotes the sign of partial derivative. The ele-
ments of the matrices fx
and fy
are calculated as fol-
lows
f
u
b
u i j=
1
02
1+3u,+0.5w2,
ui,uj,d
M()2 ui (M) 2ui,(M)
uj,(M)H(1M), (27a)
fu
b
w
i j
= 1
0
2 ui,
w,
wj,
+32w,wj,
1+u,d
+M()2
ui (M)2ui,(M)
wj, (M)
1
sgn
w
( )
2H(1M), (27b)f
w
b
u
i j
= 1
0
uj,
w,
wj,
+ 2w,wj,
1+u,d
+M()2
wi (M)
2wi,(M)
uj,(M)H(1M), (27c)f
w
b
w
i j
=
1
0
2 u,+0.5u2,+1.5w2,+0.5w,
wi,wj,
+1+u,+0.5
u2,+w2,
wi,
wj,
d+M()2
wi (M)2wi,(M)
12
k
sgn
w
()2
wj,(M)
wj,(M)H(1M), (27d)
f
u
b
u,i j = 2M
ui (M)
2ui,(M)
uj,(M)H(1M), (27e) f
u
b
w,
i j
=2M
1
sgn (
w()2
ui (M)2ui,(M)
wj,(M)H(1M),
(27f)f
w
b
u,
i j
= 2M
wi (M)
2wi,(M)
uj,(M)H(1M), (27g)
f
w
b
w,
i j
=2M
wi (M)2wi,(M) 1
2
k
sgn w()2wj,(M)
wj,(M)H(1M). (27h)
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