Dynamic response of multiple nanobeam system under a moving nanoparticle Alexandria Engineering Journal (2017) xxx, xxx–xxx HO ST E D BY Alexandria University Alexandria Engineering Journal www elsevi[.]
Trang 1Dynamic response of multiple nanobeam system
under a moving nanoparticle
Shahrokh Hosseini Hashemia,b, Hossein Bakhshi Khanikia,*
a
School of Mechanical Engineering, Iran University of Science and Technology, Narmak 16842-13114, Tehran, Iran
bCenter of Excellence in Railway Transportation, Iran University of Science and Technology, 16842-13114 Narmak, Tehran, Iran
Received 22 September 2016; revised 9 November 2016; accepted 15 December 2016
KEYWORDS
Dynamic response;
Analytical solution;
Moving particle;
Nanobeam;
Multi-layered nanobeam
Abstract In this article, nonlocal continuum based model of multiple nanobeam system (MNBS) under a moving nanoparticle is investigated using Eringen’s nonlocal theory Beam layers are assumed to be coupled by winkler elastic medium and the nonlocal Euler-Bernoulli beam theory
is used to model each layer of beam The Hamilton’s principle, Eigen function technique and the Laplace transform method are employed to solve the governing equations Analytical solutions
of the transverse displacements for MNBs with simply supported boundary condition are presented for double layered and three layered MNBSs For higher number of layers, the governing set of equations is solved numerically and the results are presented This study shows that small-scale parameter has a significant effect on dynamic response of MNBS under a moving nanoparticle Sen-sitivity of dynamical deflection to variation of nonlocal parameter, stiffness of Winkler elastic med-ium and number of nanobeams are presented in nondimensional form for each layer
Ó 2016 Faculty of Engineering, Alexandria University Production and hosting by Elsevier B.V This is an open access article under the CC BY-NC-ND license ( http://creativecommons.org/licenses/by-nc-nd/4.0/ ).
1 Introduction
One of the most important structures used in
nano-devices such as oscillators, clocks and sensor nano-devices is
Nano-beams Lots of researches have been done in order to achieve
the behavior of engineering structures such as beams, tubes,
plates and shells in various scales and static conditions such
as bending[1–11]and buckling[12–16] Also in dynamic
man-ners, with highlighting free and forced vibration analysis there
have been plenty of researches [17–28] By investigation of
nanocars in 2005 at Rice University, lots of researches had
started in different subjects to understand the behavior of nanocars Shirai et al [29] studied a controlled molecular motion on surfaces through the rational design of surface-capable molecular structures called nanocars They showed that the movement of the nanocars is a new type of fullerene-based wheel-like rolling motion, not stick-slip or slid-ing translation, due to evidence includslid-ing directional prefer-ence in both direct and indirect manipulation and studies of related molecular structures Sasaki et al.[30]reported the syn-thesis of a new nanovehicle, a porphyrin-based nanotruck The porphyrin inner core was designed for possible transportation
of metals and small molecules across a surface Akimov et al [31]developed molecular models describing the thermally initi-ated motion of nanocars, nanosized vehicles composed of two
to four spherical fullerene wheels chemically coupled to apla-nar chassis, on a metal surface The simulations were aimed
* Corresponding author.
E-mail address: h_bakhshi@mecheng.iust.ac.ir (H.B Khaniki).
Peer review under responsibility of Faculty of Engineering, Alexandria
University.
H O S T E D BY
Alexandria University Alexandria Engineering Journal
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Trang 2at reproducing qualitative features of the experimentally
observed migration of nanocars over gold crystals as
deter-mined by scanning tunneling microscopy Sasaki et al [32]
reported the synthesis of two nanocars by a process resembling
an assembly line where front and rear portions are attached
using hydrogen bonding and metal complexation Vives and
Tour[33]presented a new class of nanovehicles incorporating
trans-alkynyl(dppe) ruthenium-based wheels A four-wheeled
nanocar and a three-wheeled trimer were synthesized for future
studies at the single molecule level Khatua et al.[34]
moni-tored the mobility of individual fluorescent nanocars on three
surfaces: plasma cleaned, reactive ion etched, and
amine-functionalized glass Using single-molecule fluorescence
imag-ing, the percentage of moving nanocars and their diffusion
constants were determined for each substrate It was shown
that the nanocar mobility decreased with increasing surface
roughness and increasing surface interaction strength Ganji
et al.[35] studied nanocars motion on one-dimensional
sub-strate surfaces which provided an important contribution to
the practical goal of designing nanoscale transporters First
principles of VdW-DF calculations were performed to study
the interaction between the nanocar and the
graphene/gra-phyne surface The accuracy of this method was validated by
experimental results and the MP2 level of theory
Moreover, by having a moving nano-car, nano-truck,
three-wheeled trimer or nanoparticle on a surface, it is
neces-sary to know the behavior of the surface while the moving
mass is crossing through Understanding the dynamical
behav-ior of the surface in which the mass is moving through was an
important issue for scientist which led to a new branch of
stud-ies Sßimsßek[36]studied the forced vibration of a simply
sup-ported single-walled carbon nanotube (SWCNT) subjected
to a moving harmonic load by using nonlocal Euler–Bernoulli
beam theory The time-domain response was obtained by
using both the modal analysis method and the direct
integra-tion method Kiani[37]examined the dynamic response of a
SWCNT subjected to a moving nanoparticle in the framework
of the nonlocal continuum theory of Eringen He also studied
[38] the vibration of elastic thin nanoplates traversed by a
moving nanoparticle involving Coulomb friction using the
nonlocal continuum theory of Eringen Yas and Heshmati
[39]studied the vibrational properties of functionally graded
nanocomposite beams reinforced by randomly oriented
straight single-walled carbon nanotubes (SWCNTs) under
the actions of moving load Sßimsßek[40]presents an analytical
method for the forced vibration of an elastically connected
double-carbon nanotube system (DCNTS) carrying a moving
nanoparticle based on the nonlocal elasticity theory Yas and
Heshmati[41]studied the free and forced vibrations of
non-uniform functionally graded multi-walled carbon nanotubes
(MWCNTs) polystyrene nano-composite beams Different
MWCNTs distribution in the thickness direction was
intro-duced to improve fundamental natural frequency and dynamic
behavior of non-uniform polymer composite beam under
action of moving load Ghorbanpour et al [42]investigated
an analytical method of the small-scale parameter on the
(SWBNNT) under a moving nanoparticle SWBNNT was
embedded in bundle of carbon nanotubes (CNTs) which was
simulated as Pasternak foundation Governing equation was
derived using Euler–Bernoulli beam model, Hamilton’s
principle and nonlocal piezo-elasticity theory Heshmati and Yas[43]studied the dynamic response of functionally graded
nanocomposite beams subjected to multi-moving loads The effect of uniform, linear symmetric and unsymmetric MWCNT distributions through the thickness direction on dynamic behavior was studied Hong et al [44]analyzed the vibration of single-walled carbon nanotube embedded in an elastic medium under excitation of a moving nanoparticle based on the Winkler spring model and the Euler–Bernoulli beam model Ghorbanpour and Roudbari [45] investigated the nonlocal longitudinal and transverse vibrations of coupled boron nitride nanotube (BNNT) system under a moving nanoparticle using piezoelastic theory and surface stress based
on Euler–Bernoulli Lu¨ et al.[46]focused on the investigation
of the transverse vibration of double carbon-nano-tubes (DCNTs) which were coupled through elastic medium Both tubes were conveying moving nano-particles and their ends were simply supported The system equations were discretized
by applying Galerkin expansion method, and numerical solu-tions were obtained Li and Wang[47]investigate the nonlin-ear dynamic response characteristics of GP (graphene/ piezoelectric) laminated films in sensing moving transversal load induced by externally moving adhesive particles or mole-cules, based on the nonlocal elasticity theory and Von Ka´rma´n nonlinear geometric relations
With respect to all the researches done in this manner, pre-dicting the behavior of MNBS systems under a moving load, is
an important issue in designing nanosensors which are under different type of loadings Also, with the presentation of nanorace[48]between nanocars in 2016 and the studies around the optimization of the behavior of nanovehicles of different surfaces, understanding the reaction of MNBS systems under
a moving nanocar/particle and preventing the undesirable behaviors could be key steps in having a more efficient system
By the knowledge of the authors, there is no study based on dynamic analyzing multi-layered elastic nanobeams carrying
a moving load or nanoparticle In this paper as shown in Fig 1, dynamical behavior of multi-layered nanobeams under
a moving particle is presented while the beam layers are assumed to be coupled by winkler elastic medium and the small-scale effect is modeled by nonlocal Euler-Bernoulli beam theory for each layer Different parameter sensitivities are dis-cussed and results for multiple nanobeam systems (MNBS) are presented
2 Problem formulation
multi-layered Euler–Bernoulli beams are given by the following:
ui¼ z @wiðx; tÞ
@x
vi¼ 0
wi¼ wiðx; tÞ
eixx ¼dui
dx¼ z @
2
wi
@x2
ð1Þ
Trang 3where i presents the number of the layer u, v and w are the
dis-placement components, x is the longitudinal coordinate
mea-sured from the left end of the beam, z is the coordinate
measured from the midplane of the beam and exxis the normal
strain
The strain energy U could be written as
Ui¼1
2
Z L
0
Z
A
in which A and L are the cross-sectional area and length of the
beam and rxxis the axial stress while the strain energy due to
the shearing strain is zero
By substituting Eq.(1)into Eq.(2), the strain energy may
be expressed as
Ui¼ 1
2
Z L
0
Z
A
rixxz@2wi
@x2 dAdx¼ 1
2
Z L 0
Mi
@2wi
where Miis the bending moment defined as
Mi¼
Z
A
Also the kinetic energy Tiis given by
Ti¼1
2
Z L
0
Z
A
q @ui
@t
þ @wi
@t
dA
where q is the mass density of the beam By excluding the
rotary inertia effect, Eq.(5)may be expressed as
Ti¼1
2
Z L
0
Z
A
q @wi
@t
Transverse load on each layer with respect to Fig 2 is
defined as follows:
Fi¼
mpg ðx xpÞ Kn1ðwn wn1Þ; i ¼ n
Kiðwiþ1 wiÞ Ki1ðwi wi1Þ; 1 6 i 6 n 1
K1ðw2 w1Þ; i ¼ 1
8
>
in which Kiis the stiffness of Winkler elastic medium between ith and (i + 1)th layer, and mpand xpare the mass and posi-tion of nanoparticle moving on the upper layer of MNBS as shown inFig 1 By this definition external energy could be written as follows:
Qi¼1 2
Z L 0
Governing equation of motion is achieved by Hamilton’s principle as
Z t 0
d Tð i Ui QiÞdt ¼ 0
¼
Z t 0
Z L 0
qA @wi
@t
d @wi
@t
þ Md 2
dwi
dx2 þ Fdwi
With integrating by parts and since dW is arbitrary in
0 < x < L, the governing equation of motion is obtained as
@2
Mi
@x2 Fi¼ qA @
2
wi
To add the small-scale effects in nanoscale beams, Eringen’s nonlocal elasticity[49]is employed to model the nonlocal tinuum based of MNBS The classical elasticity does not con-flict the atomic theory of lattice dynamics and experimental observation of phonon dispersion by defining the stress at a reference point x in an elastic continuum depends only on the strain at that point while The basic assumption in the non-local elasticity theory is that the stress at a point is observed to
be a function of not only the strain at that point but also on strains at all other points of a body Eringen’s nonlocal elastic-ity involves spatial integrals which represent weighted averages
of the contributions of strain tensors of all points in the body
to the stress tensor at the given point Basic equations for a lin-ear homogenous nonlocal elastic body are given as
rnl
ij ¼ Z V
sðjx x0j; aÞrl
ijðx0ÞdVðx0Þ; 8x 2 V
rlijðxÞ ¼ Cijklekl
eij¼1
2ðui;jþ uj;iÞ
ð11Þ Figure 1 Schematic representation of multi-layered nanobeam with a moving nanoparticle on top layer
Figure 2 Free diagram of the ith layer of MNBS subjected to
external forces Fiand Fi1
Trang 4where rl
ijand el
ijare the local stress and strain tensors, Cijklis
the fourth-order elasticity tensor, |x x0| is the distance in
Euclidean form and a(|x x|, s) is the nonlocal modulus or
attenuation function incorporating into constitutive equations
the nonlocal effects at the reference point x produced by local
strain at the source x0 a is the material constant which is
defined as (e0a/l) depends on the internal (e.g lattice
parame-ter, granular distance, distance between C–C bonds) and
exter-nal (e.g crack length, wavelength) lengths Due to the
difficulty of solving the integral constitutive Eq.(13) can be
simplified to equation of differential form to fully gain the
inte-gral form results for simply supported boundary conditions
[50,51]and it is written as
For a one dimensional elastic material, Eq.(12)can be
sim-plified as
1 ðe0aÞ2 @2
@x2
where (e0a) is the scale coefficient which leads to small-scale
effect and E is the Young’s modulus of the nanobeam
Multi-plying Eq.(15)by zdA and integrating the result over the area
Alead to
M ðe0aÞ2d2M
dx2 ¼ EId
2 w
And by substituting Eq.(12)into Eq.(14), we have
Mnli ¼ EI @2wi
@x2 þ ðe0aÞ2
qA@2wi
@t2 þ Fi
ð15Þ Thus, the governing equation of motions for multi-layered
nanobeam with moving nanoparticle can be expressed in terms
of transverse displacement for nonlocal constitutive relations
as
qA @ 2 wn
@t 2 þ mgdðx x m Þ K n1 ðw n w n1 Þ
þ EI@
4
w n
@x 4 ¼ a 2 @ 2
@x 2 qA @ 2
w n
@t 2 þ mgdðx x m Þ K n1 ðw n w n1 Þ
qA @ 2 wi
@t 2 þ K i ðw iþ1 w i Þ K i1 ðw i w i1 Þ
þ EI@4wi
@x 4 ¼ a 2 @ 2
@x 2 qA @ 2 w i
@t 2 þ K i ðw iþ1 w i Þ K i1 ðw i w i1 Þ
2 6 i 6 n 1
qA @ 2 w 1
@t 2 þ K 1 ðw 2 w 1 Þ
þ EI@
4
w 1
@x 4 ¼ a 2 @ 2
@x 2 qA @ 2
w 1
@t 2 þ K 1 ðw 2 w 1 Þ
ð16Þ
3 Solution procedure
In modal form, the total transverse dynamic deflection wi(x, t)
is written as
wiðx; tÞ ¼X
1
j¼1
where Wi(t) are the unknown time-dependent generalized
coordinates and gi(x) are the eigenmodes of an undamped
simply-supported beam which are expressed as
giðxÞ ¼ sin ipx
L
ð18Þ Moreover, for the external force made by the moving par-ticle could be represented as
where d(x xp) is the Dirac delta function which could be expressed in terms of sinusoidal as follows
dðx xpÞ ¼ 2
L
X1 i¼1
sin ip
Lxp
sin ip
Lx
ð20Þ
where xpdenotes the coordinate of moving particle from the left end of the upper layer Substituting Eqs (17)–(20) into
Eq.(16)leads to set of Equations as follows:
qAL4@2
Wn
@t2 þ2mgL3sinðipXpÞKn1L4ðWnWn1ÞþðipÞ4
EIWn
¼a2 qAL4ðipÞ2@2Wn
@t2 2ðipÞ2
mgL3sinðipXpÞ
þKn1L4ðipÞ2
ðWnWn1Þi
qAL4@2
wi
@t2 þKiL4ðwiþ1wiÞ
Ki1L4ðwiwi1ÞþðipÞ4
EI@4
wi
@x4
¼ðipÞ2a2
L2 qAL4@2
wi
@t2 þKiL4ðwiþ1wiÞ
Ki1L4ðwiwi1Þ
qAL2@2W1
@t2 þK1L2ðW2W1Þ þðipÞ4
EIW1¼ðipÞ2a2
L2 qAL2@2
W1
@t2 þK1L2ðW2W1Þ
ð21Þ
By defining new set of parameters as follows
X¼x
L; W ¼w
L; a ¼e0a
L ; f ¼L
ffiffiffiffi A
p ffiffi I
p ; c2¼qAL2
P2¼mgL2
EI ; j ¼KL2
The set of equations of motions could be represented in a matrix form as
c 2 @ 2
@t 2
W 1
W 2
W 3
: : :
W n1
W n
8
>
>
>
<
>
>
>
>
9
>
>
>
=
>
>
>
>
þ
j 1 j 1 þ v 2 þ j 2 j 2 0 0 0
0 j 2 j 2 þ v 3 þ j 3 j 3 0 0
0 0 0 j n2 j n2 þ v n1 þ j n1 j n1
2 6 6 6 6 6
3 7 7 7 7 7
w 1
w 2
w 3
: : :
w n1
w n
8
>
>
>
>
>
>
:
9
>
>
>
>
>
>
;
¼
0 0 0 : : : 0
C i
8
>
>
>
>
>
>
:
9
>
>
>
>
>
>
;
ð23Þ where viand C are defined as
vi¼ ðipÞ
4
1þ ðipÞ2 a 2
L 2
Ci¼ 2mgL3
EI sinðipXpÞ
ð24Þ
Trang 5With assuming that the moving particle starts moving from
the left end of the first beam at t = 0 by having a constant
velocity through the path, the dimensionless location of the
moving nanoparticle would be
Xp¼V
where V is the velocity of the moving nanoparticle At t = L/V
moving mass reaches the end of the beam By having the same
material of elastic modulus, mass density, uniform cross
sec-tion A and same continuous linear Winkler elastic medium
of stiffness per length K the set of equations of motions could
be represented as
c2@2
@t2
w1
w2
w3
:
:
:
wn1
wn
8
>
>
>
>
>
>
>
>
>
>
9
>
>
>
>
>
>
>
>
>
>
þ
2
6
6
6
6
6
3 7 7 7 7 7
w1
w2
w3
:
:
:
wn1
wn
8
>
>
>
>
>
>
>
>
>
>
9
>
>
>
>
>
>
>
>
>
>
¼
0
0
0
0
0
0
0
C
8
>
>
>
>
>
>
>
>
>
>
9
>
>
>
>
>
>
>
>
>
>
ð26Þ For solving Eq (26) in time domain, Laplace transform
method is employed so the following set of equations is
obtained:
c 2
S 2 þ v þ 2j j
2
6
6
6
6
6
3 7 7 7 7 7
LfW 1 g
LfW 2 g
LfW 3 g
:
:
:
LfW n1 g
LfW n g
8
>
>
>
<
>
>
>
>
9
>
>
>
=
>
>
>
>
¼
0 0 0 0 0 0 0
2P 2 ipV p ðipV p Þ 2 þS 2
8
>
>
>
>
>
>
>
>
9
>
>
>
>
>
>
>
>
ð27Þ
By inversing the coefficient matrix and evaluating the
inverse Laplace transform of Eq.(27), the results of transverse
displacement with respect to time for each layer could be
achieved Further solution depends on the number of layers
used for MNBS where the process is presented for double
lay-ered and three laylay-ered MNBS with moving nanoparticle and
for MNBS with more layers the procedure is the same
3.1 Double layered MNBS with moving nanoparticle
For two layered MNBS, Eq.(27)could be rewritten as
c2S2þ v þ j j
j c2S2þ v þ j
LfW1g LfW2g
¼ 2P2 0ipV p
ðipV p Þ 2 þS 2
ð28Þ
By inversing the coefficient matrix Eq.(28)may be written as
LfW1g LfW2g
c4S4þ 2jc2S2þ 2c2vS2þ 2jv þ v2
j c2S2þ v þ j
2 4
3
2P2 ipV p
ðipV p Þ 2 þS 2
8
<
:
9
=
; ð29Þ
By evaluating the inverse Laplace transforms and doing some calculations, the values of W1(t) and W2(t) are obtained as
W1ðtÞ ¼ P 2
(
2j sinðnpVtÞ
ðV 2 c 2 n 2 p 2 vÞðV 2 c 2 n 2 p 2 þ 2j þ vÞ
þ
c sinh
ffiffiffiffiffiffiffiffiffiffi
2jv p
c t
npV ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2j v p
ðV 2 c 2 n 2 p 2 2j vÞþ
c sinh p ffiffiffiffiv
c t npV ffiffiffiffiffiffiffi
v
c 2 n 2 p 2 2j vÞ
) ð30Þ
W 1 ðtÞ ¼ P 2
( c sinh pffiffiffiffiffiffiffiffiffiffi2jv
npV ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2j v p ðV 2 c 2 n 2 p 2 þ 2j þ vÞ
þ c sinh
ffiffiffiffi
v p
c t npV ffiffiffiffiffiffiffi
v
p ðV2
c 2 n 2 p 2 þ 2j þ vÞþ
2 sinðnpVtÞðV 2
c 2
n 2
p 2 þ j þ vÞ
ðV 2
c 2 n 2 p 2 vÞðV 2
c 2 n 2 p 2 þ 2j þ vÞ
9
;ð31Þ
Substituting Eqs.(30) and (31)into Eq.(17)the result for transverse displacement of double layered nanobeams with moving nanoparticle with respect to time is achieved as
w 1 ðX;tÞ ¼X
1 i¼1
P 2
(
2j sinðnpVtÞ
ðV 2 c 2 n 2 p 2 vÞðV 2 c 2 n 2 p 2 þ 2j þ vÞ
þ
c sinh
ffiffiffiffiffiffiffiffiffiffi
2jv
p
npV ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2j v p
ðV 2 c 2 n 2 p 2 2j vÞþ
c sinh p ffiffiffiffiv
c t npV ffiffiffiffiffiffiffi
v
p ðV2 c 2 n 2 p 2 2j vÞ
) sinðipXÞ ð32Þ
w2ðX;tÞ ¼X
1 i¼1
P2
( c sinh pffiffiffiffiffiffiffiffiffiffi2jv
c t
npV ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2j v p
ðV2c2n2p2þ 2j þ vÞ
ffiffiffiffi
v p
c t
npV ffiffiffiffiffiffiffi
v
p ðV2c2n2p2þ 2j þ vÞ
þ 2 sinðnpVtÞðV2c2n2p2þ j þ vÞ
ðV2c2n2p2 vÞðV2c2n2p2þ 2j þ vÞ
sinðipXÞ ð33Þ
3.2 Three layered MNBS with moving nanoparticle
Following the same procedure, for three layered MNBS, Eq (27)is expressed as
Trang 6c2S2þ v1þ j j 0
2
6
3
7 LfWLfW1g
2g LfW3g
8
>
>
9
>
>
¼
0
0
2P2 ipV p
ðipV p Þ 2 þS 2
8
>
>
9
>
Inversing the coefficient matrix in Eq.(34)leads to
where D1is
D1¼ c4S6þ jc4S4þ vc4S4þ 3jc2S4þ 2vc2S4þ 2j2c2S2
þ 5vjc2S2þ 2v2c2S2þ j2S2þ 3vjS2þ v2S2
By evaluating the inverse Laplace transforms, the values of
W1(t), W2(t) and W3(t) are obtained as
W1ðtÞ ¼P2j2
D2i 2 sinðipVtÞ ipV X
n i
en i tg1i=g2i
!
ð37Þ
W 2 ðtÞ ¼P2j
D2i 2ðV 2 c 2 i 2 p 2 þ j þ vÞsinðipVtÞ ipV X
n i
e n i t g3i=g 4i
!
ð38Þ
W 3 ðtÞ¼P2
D 2i
2ðV 4 c 4 i 4 p 4 3jV 2 c 2 i 2 p 2 2vV 2 c 2 i 2 p 2 þj 2
n i
e n i t
n i
ðg 5i þg 6i þg 7i Þ=g 8i
ð39Þ
where g1ito g8iare defined inAppendix Aand niand D2iare
defined as
ni¼ Roots of equation ðc4n6þ ðjc4þ vc4þ 3jc2
þ 2vc2Þn4þ ð2j2c2þ 5jvc2þ 2v2c2þ j2þ 3jv
D2i¼ V6
c4i6p6 jV4
c4i4p4 vV4
c4i4p4 3jV4
c2i4p4
2vV4
c2i4p4þ 2j2
V2c2i2p2þ 5jvV2
c2i2p2
þ 2v2
V2c2i2p2þ j2
V2i2p2þ 3jvV2
i2p2 3j2
v
In the same way by Substituting Eqs.(36) and (37)into Eq (17) the result for transverse displacement of three layered nanobeams with moving nanoparticle with respect to time is calculated as
w1ðX;tÞ ¼X1 i¼1
P 2 j 2
n i
e n i t g1i=g2i
!
w2ðX;tÞ ¼X1 i¼1
P 2 j D2i 2ðV 2 c 2 i 2 p 2 þ j þ vÞsinðipVtÞ ipV X
n i
e n i t g3i=g4i
!
sinðipXÞ ð43Þ
3.3 Higher layered MNBS with moving nanoparticle
For more than three layered MNBS with moving nanoparticle, the same calculation procedure is done which causes long complex equations due to the inverse of matrix coefficient For having a more accurate results and forbidding the errors, numerical solution is used to obtain the deflection of each layer in higher number of layers by solving Eq.(45)numerically
LfW 1 g
LfW 2 g
LfW 3 g
8
>
>
9
>
>¼
1
D 1
2
6
3 7 0 0
2P 2 ipV p
ðipV p Þ 2 þS 2
8
>
>
9
>
> ð35Þ
w3ðX; tÞ ¼X
1
i¼1
P2
D2i
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1 C C C C C C C A
Trang 74 Results and discussions
For different number of layers, stiffness and nonlocal
param-eter the dynamical behavior of MNBS under a moving
nanoparticle is illustrated Wang and Wang [52] has shown
that the value of e0ashould be smaller than 2.0 nm for carbon
nanotubes and also the exact value of nonlocal parameter is
not exactly known The external characteristic length varies
so the nonlocal or scale coefficient parameter is assumed to
be a¼e 0 a
l ¼ 0 to 1 The geometrical and mechanical properties
of the multi-layered nanoribbons are considered as [47]:
E= 1.0 TPa, q = 2.25 g/cm3, t = 0.34 nm, L = 10t In order
to achieve a nondimensional dynamical deflection parameter, static deflection[46]is assumed as wst¼m p gL 3
48EI The presented analysis, describes the dynamical behavior
of simply supported Euler–Bernoulli Multi-layered nanobeam
0
0.5
1
0 0.5 1
0.5
1
1.5
T X
W1
0 0.2 0.4 0.6 0.8 1 1.2
0
0.5
1
0 0.5 1 0 0.2 0.4 0.6 0.8
T X
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
Figure 4 Dynamic respond of double layered nanobeam with nondimensional nonlocal parameter = 0.3: (a) second layer nondimensional transverse displacement and (b) first layer nondimensional transverse displacement
Table 1 Nondimensional maximum deflection of double layered carbon nanotubes
0
0.5
1
0 0.5 1
0
0.2
0.4
0.6
0.8
T X
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
0
0.5
1
0 0.5 1 0 0.1 0.2 0.3 0.4
T X
0.05 0.1 0.15 0.2 0.25 0.3 0.35
Figure 3 Dynamic respond of double layered nanobeam with nondimensional nonlocal parameter = 0.1: (a) second layer nondimensional transverse displacement and (b) first layer nondimensional transverse displacement
Trang 8carrying a moving nanoparticle The Eigen function
tech-nique and the Laplace transform method are employed to
solve the governing equations of the nanobeams In order
to verify the validation of present solution procedure the
number of layers is assumed to be two (double layered)
and the analysis is done for carbon nanotubes to compare
the present solution with the forced vibration of an elastically
connected double-carbon nanotube system under a moving nanoparticle presented by Sßimsßek [40] In Table 1the maxi-mum non-dimensional deflection of the first and second layer
of double carbon nanotube for various values of stiffness and nonlocal parameter is presented and compared to those achieved by Sßimsßek [40] which shows a great equality in the results
0
0.5
1
0 0.5 1 0 0.5 1 1.5 2
T X
W 2
0 0.5 1 1.5
0
0.5
1
0 0.5 1
0
0.5
1
1.5
2
2.5
T X
W 1
0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2
Figure 5 Dynamic respond of double layered nanobeam with nondimensional nonlocal parameter = 0.5: (a) second layer nondimensional transverse displacement and (b) first layer nondimensional transverse displacement
0 0.5 1 0
0.5 1 0.1 0.2 0.3 0.4
T X
W 2
0 0.05 0.1 0.15 0.2 0.25
0 0.5 1 0
0.5 1 0.2 0.4 0.6 0.8
T X
W1
0 0.1 0.2 0.3 0.4 0.5 0.6
0 0.5 1 0
0.5
1
0.05
0.1
0.15
0.2
T X
W 3
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14
Figure 6 Dynamic respond of three layered nanobeam with nondimensional nonlocal parameter = 0.1: (a) first layer nondimensional transverse displacement, (b) second layer nondimensional transverse displacement, and (c) third layer nondimensional transverse displacement
0 0.5 1 0
0.5
1
0.5
1
1.5
T X
W1
0 0.2 0.4 0.6 0.8 1
0 0.5 1 0
0.5 1 0.2 0.4 0.6 0.8
T X
W 2
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
0 0.5 1 0
0.5 1 0.1 0.2 0.3 0.4
T X
W3
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35
Figure 7 Dynamic respond of three layered nanobeam with nondimensional nonlocal parameter = 0.3: (a) third layer nondimensional transverse displacement, (b) second layer nondimensional transverse displacement, and (c) first layer nondimensional transverse displacement
Trang 9Dynamic response of double layered MNBS with
nondi-mensional nonlocal parameter = 0.1, 0.3, 0.5 is shown in
Figs 3–5by having a nanoparticle moving on the upper layer
Results are shown with respect to the nondimensional time
parameter which in time T = 0 nanoparticle enters the system
and at T = 1 leaves it Dynamical deflection is presented for each layer separately The same analysis has been done for three layered MNBS and the results of each layer for different nonlocal parameter are presented inFigs 6–8 It can be seen that for both cases, increasing the nonlocal term leaded to a
0 0.5 1 0
0.5
1
0.5
1
1.5
2
2.5
T X
W1
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
0 0.5 1 0
0.5 1 0.5 1 1.5
T X
W2
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0 0.5 1 0
0.5 1 0.2 0.4 0.6 0.8 1
T X
W3
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
Figure 8 Dynamic respond of three layered nanobeam with nondimensional nonlocal parameter = 0.5: (a) third layer nondimensional transverse displacement, (b) second layer nondimensional transverse displacement, and (c) first layer nondimensional transverse displacement
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
T
W 1m
k = 1
k = 5
k = 10
k = 50
k = 100
k = 500
k = 1000
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0
0.5 1 1.5 2 2.5
T
W 2m
k = 1
k = 10
k = 50
k = 100
k = 500
k = 1000
Figure 9 Nondimensional maximum deflection of double layered MNBS with respect to nondimensional time parameter from the time which nanoparticle enters the surface of the first layer till it leaves for different stiffness parameters: (a) second layer and (b) first layer
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
T
k = 1
k = 10
k = 100
k = 1000
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
T
k = 1
k = 10
k = 100
k = 1000
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
T
W3m
k = 1
k = 10
k = 100
k = 1000
Figure 10 Nondimensional maximum deflection of three layered MNBS with respect to nondimensional time parameter from the time which nanoparticle enters the surface of the first layer till it leaves for different stiffness parameters: (a) third layer, (b) second layer, and (c) first layer
Trang 10higher deformation in MNBS system In order to show the
effects of the Winkler elastic medium between each layer on
the dynamical behavior and the maximum deflection of each
layer, stiffness parameter varies from 1 to 1000 As shown in
Figs 9and10, by having j = 1 each layer almost acts
inde-pendently from others By increasing the stiffness parameter
to higher orders, deflection is more shared between layers also
by having more layers and deflection will decrease in the layer carrying the nanoparticle which is caused by the incorporation
of other layers in the MNBS system
Also in Fig 11 by changing the nonlocal parameter, maximum deflection parameter is presented through the beam
0
0.5
1
1.5
2
2.5
X
Wmax
Double layered
First layer, α = 0.1 First layer, α = 0.3 First layer, α = 0.5 Second layer, α = 0.1 Second layer, α = 0.3 Second layer, α = 0.5
0 0.5 1 1.5 2 2.5
X
Wmax
Three layered
First layer, α = 0.1 First layer, α = 0.3 First layer, α = 0.5 Second layer, α = 0.1 Second layer, α = 0.3 Second layer, α = 0.5 Third layer α = 0.1 Third layer, α = 0.3 Third layer, α = 0.5
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
X
Wmax
Four layered
First layer, α = 0.1 First layer, α = 0.3 First layer, α = 0.5 Second layer, α = 0.1 Second layer, α = 0.3 Second layer, α = 0.5 Third layer α = 0.1 Third layer, α = 0.3 Third layer, α = 0.5 Fourth layer α = 0.1 Fourth layer, α = 0.3 Fourth layer, α = 0.5
Figure 11 Nondimensional maximum deflection of MNBS for different amounts of nondimensional nonlocal parameter: (a) double layered, (b) three layered, and (c) four layered
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0
1 2 3 4 5 6 7 8
α
W max
k = 50
k = 100
k = 500
k = 1000
k = 5000
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0
1 2 3 4 5 6 7
α
W max
k = 50
k = 100
k = 500
k = 1000
k = 5000
Figure 12 Nondimensional maximum deflection of double layered MNBS for different stiffness parameters with respect to nonlocal parameter: (a) first layer and (b) second layer
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0
1 2 3 4 5 6
α
k = 50
k = 100
k = 1000
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0
0.5 1 1.5 2 2.5 3 3.5 4 4.5
α
k = 50
k = 100
k = 1000
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
α
k = 50
k = 100
k = 1000
Figure 13 Nondimensional maximum deflection of three layered MNBS for different stiffness parameters with respect to nonlocal parameter: (a) first layer, (b) second layer, and (c) third layer
... MNBS under a movingnanoparticle is illustrated Wang and Wang [52] has shown
that the value of e0ashould be smaller than 2.0 nm for carbon
nanotubes and also the... displacement, and (c) first layer nondimensional transverse displacement
Trang 9Dynamic response of. .. nondimensional time
parameter which in time T = nanoparticle enters the system
and at T = leaves it Dynamical deflection is presented for each layer separately The same analysis has been