Application of Elastically Supported Single-walled Carbon Nanotubes for Sensing Arbitrarily Attached Nano-objects

Application of elastically supported single-walled carbon nanotubes for sensingarbitrarily attached nano-objects

Keivan Kiani a,*, Hamed Ghaffari a, Bahman Mehri b,c

aDepartment of Civil Engineering, Islamic Azad University, Chalous Branch, P.O. Box 46615-397, Chalous, Mazandaran, IranbDepartment of Mathematical sciences, Ghiaseddin Jamshid Kashani University, Abiek, Qazvin, IrancDepartment of Mathematical sciences, Sharif University of Technology, Tehran, Iran

a r t i c l e i n f o

Article history:Received 18 May 2012Received in revised form19 June 2012Accepted 26 June 2012Available online 7 July 2012

Keywords:Single-walled carbon nanotube (SWCNT)VibrationNanomechanical sensorNonlocal continuum-based beam modelsMeshless method

a b s t r a c t

The potential application of SWCNTs as mass nanosensors is examined for a wide range of boundaryconditions. The SWCNT is modeled via nonlocal Rayleigh, Timoshenko, and higher-order beam theories.The added nano-objects are considered as rigid solids, which are attached to the SWCNT. The massweight and rotary inertial effects of such nanoparticles are appropriately incorporated into the nonlocalequations of motion of each model. The discrete governing equation pertinent to each model is obtainedusing an effective meshless technique. The key factor in design of a mass nanosensor is to determine theamount of frequency shift due to the added nanoparticles. Through an inclusive parametric study, theroles of slenderness ratio of the SWCNT, small-scale parameter, mass weight, number of the attachednanoparticles, and the boundary conditions of the SWCNT on the frequency shift ratio of the rst exuralvibration mode of the SWCNT as a mass sensor are also discussed.

2012 Elsevier B.V. All rights reserved.

1. Introduction

The remarkable properties of carbon nanotubes (CNTs) havebeen brought a wide range of applications for them includingnanouidic conveying [1e4], drug delivery [5e7], hydrogen storage[8e11], and nanosensors [12e15]. To date, vibration characteristicsof single-walled carbon nanotubes (SWCNTs) have been fairly wellexamined via classical and nonlocal continuum models. The freevibration and the induction of vibration in SWCNTs due to theexternally applied forces are also investigated bymany researchers.Free vibration of SWCNTs [16e18], their vibrations due to movinginside uids [19e21] and nanoparticles [22e27], dynamical loads[28,29], and magnetic elds [30e32] have been theoreticallystudied.

The high ratio of elasticity modulus of CNTs to their massdensity is suggesting highly effective nanostructures as resonatorsof giga- or even tera-hertz bending frequencies [33e36]. Sucha brilliant characteristic of CNTs and SWCNTs as well, is the majorreason for that the vibration of such nanostructures has been infocus of attention of various scientic disciplines. It implies thatSWCNTs would be also excellent nanomechanical sensors since

many attached nano-sized particles can change their resonantfrequencies. Therefore, the applicability of SWCNTs as physicalnanosensors has been investigated theoretically [37e45] andexperimentally [46e48] as well. Some inclusive review studiesregarding chemical, physical, and biological sensors based on CNTscould be found in the literature [49e53]. Recently, the potentialapplication of graphene sheets in detecting gas atoms has beennumerically investigated [54].

A brief review of the literature reveals that the conducted theo-retical works on nanotubes as nanosensors were limited to somespecial cases. For example, a slender SWCNT based nanomechanicalsensor with clamped-free boundary conditions and an added massat its tip [41,46], or a slender SWCNT with clamped-free/clampedeclamped conditions and a mass at its tip/midspan point[39,40,42], or a slender boron-nitride nanotube for sensing nano-bioobjects [55], or a stocky clamped-free SWCNT as a nanosensor withamass at its tip [56,57]. Further, most of the undertakenworks werebased on the classical continuum mechanics [41,42,46] which maynot accurately interpret the vibrationmechanisms of such nanoscalestructures for sensing nanoparticles. Such lack of knowledgeregarding stocky SWCNTs-based nanomechanical sensors encour-aged the author to investigate the problem in a more generalframework: nonlocal elastically supported stocky SWCNTs for sensingarbitrarily attached nanoparticles (for more awareness regarding theapplication of nonlocal continuum theories in modeling CNTs and

* Corresponding author. Tel.: 98 191 2223796; fax: 98 191 2220536.E-mail addresses: [email protected], [email protected] (K. Kiani).

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Current Applied Physics

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1567-1739/$ e see front matter 2012 Elsevier B.V. All rights reserved.http://dx.doi.org/10.1016/j.cap.2012.06.023

Current Applied Physics 13 (2013) 107e120

graphenes, the readers are refereed to Ref. [58]). In the presentwork,an efcient meshless technique, namely reproducing kernel particlemethod (RKPM), is employed. This numerical scheme was devel-oped by Liu and his coworkers [59e61] at Northwestern universityin the last decade of twentieth century. In this numerical technique,the spatial domain is discretized using particles. Each particle has itsown numerically constructed shape functions. The shape functionsare evaluated according to the chosen window and base functionsfor each problem. In contrast to the nite-element method (FEM),the magnitudes of the RKPM shape function of a particle generallydo not vanish at its surrounding particles; thereby, special treat-ments should be implemented for satisfaction of essential boundaryconditions. Furthermore, the shape functions of RKPM possesshigher continuity in compare to those of FEM. This matter would bebenecial in solving the problems suffer from moving boundaries,sharply varied elds, and higher-order governing equations. To date,the application of RKPM to many one- and two-dimensional math-ematical-mechanical models has been led to fairly accurate results[18,62e65].

Herein, the potential application of SWCNTs as nanosensors isnumerically explored in the context of the nonlocal continuumtheory of Eringen. Using nonlocal beam theories and an efcientnumerical scheme, the equations of motion of a SWCNT withattached nanoparticles are constructed in both strong and weakforms. The alteration to the rst exural resonant frequency due tothe added nanoparticles is studied by the proposed nonlocal beammodels. The effects of the important parameters on the practica-bility of SWCNTs as nanomechanical sensors as well as the capa-bilities of the proposed models in capturing their resonantfrequencies are also of interest. It is hoped that the obtained resultsin this article could provide new insights to the researchers who areconducting theoretical and experimental works on SWCNTs asnanomechanical sensors.

2. Assumptions and denition of the problem

Consider an ECS pertinent to the SWCNT of length lb as illus-trated in Fig. 1. It is assumed that the SWCNT contains Np nano-particles which are perfectly attached to the nanotube structure.It implies that no movement or separation of nanoparticles wouldoccur during free vibration. The nanoparticles are considered asrigid solid objects. It is also assumed that the distance betweeneach two nanoparticles is large enough that no atomic bondbetween them exists. The position, mass weight, and massmoment of inertia of the ith nanoparticle are represented by xpi,Mpi, and Ipi, respectively. The SWCNT is embedded in an elasticmedium and experiences an initially axial force of magnitude Nb.

Its interaction with the surrounding medium is modeled throughthe lateral and rotational continuous springs whose constants aredenoted by Kt and Kr, respectively. In order to investigate theproblem for a more general form of boundary conditions, bothends of the ECS are attached to the laterally and rotationallypointed springs. The constants of such lateral and rotationalsprings are represented by Kz(xk) and Ky(xk);k 1,2, respectively,where x1 0 and x2 lb. The change of resonant frequencies ofthe SWCNT due to addition of nanoparticles as well as the role ofinuential parameter on such a fact is the main goal of thisresearch work. For this purpose, the ECS is modeled according tothe nonlocal Rayleigh beam theory (NRBT), nonlocal Timoshenkobeam theory (NTBT), and nonlocal higher-order beam theory(NHOBT). Both the lateral and rotational inertial effects of theattached nanoparticles are incorporated into the governingequations of each model.

In the following parts, the explicit forms of governing equationsof SWCNTs as nanomechanical sensors are derived based on thetheories of nonlocal Rayleigh beam (NRB), nonlocal Timoshenkobeam (NTB), and nonlocal higher-order beam (NHOB). Since ndingan appropriately analytical solution to such equations are not aneasy task, particularly when studying of the problem for a moregeneral boundary condition is of interest, an efcient numericalscheme is exploited.

3. Modeling SWCNT-based mass sensor via NRBT

3.1. Governing equations

The explicit expressions of the equations of motion of a nano-tube structure with attached masses is of concern in the context ofthe NRBT. The governing equation in terms of nonlocal bendingmoment, Mnlb R, and transverse displacement of the SWCNT, wR, isas

rb

Ab w

R Ib wR;xxXNpi1

Mpi w

R Ipi wR;xxdx xpi

Mnlb R;xxNbw

R;x

;xKtwR KrwR;xx

X2k1

KzxkwR KyxkwR;xx

dx xk 0; 1

where rb is the density, Ab is the cross-sectional area, Ib is thesecond inertia moment of the cross-section of the ECS, d denotesthe Dirac delta function, and the over dot sign represents thedifferentiation with respect to time. Based on the nonlocal

Fig. 1. Schematic illustration of an elastically supported embedded ECS with attached nanoparticles.

K. Kiani et al. / Current Applied Physics 13 (2013) 107e120108

continuum theory of Eringen [66e68], the nonlocal bendingmomentwithin the ECSmodeled based on the NRBT is simply givenby

Mnlb

Re0a2Mnlb R;xx EbIbwR;xx; (2)where a denotes an internal characteristic length, e0 is determinedby comparing the predicted dispersion curves with those ofanother atomic model, and e0a is called small-scale parameter. Bycombining Eq. (1) and Eq. (2),

Mnlb

REbIbwR;xxe0a2"rb

Ab w

RIb wR;xx

XNpi1

Mpi w

RIpi wR;xxdxxpi

NbwR;x;x

KtwRKrwR;xxX2k1

kzxkwRkyxkwR;xx

dxxk

#;

(3)

substitution of Eq. (3) into Eq. (1) leads to the governing equation ofa SWCNT-based mass sensor according to the NRBT as in thefollowing form

3.2. Application of RKPM to the NRB model

The kinetic energy, TR(t), and the elastic strain energy, UR(t), ofan embedded nanotube structure with attached nanoparticlesaccording to the NRBT could be expressed by

TRt 12

Zlb0

rb

hAb_wRx; t2Ib _wR;xx; t2idx

12

XNpi1

Mpi

_wRxpi ; t

2Ipi _wR;xxpi ; t2; (5a)

URt 12

Zlb0

hwR;xxx; t

Mnlb

Rx;tNbwR;xx;t2idx

12

Zlb0

hKtwRx; t

2KrwR;xx;t2idx

12

X2k1

Kzxk

wRxk; t

2KyxkwR;xxk;t2; (5b)

by substituting Eq. (3) into Eqs. (5a) and (5b), employing Hamlitonsprinciple, and taking the required integration by parts, the varia-tional form of the governing equation of the SWCNT-based masssensor based on the NRBT is obtained as follows

The only unknown of the problem is now discretized inthe spatial domain as: wRx; t PNPI1 fwI xwRI t, where NP isthe total number of the RKPM particles, fwI x represents the

RKPM shape function pertinent to the Ith RKPM particle, andwRI t denotes the nodal parameter value of the Ith RKPMparticle pertinent to the NRB. For more convenience in analyzingof the problem, the following dimensionless parameters aredened

rb

Ab w

R Ib wR;xx e0a2rb

Ab w

R;xx

Ib wR;xxxxXNpi1

Mpi w

R Ipi wR;xxdx xpi

e0a2XNpi1

Mpi

wRd

x xpi

;xx

IpiwR;xxdx xpi

;xx

EbIbwR;xxxx

Nbw

R;x

;x e0a2

Nbw

R;x

;xxx

Kt

wR e0a2wR;xx

Kr

wR;xx e0a2wR;xxxx

X2k1

hKzxk

wRdx xk e0a2

wRdx xk

;xx

Kyxk

wR;xxdx xk e0a2

wR;xxdx xk

;xx

i 0:

(4)

Zlb0

(dwRrbAb w

R dwR;xrbIb wR;xXNpi1

dwRMpi w

R dwR;xIpi wR;xdx xpi

dwR;xxEbIbwR;xx dwR;xNbwR;x dwRKtwR dwR;xKrwR;xX2k1

dwRKzxkwR dwR;xKyxkwR;x

dx xk e0a2dwR;xx

"rb

Ab w

R Ib wR;xxXNpi1

Mpi w

R Ipi wR;xxdx xpi

NbwR;x;x

KtwR KrwR;xxX2k1

KzxkwR KyxkwR;xx

dx xk

#)dx 0:

(6)

K. Kiani et al. / Current Applied Physics 13 (2013) 107e120 109

where rb and rpi represent the radius of gyration of the cross-section of the ECS and the ith attached nanoparticle, respectively.By introducing the dimensionless quantities in Eq. (7) to Eq. (6), thedimensionless equations of motion of the SWCNT with attachednanoparticles based on the NRBT are obtained as

MRbw

R;ss K

Rbw

R 0; (8)

where

hM

Rb

iIJZ10

fwI f

wJ l2fwI;xfwJ;x m2fwI;xx

fwJ l2fwJ;xx

dx

XNpi1

MpifwI

xpi

fwJ

xpi

l2pi f

wI;x

xpi

fwJ;x

xpi

m2fwI;xxxpi

fwJ

xpi

l2pi f

wJ;xx

xpi

; (9a)

hKRb

iIJZ10

fwI;xxf

wJ;xxN

Rb

fwI;xf

wJ;xm2fwI;xxfwJ;xx

dx

Z10

KRt

fwI m2fwI;xx

fwJ K

Rr

fwI;xf

wJ;xm2fwI;xxfwJ;xx

dx

X2k1

KRz xk

fwI xkm2fwI;xxxk

fwJ xk

KRyxkfwI;xxkfwJ;xxkm2fwI;xxxkfwJ;xxxk

; (9b)

wRs < wR1s; wR2s;.;wRNPs>T: (9c)In order to determine the natural frequencies of the SWCNT-

based mass sensor, it is assumed that wRs ~wR0ei6Rs where

i 1

pand ~wR0 is a vector represents the nodal parameter values

of the RKPM particles associated with the initial deection of thenanosensor, and 6R denotes the dimensionless exural frequencyof the nanosensor. By substituting this relation into Eq. (8),

h 6R2MRb KRbi ~wR0 0; (10)

by solving the set of eigenvalue equations in Eq. (10), the dimen-sionless natural frequencies of the SWCNT-based mass sensor areobtained based on the hypotheses of the NRBT.

4. Modeling SWCNT-based mass sensor via NTBT


Herein, realizing the characteristics of free transverse vibrationof a SWCNT with attached nanoparticles is of concern in theframework of the NTBT. For this purpose, the governing equation ofthe ECS with attached nanoparticles as a function of nonlocalinternal forces and displacements are expressed by

rbIbqT PNp

i1Ipi

qTdx xpi

Qnlb TMnlb T;x KrqTX2k1

KyxkqTdx xk 0; (11a)

rbAb wT PNp

i1Mpi w

Tdx xpi

Qnlb T;x Nbw

T;x

;x KtwT

X2k1

KzxkwTdx xk 0; (11b)

where wT, qT, Qnlb T, and Mnlb T are the deection, the deformationangle, resultant shear force, and the resultant bending momentwithin the ECS based on the NTBT, respectively. According to thenonlocal continuum theory of Eringen [66,67], the nonlocal resul-tant shear force and bending moment within the ECS are as,

QnlbTe0a2Qnlb T;xx ksGbAb

wT;x qT

; (12a)

Mnlb

Te0a2Mnlb T;xx EbIbqT;x; (12b)in Eq. (12a), Gb denotes the shear modulus of elasticity in whichexpressed by Gb Eb/(2(1n)) where n is the Poissons ratio of theECS associated with the SWCNT. The ks is the shear correction factorof the cross-section of the ECS. By merging Eqs. (11) and (12), thenonlocal internal forces within the ECS based on the NTBT are ob-tained as

QnlbT ksGbAbwT;xqTe0a2

"rbAb w

TXNpi1

Mpi wTdxxpi

Nbw

T;x

;xKtwT

X2k1

KzxkwTdxxk#;x

; (13a)

Mnlb

T EbIbqT;x e0a2"rbAb w

T rbIbqT

;x

XNpi1

Mpi w

Tdx xpi

IpiqTdx xpi;x

Nbw

T;x

;x KtwT KrqT;x

X2k1

KzxkwTdx xk

KyxkqTdx xk

;x

#; 13b

by substituting Eqs. (13a) and (13b) into Eqs. (11a) and (11b), theexplicit expressions of the governing equations of the SWCNT-based mass sensor according to the NTBT are obtained as

x xlb; xpi

xpilb; wR w

R

lb; s 1

l2b

EbIbrbAb

st; m e0a

lb; l lb

rb; Mpi

MpirbAblb

; lpi lbrpi

;

KRz

Kzl3bEbIb

; KRy xk

KyxklbEbIb

; KRt

Ktl4bEbIb

; KRr

Krl2bEbIb

; NRx

Nbl2b

EbIb;

(7)


rbIbqT e0a2q

T

;xx

XNpi1

IpiqTdx xpi

e0a2qTdx xpi;xx ksGbAbwT;x qT

EbIbqT;xx KrqT e0a2qT;xx

X2k1

KyxkqTdx xk

e0a2qTdx xk

;xx

0; 14a

rbAbwT e0a2 wT;xx

XNpi1

MpiwTd

x xpi

e0a2

wTd

x xpi

;xx

ksGbAb

wT;xx qT;x

NbwT;x

;x e0a2

Nbw

T;x

;xxx

Kt

wT e0a2wT;xx

X2k1

KzxkwTdx xk e0a2

wTdx xk

;xx

0:

(14b)

4.2. Application of RKPM to the NTB model

In the context of the NTBT, the total kinetic energy and theelastic strain energy of the ECS with added nanoparticles, respec-tively, denoted by TT(t) and UT(t), are provided by

TTt 12

Zlb0

rb

hAb_wTx; t2Ib _qTx; t2idx

12

XNpi1

Mpi _w

Txpi Ipi _qTxpi; (15a)

UTt 12

Zlb0

qT;xx; t

Mnlb

Tx; t wT;xx; t qTx; t

QnlbTx; t NbwT;xx; t2KtwTx; t2

KrqTx; t

2X2k1

Kzxk

wTx; tdx xk

2

KyxkqTx; tdx xk

2!dx; (15b)

by substituting Mnlb T and Qnlb T fromEq. (13) into Eq. (15), andusingHamiltons principle, after taking the necessary integration by parts,

Now the unknown elds of the ECS based on the NTBT are dis-cretized in terms of RKPM shape functions as

wTx; t PNPI1fwI xwTI t and qTx; t PNPI1fqI xqTI t. Byintroducing such discretized forms of the unknown elds and thefollowing dimensionless quantities to Eq. (16),

wT wT

lb; q

T qT; s 1lb

ksGbrb

st; h EbIb

ksGbAbl2b;

KTy

KyksGbAblb

; KTz

KzlbksGbAb

; KTr

KrksGbAb

;

KTt

Ktl2bksGbAb

; NTb

NbksGbAb

; (17)

the discrete form of the governing equations of the SWCNT-basedmass sensor according to the NTBT are obtained as follows2664hM

Tb

iww hM

Tb

iwqhM

Tb

iqw hM

Tb

iqq3775(wT;ssQ

T;ss

)

2664hKTb

iww hKTb

iwqhKTb

iqw hKTb

iqq3775wT

QT

00

;

(18)

where

hM

Tb

iwwIJ

Z10

fwI f

wJ m2fwI;xfwJ;x

dx

XNpi1

MpifwI

xpi

fwJ

xpi

m2fwI;xxpi

fwJ;x

xpi

; (19a)

hM

Tb

iqqIJ

Z10

l2fqI f

qJ m2fqI;xfqJ;x

dx

XNpi1

l2pifqI

xpi

fqJ

xpi

m2fqI;x

xpi

fqJ;x

xpi

;

(19b)

hKTb

iwwIJ

Z10

fwI;xf

wJ;xN

Tb

fwI;xf

wJ;xm2fwI;xxfwJ;xx

dx

Z10

KTt

fwI f

wJ m2fwI;xfwJ;x

fwJ dx

X2k1

KTzxk

fwI xkfwJ xkm2fwI;xxkfwJ;xxk

;

(19c)

ZIb0

(rbdq

TqT rbAbdwT wT dwTKtwT dwT;xNbwT;x

X2k1

dqTKyxkqT dwTKzxkwT

dx xk dqTKrqT

dwT;x dqT

"ksGbAb

wT;x qT

e0a2

rbAb w

T;xXNpi1

Mpi wTdx xpi

NbwT;x;xx

KtwT;x X2k1

Kzxkwdx xk;x!#

dqT;x" EbIbqT;x e0a2

rbAb w

T rbIbqT

;xNbw

T;x

;x KtwT KrqT;x

XNpi1

Mpi w

Tdx xpi

IpiqTdx xpi;x

X2k1

KzxkwTdx xk Kyxk

qTdx xk

;x

!#)dx 0: (16)


hKTb

iwqIJ

Z10

fwI;xfqJ dx; (19d)

hKTb

iqwIJ

Z10

fqI fwJ;xdx; (19e)

hKTb

iqqIJZ10

fqI f

qJ hfqI;xfqJ;x

dx

Z10

KTr

fqI f

qJ m2fqI;xfqJ;x

dx

X2k1

KTyxk

fqI xkfqJ xkm2fqI;xxkfqJ;xxk

; (19f)

wTs < wT1s;wT2s;.;wTNPs>T;Q

Ts < qT1s; qT2s;.; q

TNPs>T: (19g)

5. Modeling SWCNT-based mass sensor via NHOBT


The nal aim of this part is to provide the discrete equations ofmotion of a SWCNT as a mass sensor according to the hypotheses ofthe NHOBT. In such a context, the governing equations in terms ofnonlocal internal forces and deformation elds are stated by thefollowing equations

I2 2aI4 a2I6

jH

XNpi1

IpijHdx xpi

a2I6 aI4 wH;xQnlbHaPnlb H;x

Mnlb

H;x KrjH

X2k1

KyxkjHdx xk 0; (20a)

I0 wH

XNpi1

Mpi wHdxxpi

a2I6aI4jH;xa2I6 wH;xxQnlbH;x

aPnlbH;xxNbw

H;x

;xKtwH

X2k1

KzxkwHdxxk 0;

where wH and jH denote the deection and the deection angle ofthe ECS based on the NHOBT, respectively, and a 4=3D2o whereDo is the outer diameter of the ECS. According to the nonlocalcontinuum theory of Eringen, the nonlocal resultant forces in Eqs.(20a) and (20b) are expressed by

Mnlb

He0a2Mnlb H;xx J2jH;x aJ4jH;x wH;xx

; (21a)

Qnlb aPnlb;x

He0a2Qnlb aPnlb;xH;xx k

jH wH;x

aJ4jH;xx a2J6

jH;xx wH;xxx

; (21b)

where

kZAb

Gb13az2

dA; In

ZAb

rbzndA; Jn

ZAb

EbzndA; (22)

by mixing the governing equations in Eqs. (20a) and (20b) with thenonlocal resultant forces in Eqs. (21a) and (21b), the nonlocal

resultant forces in terms of the deformation elds of the ECS basedon the NHOBT are obtained as,

Mnlb

H J2jH;x aJ4jH;x wH;xx e0a2"I2 aI4j

H

;x

XNpi1

IpijHdx xpi;x I0 wH X

Np

i1Mpi w

Hdx xpi

aI4 wH;xx

Nbw

H;x

;x KtwH KrjH;x

X2k1

KzxkwHdx xk Kyxk

jHdx xk

;x

#; (23a)

Qnlb aPnlb;x

H kjH wH;x aJ4jH;xx a2J6jH;xx wH;xxx

e0a2"I0 w

H;xXNpi1

MpiwHd

x xpi

;x

Nbw

H;x

;xx

KtwH;x a2I6 aI4

jH

;xx

a2I6 wH;xxxx X2k1

KzxkwHdx xk

;x

#;

(23b)

by substituting Eqs. (23a) and (23b) into Eqs. (20a) and (20b), theexplicit expressions of the nonlocal equations of motion areobtained

I2 2aI4 a2I6

jH e0a2j

H

;xx

XNpi1

IpijHdx xpi

e0a2jHdx xpi;xx a2I6 aI4wH;x e0a2 wH;xxx

kjH wH;x

J2 2aJ4 a2J6

jH;xx

aJ4 a2J6

wH;xxx

KrjH e0a2jH;xx

X2k1

KyxkjHdx xk

e0a2jHdx xk

;xx

0; (24a)

I0wH e0a2 wH;xx

XNpi1

MpiwHd

x xpi

e0a2wHd

x xpi

;xx

a2I6 aI4

jH

;x e0a2jH;xxx

a2I6

wH;xx

e0a2 wH;xxxx kjH;x wH;xx

aJ4jH;xxx

a2J6jH;xx wH;xxx

Nb

wH;xx e0a2wH;xxxx

KtwH e0a2wH;xx

X2k1

KzxkwHdx xk

e0a2wHdx xk

;xx

0; (24b)

Eqs. (24a) and (24b) show the incorporation of the small-scaleparameter, interaction of the SWCNT with its surroundingmedium, and the initially axial force into the governing equationsof an elastically supported SWCNT as a nanomechanical sensor onthe basis of the NHOBT. In general, seeking an analytical solution tothese equations is not an easy task. Therefore, developing efcientnumerical schemes is of great advantageous for analyzing of theproblem for a wide range of boundary conditions.


5.2. Application of RKPM to the NHOB model

The total kinetic energy, TH, and the total stain energy, UH, of theelastically supported SWCNT-mass sensor based on the NHOBT areexpressed by

THt 12

Zlb0

hI0_wH;xx;t

2I2_jHx;t

22aI4 _jHx;t _jHx;t _wH;xx;t

a2I6

_jHx; t _wH;xx; t

2idx

12

XNpi1

Mpi _w

Hxpi Ipi _jHxpi; (25a)

UHt 12

Zlb0

"jH;xx;tMHb x;t

jHx;twH;xx;t

aPHb;xx;tQHb x;t

Nb

wH;xx;t

2KrjHx;t2

KtwHx;t

2X2k1

Kzxk

wHx;tdxxk

2

KyxkjHx;tdxxk

2#dx; (25b)

by substituting Mnlb H and Qnlb aPnlb;xH from Eqs. (23a) and (23b)into Eqs. (25a) and (25b), and using Hamiltons principle,

Zlb0

(dwHI0 w

H XNpi1

MpidwH wHd

x xpi

dwH;xa2I6 aI4jH

dwH;xa2I6 wH;x djHI2 2aI4 a2I6

jHXNpi1

IpidjHj

H

dx xpi djHa2I6 aI4wH;x kdjH dwH;x

jH wH;x

djH;x

J2 2aJ4 a2J6

jH;x

aJ4 a2J6

wH;xx

dwH;xxaJ4j

H;x a2J6

jH;x wH;xx

dwH

X2k1

KzxkwHdx xk

djHX2k1

KyxkjHdx xk dwHKtwH djHKrjH

dwH;xNbwH;x e0a2djH dwH;x

"I0 w

H;xNbw

H;x

;xx

KtwH;x KrjH;xx a2I6 aI4

jH

;xx a2I6 wH;xxx

X2k1

Kzxk

wHdx xk

;x

# e0a2djH;x

"I2 aI4j

H

;x

I0 wH Nbw

H;x

;x KtwH aI4 wH;xx Krj

H;x

X2k1

KzxkwHdx xk Kyxk

jHdx xk

;x

#)dx 0:

(26)

Let discretize the unknown deformation elds of the SWCNT-based mass sensor according to the NHOBT as:wHx; t PNPI1 fwI xwHI t and jHx; t PNPI1 fjI xjHI t. Bysubstituting such discretized elds into Eq. (26) and introducingthe following dimensionless quantities to the resulting expression,

wH wH

lb; j

H jH; g21 aI4 a2I6

I0l2b; g22

a2I6I0l2b

; g23 kl2ba2J6

;

g24 aJ4 a2J6

a2J6; g26

aI4 a2I6I2 2aI4 a2I6

;

g27 kI0l4b

I2 2aI4 a2I6a2J6

; g28 J2 2aJ4 a2J6

I0l2b

I2 2aI4 a2I6a2J6

;

g29 aJ4 a2J6

I0l2b

I2 2aI4 a2I6a2J6

; KHy xk

KyxkI0l3ba2I2 2aI4 a2I6

J6;

KHz xk

Kzxkl3ba2J6

; KHr

KrI0l4ba2I2 2aI4 a2I6

J6;

KHt

Ktl4ba2J6

; NHb

Nbl2b

a2J6; Ipi

IpiI2 2aI4 a2I6

lb;

(27)

the dimensionless discrete equations of motion of the SWCNT-based mass sensor according to the NHOBT are derived as follows

hMHb iww hMHb iwjhMHb ijw hMHb ijj (wH;ssJH;ss)hKHb iww hKHb iwjhKHb ijw hKHb ijj wHJH00;(28)

where

hM

Hb

iwwIJ

Z10

fwI f

wJ g22fwI;xfwJ;x m2

fwI;xf

wJ;x g22fwI;xxfwJ;xx

dx

XNpi1

MpifwI

xpi

fwJ

xpi

m2fwI;x

xpi

fwJ;x

xpi

;

(29a)

hM

Hb

iwjIJ

Z10

g21

fwI;xf

jJ m2fwI;xxf

jJ;x

dx; (29b)

hM

Hb

ijwIJ

Z10

g26

fjI f

wJ;x m2fjI;xf

wJ;xx

dx; (29c)

hM

Hb

ijjIJ

Z10

fjI f

jJ m2f

jI;xf

jJ;x

dx

XNpi1

IpifjI

xpi

fjJ

xpi

m2fjI;xxpi

fjJ;x

xpi

; (29d)

hKHb

iwwIJ

Z10

g23f

wI;xf

wJ;xfwI;xxfwJ;xxN

Hb

fwI;xf

wJ;xm2fwI;xxfwJ;xx

dx

Z10

KHt

fwI f

wJ m2fwI;xfwJ;x

dx

X2k1

KHz xk

fwI xkfwJ xkm2fwI;xxkfwJ;xxk

;

(29e)

hKHb

iwjIJ

Z10

g23f

wI;xf

jJ g24fwI;xxf

jJ;x

dx; (29f)


hKHb

ijwIJ

Z10

g27f

jI f

wJ;x g29fjI;xf

wJ;xx

dx; (29g)

hKHb

ijjIJ

Z10

g27f

jI f

jJ g28f

jI;xf

jJ;x

dx

Z10

KHr

fjI f

jJ m2f

jI;xf

jJ;x

dx

X2k1

KHy xk

fjI xkf

jJ xkm2f

jI;xxkf

jJ;xxk

;

(29h)

wHs < wH1 s; wH2 s;.; wHNPs>T;J

Hs < jH1 s; jH2 s;.;j

HNPs>T: (29i)

6. Results and discussion

6.1. Validation of the proposed models

6.1.1. Convergence checkConsider a clamped-free SWCNT with a mass at its tip.

According to the work of Chowdhury et al. [40], the dimensionlessrst resonant frequency of the system based on the assumptions ofthe classical Euler-Bernoulli beam theory could be analytically

evaluated as: UA1 420=33 140Mp1 14. Such a resonant

frequency associated with the lateral vibration of the cantileverSWCNT with a tip mass, is considered as a benchmark value for theconvergence check of the proposed numerical models. For thispurpose, three case studies are investigated according to the givendata in Table 1. The slenderness ratios of cases I, II, and III in orderare 231.7, 38.9, and 14.4 in which denoting a slender, fairly slender,and stocky SWCNT, respectively. According to the above-mentionedformulas, the resonant frequencies of the cases I, II, and III arecalculated as 0.9704, 0.6950, and 0.5713, respectively. The pre-dicted dimensionless resonant frequency of the proposed numer-ical models are summarized in Table 2 for different number ofRKPM particles. Hereafter, the ith dimensionless natural frequency

of the SWCNT-based mass sensor is denoted by U:i and determined

by U:i

rbAbl4b

u:i

2EbIb

1=4where u:i is the ith

natural frequency of the nanosensor and [.] R or T or H. As it isseen in Table 2, for all considered cases, the predicted results by theNRBT are converging to the analytically calculated results as thenumber of RKPM particles increases. In the case II, the discrep-ancies between the results of the NTBT/NHOBT and those of theNRBT are decreasing with the number of RKPM particles; however,the predicted results by such nonlocal shear deformable beamtheories do not approach to the analytically predicted values. It ischiey related to the incorporation of the shear strain energy intothe total strain energy of the SWCNT with a tip mass. This matter ismore obvious in the case III.

6.1.2. Comparison of the obtained resultsIn the rst comparison study, the predicted rst four dimen-

sionless frequencies of a clamped-free SWCNT with a mass at thetip by the proposed numerical models as well as those of theanalytical solution by Mehdipour et al. [41] are provided in Table 3.Themechanical and geometry properties of the ECS associatedwiththe SWCNT are as: lb 5500 nm, Do 33 nm, Di 18.8 nm,Eb 32 GPa, and rb 1330 kg/m3. The results of Ref. [41] wereobtained analytically based on the local Euler-Bernoulli beamtheory. Moreover, the predicted results by the proposed numericalmodels are given for NP 21 and e0a 2 nm. As it is obvious fromTable 3, the predicted results by the NRBT are close to those ofRef. [41] for different levels of mass weight of the attached nano-particle at the tip. Further, the discrepancies between the predicteddimensionless frequencies by the NRBT/NTBT and those of Ref. [41]decreasewith vibrationmode number. For all values of the attachedmass at the tip, the predicted results by the NRBT and those of theNTBT are in line with those of Ref. [41].

In another examination, the predicted resonant frequencies bythe proposed models based on the NRBT and NTBT are comparedwith those of experimentally obtained results by Wang et al. [46]and the results of three-dimensional FEM by Joshi et al. [42]. Therst resonant frequency of cantilevered SWCNTs without anyattached nanoparticle of the above-mentioned works are providedfor various SWCNTs in Table 4. The predicted results by the NRBTand the NTBT are calculated using RKPM for NP 31 ande0a 2 nm; however, for such lengthy SWCNTs, the effect of small-scale parameter on exural behavior of the SWCNTs would berationally negligible. As it is seen in Table 4, the predicted resonantfrequencies by the proposed numerical models are generally closerto those of experimentally results of Wang et al. [46] in compare to

Table 1Material and geometry properties of the SWCNT as well as the mass weight of theattached nanoparticle for different case studies.

Case study Eb (TPa) rb (kg/m3) Dyo (nm) D

yyi (nm) lb (nm) Mp1 (fg)

I 1.2 2500 33 18.8 2200 10II 1.1 2300 29 10.5 300 5III 1.2 2400 24 14 100 2

y yy: Do and Di in order are the outer and the inner diameters of the ECS.

Table 2The predicted dimensionless resonant frequency of the SWCNTas a nanomechanicalsensor using the proposed nonlocal beam theories for different number of RKPMparticles.

Case study Nonlocal model NP 11 NP 21 NP 51 NP 101I NRBT 0.9937 0.9817 0.9747 0.9724

NTBT 1.0039 0.9783 0.9712 0.9704NHOBT 0.9269 0.9097 0.9048 0.9044

II NRBT 0.7113 0.7029 0.6980 0.6964NTBT 0.6961 0.6941 0.6938 0.6937NHOBT 0.6819 0.6798 0.6776 0.6757

III NRBT 0.5847 0.5778 0.5738 0.5724NTBT 0.5632 0.5603 0.5630 0.5630NHOBT 0.5437 0.5405 0.5353 0.5320

Table 3Comparison of the predicted rst four dimensionless exural frequencies of a can-tilevered SWCNT with a tip mass via the proposed numerical models with those ofMehdipour et al. [41].

Mp1 (fg) Approach U1 U2 U3 U4

0 Ref [41]. 1.8750 4.6941 7.8548 10.9955NRBT 1.9199 4.8042 8.0367 11.2480NTBT 1.9720 4.9394 8.2712 11.5909

20 Ref [41]. 0.8815 3.9513 7.0785 10.2204NRBT 0.8975 4.0438 7.4594 10.4557NTBT 0.9162 4.1564 7.4594 10.7784

30 Ref [41]. 0.7997 3.9432 7.0785 10.2170NRBT 0.8141 4.0357 7.2429 10.4523NTBT 0.8301 4.1483 7.4546 10.7749

40 Ref [41]. 0.7457 3.9391 7.0760 10.2153NRBT 0.7591 4.0316 7.2404 10.4506NTBT 0.7748 4.1442 7.4519 10.7730

50 Ref [41]. 0.7061 3.9366 7.0745 10.2143NRBT 0.7188 4.0292 7.2389 10.4496NTBT 0.7336 4.1417 7.4504 10.7718


those of Joshi et al. [42]. In all cases, both NRBT and NTBT couldcapture the results of Wang et al. [46] with accuracy lowerthan 3.5%.

In another justication, the capabilities of the proposed modelsin predicting the amount of change of the fundamental frequenciesof bridged and cantilevered SWCNTs are studied. In the case ofa bridged SWCNT (i.e., a SWCNT with CC boundary conditions), theadded masses of distributed length gmlb, are symmetrically placedabout the midspan point of the SWCNT. For the case of cantileveredSWCNT, the attached masses are located at the end of the SWCNTwhose distributed length is identical to the previous case. In orderto examine the trend of the fundamental frequency of the SWCNTin terms of gm, the calibration constant of mass, cm, is dened as[69]

cm 1PNpi1Mpi

f0nfn

21

; (30)

where f0n is the fundamental frequency of the SWCNT in theabsence of any attached mass, and fn denotes the fundamentalfrequency of the SWCNT with attached masses. Using a localversion of Euler-Bernoulli beam theory, Adhikari and Chowdhury[69] studied the shift in the exural fundamental frequencies of thebridged and cantilevered SWCNTs due to distributed masses ontheir surfaces via energy and molecular dynamic methods. Thepredicted values of cm by the analytical solution in Ref. [69] as wellas those obtained from the proposed models are now provided inTable 5. For RKPM analysis, 51 particles with equal distances fromthe neighboring particles are used. The shape functions of theRKPMs particles are constructed based on quadratic base function,and exponential window function with dilation parameter equal to3.2. According toTable 5, for both cantilevered and bridged SWCNTswith attached masses, there is a reasonably good agreementbetween the predicted results by the proposed models and those ofRef. [69].

Since we are now condent about the calculations of theproposed numerical models, we can proceed in parametric analysisof the problem to explore the effects of inuential factors on thefrequency shift of the SWCNT-based mass sensor due to the addi-tion of nanoparticles.

6.2. Numerical studies

From applied mechanics point of view, the key characteristic ofa SWCNT-based mass sensor is the change in natural frequencies ofthe SWCNT. Since the lateral vibration of the nanosensor is ofconcern in this study, we focus on the transform of or shift in thenatural exural frequencies of the SWCNT. Such a frequency shiftpertinent to the ith vibration mode of the SWCNT is dened byRi(1Ui)/U0i, called frequency shift ratio. In this relation, U0irepresents the dimensionless natural frequency of the SWCNT-based mass sensor pertinent to the ith vibration mode in theabsence of the attached nanoparticles (i.e., Mpi 0; wi 1,.,Np).The so-called frequency shift ratio contains the information aboutthe change in the natural frequency of the SWCNT due to theaddition of nanoparticles. If the mass weight of the connectedparticle to the SWCNT would be negligible in compare to the massweight of the SWCNT, with a good accuracy Ri z 0. If the massweight of the attached particle would be extremely large withrespect to themass weight of the SWCNT, the natural frequencies ofthe nanosensor approach zero, and thereby, Ri z 1.

In RKPM analysis of the problem, 11 uniformly distributedparticles in the length of the SWCNT with exponential windowfunction, linear base function, and dilation parameter equal to 3.2are used. The dimensionless values of the constants of transverseand rotational springs at both ends of the SWCNT for differentboundary conditions have been summarized in Table 6. Eachboundary condition has been represented by two letters. The rstletter and the second one are, respectively, associated with the leftend and the right end boundary conditions of the SWCNT. Theletters S, C, Sf, and F are denoting a simple, clamp, shear-free, andfree end, respectively.

6.2.1. Effect of slenderness ratio on the frequency shiftIn Fig. 2(a)e(f), the plotted results of the frequency shift ratio of

a SWCNT with an attached nanoparticle at its midspan point interms of slenderness ratio of the ECS are provided. The graphs arebased on the predictions of the proposed models for differentboundary conditions of the SWCNT as a sensor as well as variouslevels of the mass weight of the attached nanoparticle (i.e., Mp1 0.1, 0.2, and 0.3). As it is seen in Fig. 2(a)e(f), for low levels of theslenderness ratio of the ECS, the NRBT could not predict thedynamic behavior of the SWCNT as a nanosensor at all. Moreover,the discrepancies between the predicted results by the NRBT andthose of the nonlocal shear deformable beam models generallyincrease as the mass weight of the attached nanoparticle decreases,irrespective of the boundary conditions of the SWCNT. Thediscrepancies between the predicted results by various modelswould commonly lessen with the slenderness ratio, irrespective ofthe mass weight of the attached nanoparticle. For a fairly slender

Table 5Justication of the obtained fundamental frequencies for the cantilevered andbridged SWCNTs with attached masses with those of Adhikari and Chowdhury [69].

Size ofmass(gm)

Evaluation of cm for cantileveredSWCNTs

Evaluation of cm for bridgedSWCNTs

Ref [69]. NRBT NTBT NHOBT Ref [69]. NRBT NTBT NHOBT

0.0 4.0000 4.0523 4.0470 3.9609 2.5222 2.5477 2.5374 2.43810.1 3.4747 3.5001 3.4964 3.4329 2.4866 2.5075 2.4965 2.40230.2 3.0008 3.0113 3.0088 2.9640 2.3839 2.3959 2.3856 2.30380.3 2.5796 2.5829 2.5813 2.5516 2.2261 2.2300 2.2214 2.15650.4 2.2123 2.2128 2.2118 2.1935 2.0308 2.0294 2.0231 1.97650.5 1.8985 1.8983 1.8978 1.8877 1.8181 1.8141 1.8101 1.78070.6 1.6363 1.6363 1.6361 1.6313 1.6075 1.6026 1.6006 1.58500.7 1.4218 1.4222 1.4221 1.4203 1.4144 1.4095 1.4088 1.40240.8 1.2492 1.2499 1.2498 1.2494 1.2481 1.2435 1.2435 1.2418

Table 6The values of K

:yi and K

:zi for different boundary conditions of the SWCNT.

SS CC SC SfS SfC CF

K:zi 0 108 108 108 0 0 108

K:zi 1 108 108 108 108 108 0

K:yi0 0 108 0 108 108 108

K:yi1 0 108 108 0 108 0

Table 4Comparison of the predicted resonant frequencies by the proposed nonlocal modelswith those of Wang et al. [46] and Joshi et al. [42] for different SWCNTs with

clamped-free boundary conditions f :1 u:1 =2p.

Do (nm) Di (nm) lb (nm) Eb (GPa) Experimentalresults [46](MHz)

FEM results[42] (MHz)

RKPMresults(MHz)

NRBT NTBT

33 18.8 5500 32 0.658 0.854 0.641 0.64439 19.4 5700 26.5 0.644 0.830 0.623 0.62539 13.8 5000 26.3 0.791 1.020 0.766 0.76845.8 16.7 5300 31.5 0.908 1.170 0.893 0.91250 27.1 4600 32.1 1.420 1.830 1.401 1.41864 27.8 5700 23 0.968 1.230 0.943 0.960


SWCNT (i.e., l 35), frequency shift ratio of the nanostructure ismore obvious for the SWCNT with CC boundary conditions. In sucha case, as one moves from CC to SC, SS, SfS, then SfC, and nally CFboundary conditions, the sensitivity of the rst exural frequencyof the SWCNT to the added nanoparticles decreases. Based on thepredicted results by the NHOBT for a very stocky SWCNT (i.e., l 5),the SWCNT with CC boundary conditions has the most observablefrequency shift ratio in compare to other cases, irrespective of thesmall-scale parameter. For such a case, the predicted R1 by theNHOBT for SS and SC conditions are very close to each other.Further, the predicted R1 by the NHOBT for CF, SfS, and SfC boundaryconditions are very close to each other. The sensitivity of the verystocky SWCNT to the attached nanoparticle is roughly identical tothe studied case of fairly slender SWCNTs. As it is also obvious inFig. 2(a)e(f), the predicted results of R1 by the NTBT are generallyclose to those of the NHOBT in most of the cases. In the cases of SS,SC, and CF boundary conditions, the NRBT could predict the resultsof the NHOBT with relative error lower than 10 percent for l > 22,l > 20, and l > 12, respectively.

6.2.2. Effect of small-scale parameter on the frequency shiftThe effect of the small-scale parameter on the frequency shift

ratio of the SWCNT is of particular importance in design of SWCNTsas nanomechanical sensors. In Fig. 3(a)e(f), the frequency shiftratio of the rst exural vibration mode of the SWCNT with l 10under different boundary conditions is plotted in terms of thesmall-scale parameter. In the cases of SS, CC, and SC boundaryconditions, both the NTBT and the NHOBT predict that thefrequency shift ratio commonly decreases with the small-scaleparameter (see Fig. 3(a)e(c)). In other words, the vibration sensi-tivity of the SWCNT due to the added mass would reduce as theeffect of the small-scale parameter becomes highlighted. A more

close scrutiny also reveals that the discrepancies between thepredicted results by the NTBT and those of the NHOBT are generallylower than 10 percent. As it is observed in Fig. 3(a)e(c), the NRBTshows ascending curves for the plots of R1 -e0a. As it was discussedin the previous part, the predictions of the NRBT would not betrustable for the nanotubewith such a slenderness ratio. In the caseof SfS boundary conditions (see Fig. 3(d)), no obvious variation of R1in terms of the small-scale parameter is detectable for the consid-ered range of mass weight of the attached nanoparticle (i.e., 0:1 Mp1 0:3). For the cases of SfS and CF boundary conditions, boththe NTBT and the NHOBT predict that the frequency shift ratio ofthe rst vibration mode increases with the small-scale parameter.Additionally, the rate of variation of the frequency shift ratio asa function of the small-scale parameter intensies with the massweight of the attached nanoparticle (see Fig. 3(e) and (f)). In suchcases, the NRBT exhibits a descending behavior for the plots of R1-e0a, in which would not be realistic at all.

6.2.3. Effect of mass weight of the attached nanoparticle on thefrequency shift

We are also interested in investigating the inuence of the massweight of the attached nanoparticle on the frequency shift ratio ofthe nanotube under different boundary conditions as well asvarious levels of the small-scale parameter. The predicted results byvarious nonlocal beam models for a SWCNT with l 10 have beendemonstrated in Fig. 4(a)e(f). For all SWCNTs boundary conditions,the frequency shift ratio of the rst vibration mode increases withthe mass weight of the attached nanoparticle. In the case of theSWCNT under SS conditions with e0a 0 nm, both the NRBT andthe NTBT could track the predicted frequency shift ratio of theNHOBT with accuracy lower than 5 percent for the studied range ofnormalizedmass weight of the attached nanoparticle (see Fig. 4(a)).

5 20 350

0.110.22

R 1

5 20 350

0.110.22

5 20 350

0.110.22

5 20 350.020.190.36

R 1

5 20 350.020.190.36

5 20 350.020.190.36

5 20 350.020.16

0.3

R 1

5 20 350.020.160.3

5 20 350.020.160.3

5 20 350.020.06

0.1

R 1

5 20 350.020.060.1

5 20 350.020.060.1

5 20 350.02

0.080.18

R 1

5 20 350.02

0.080.18

5 20 350.02

0.080.18

5 20 350.02

0.090.2

R 1

5 20 350.02

0.090.2

5 20 35

0.020.090.2

a

b

c

d

e

f

Fig. 2. Frequency shift ratio of the SWCNT with a mass at the midspan point as a function of slenderness ratio for different boundary conditions: (a) SS, (b) CC, (c) SC, (d) SfS, (e) SfC,(f) CF; ((B)Mp1 0:1, ()Mp1 0:2, (6)Mp1 0:3 ; (.) NRBT, (.) NTBT, () NHOBT; Kt Kr]Nb 0; e0a 2 nm).


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