Zhi-Bin Shene-mail: zb_shen@yeah.net

Bin Denge-mail: dbhj0@sina.com

College of Aerospace and Material Engineering,

National University of Defense Technology,

Changsha, Hunan 410073, China

Xian-Fang LiSchool of Civil Engineering,

Central South University,

Changsha, Hunan 410075, China

e-mail: xfli00@hotmail.com

Guo-Jin Tang1

College of Aerospace and Material Engineering,

National University of Defense Technology,

Changsha, Hunan 410073, China

e-mail: tanggj@nudt.edu.cn

Vibration of Double-WalledCarbon Nanotube-Based MassSensor via Nonlocal TimoshenkoBeam TheoryThe potential of double-walled carbon nanotubes (DWCNTs) as a micromass sensor isexplored. A nonlocal Timoshenko beam carrying a micromass at the free end of the innertube is used to analyze the vibration of DWCNT-based mass sensor. The length of theouter tube is not equal to that of the inner tube, and the interaction between two tubes isgoverned by van der Waals force (vdW). Using the transfer function method, the naturalfrequencies of a nonlocal cantilever with a tip mass are computed. The effects of theattached mass and the outer-to-inner tube length ratio on the natural frequencies are dis-cussed. When the nonlocal parameter is neglected, the frequencies reduce to the classicalresults, in agreement with those using the finite element method. The obtained resultsshow that increasing the attached micromass decreases the natural frequency butincreases frequency shift. The mass sensitivity improves for short DWCNTs used in masssensor. The nonlocal Timoshenko beam model is more adequate than the nonlocal Euler-Bernoulli beam model for short DWCNT sensors. Obtained results are helpful to thedesign of DWCNT-based resonator as micromass sensor. [DOI: 10.1115/1.4005489]

Keywords: DWCNT, Timoshenko beam theory, mass sensor, nonlocal elasticity, transferfunction method, free vibration

1 Introduction

Since the discovery of carbon nanotubes (CNTs) by Iijima [1]in 1991, they have demonstrated a significant potential for use in adiverse range of new and evolving applications [2,3]. In particu-lar, the experimental evidence shows that CNTs have low weight,high aspect ratio, extremely high stiffness and are highly sensitiveto their environment change [4]. Such features of CNTs makethem promising candidates for atomic-resolution mass sensor [5].

Poncharal et al. [6] first proposed the idea of using individualCNTs as high sensitivity nanobalances in 1999. In general, theprinciple of mass detection using CNT-based sensor from a vibra-tion analysis is based on the fact that the resonant frequency issensitive to the attached mass [7]. Many researchers haveexplored the potential of using single-walled carbon nanotubes(SWCNTs) as nanomechanical resonators in atomic scale [8–11].Multiwalled carbon nanotubes (MWCNTs) resonators have differ-ent mechanical structures than SWCNT ones, due to the interac-tion between intertubes such as the van der Waals (vdW) force.Mateiu et al. [12,13] proposed an approach for building a masssensor based on MWCNTs. Elishakoff et al. [14] investigatedvibration of double-walled carbon nanotubes (DWCNTs) cantile-ver with attached bacterium on the tip. Recently, synthesis tech-nique has made DWCNTs with different wall lengths possible bemanufactured [15]. Kang et al. [16,17] examined frequencychange of ultrahigh frequency nanomechanical resonators basedon DWCNTs with different wall lengths using molecular dynam-ics simulations.

Continuum models have been widely used to study the vibra-tional behavior of CNT-based mass sensor to avoid the difficultiesencountered during the experimental characterization of nanotubesas well as the time-consuming nature of computational atomisticsimulations. For example, the classical Euler-Bernoulli beam

theory (EBT) was utilized to model a nanomechanical resonator[9,18,19]. However, the effects of shear deformation and rotaryinertia are neglected in these analyses. When these two factors aretaken into account simultaneously, the Timoshenko beam theory(TBT) Ref. [20] is necessary and also more effective. Some theo-retical analyses involving wave propagation and free vibration ofCNTs have also been presented [21–23] using the TBT.

In addition, owing to the fact that the classical continuumapproach is scale-independent and then cannot describe the sizeeffects arising from small scale. Although the classical continuumtheory is sometimes applied to analyze the mechanical behaviorof nanostructures such as CNTs, it is found to be inadequatebecause of ignoring the size effects. To improve this situation, thenonlocal elasticity theory presented by Eringen [24,25] has beendeveloped to tackle scale-dependent problems. Along this line,Lee et al. [26] used the nonlocal EBT to analyze the frequencyshift of CNT-based mass sensors. As mentioned above, the EBTneglects the influences of shear deformation and rotary inertia. Toconsider these two factors, application of the nonlocal TBT inanalyzing the frequency shift of CNT-based mass sensors is a pre-requisite. To the best of the authors’ knowledge, this topic has notbeen studied, in particular for the vibration of DWCNT basedmass sensor with inner and outer nanotubes of different lengths.Such DWCNTs with different wall lengths have been reported tohave some specific practical applications [27].

In the present study, the dynamic behavior of a DWCNT basedmass sensor with different wall lengths is studied. The DWCNT ismodeled as two Timoshenko cantilevers and their interaction isdescribed by the vdW force, and the size effects are described bythe nonlocal elasticity theory. For the case of a micromass at thefree end of the inner tube, the transfer function method (TFM)Ref. [28] is used to determine the natural frequencies of themicromass sensor. The effects of the attached tip mass, nonlocalparameter, length ratio, DWCNT length, and rotary inertia on thefundamental frequencies are analyzed. The natural frequencies ofDWCNT-based mass sensors when using the nonlocal TBT arecompared with those when using nonlocal EBT and the classical

1Corresponding author.Manuscript received July 14, 2011; final manuscript received August 25, 2011;

published online January 9, 2012. Editor: Boris Khusid.

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TBT. Our analysis shows novel characteristics of the fundamentalfrequency shift.

2 Basic Equations

2.1 Nonlocal TBT. As in the classical TBT, there are onlytwo independent variables, the transverse displacement w and therotation of cross-section h, both of which depend on the longitudi-nal coordinate x and time t. According to the Hamilton’s principleand Eringen’s nonlocal elasticity theory [24,25], governing equa-tions of the nonlocal TBT can be expressed as [21]

qA 1� ðe0aÞ2 @2

@x2

� �@2w

@t2� jGA

@h@xþ @

2w

@x2

� �

¼ 1� ðe0aÞ2 @2

@x2

� �pðx; tÞ (1)

qI 1� ðe0aÞ2 @2

@x2

� �@2h@t2þ jGA hþ @w

@x

� �� EI

@2h@x2¼ 0 (2)

where q is the mass density, A and I are the cross-sectional areaand its second moment, respectively, p x; tð Þ is the distributed forceper unit length, E and G are Young’s modulus and shear modulus,respectively. j is the shear correction factor, which is introducedto account for the relaxation of the inconsistency of the usualshear-free boundary condition at the beam surface and to compen-sate for the error in assuming a constant shear stress over thewhole cross-section of the beam. It depends on the material andgeometric parameters of the beam. e0a is a small scale parameterwith length unit which can be used to describe the size effects, e0

being a nondimensional material constant that can be determinedby experiments or numerical simulations from molecular dynam-ics simulation [29] and a being the C–C bond length. Here, thenonlocal effects are assumed to be present for both normal andshear stresses. When e0a ¼ 0, the nonlocal elasticity reduces tothe classical TBT.

In addition, the nonlocal bending moment M and the nonlocalshearing force Q can be expressed below, respectively, as

Mðx; tÞ ¼ EI@h@xþ ðe0aÞ2 qA

@2w

@t2þ qI

@3h@x@t2

� pðx; tÞ� �

(3)

Qðx; tÞ ¼ jGA hþ @w

@x

� �þ ðe0aÞ2 qA

@3w

@x@t2� @pðx; tÞ

@x

� �(4)

For a DWCNT, two nanotubes are interacted through the vdWforce, which for simplicity can be described by the following lin-ear relation [23]

pðx; tÞ ¼ c½w1ðx; tÞ � w2ðx; tÞ� (5)

where c is the vdW interaction coefficient between the inner andouter nanotubes. Here, we take [22]

c ¼ 320� ð2R1Þerg=cm2

0:16a2; ða ¼ 0:142 nmÞ (6)

where R1 is the inner radius of the DWCNT. For the presentDWCNT with R1¼ 0.35 nm, Eq. (6) gives c¼ 6:943� 1011erg=cm

3

¼ 69:43 GPa.Now, let us consider transverse vibration of a DWCNT carrying

an atomic-resolution mass at the free tip, as shown in Fig. 1. With-out loss of generality, we assume that the inner and outer nano-tubes have different lengths, Lj and dj representing the length anddiameter of the inner ( j¼ 1) and outer ( j¼ 2) tubes, respectively,and the atomic-resolution mass is attached to the free end of theinner tube. In the following, we consider the case where the innertube is longer than the outer tube, as shown in Fig. 1, the govern-ing equations for the DWCNT carrying an atomic-resolution massread

qA1 1� ðe0aÞ2 @2

@x2

� �@2w1

@t2� jGA1

@h1

@xþ @

2w1

@x2

� �

¼ 1� ðe0aÞ2 @2

@x2

� �cðw2 � w1Þ 0 � x � L2ð Þ (7)

qI1 1� ðe0aÞ2 @2

@x2

� �@2h1

@t2þ jGA1 h1 þ

@w1

@x

� �

� EI1

@2h1

@x2¼ 0 0 � x � L2ð Þ (8)

qA2 1� ðe0aÞ2 @2

@x2

� �@2w2

@t2� jGA2

@h2

@xþ @

2w2

@x2

� �

¼ � 1� ðe0aÞ2 @2

@x2

� �cðw2 � w1Þ 0 � x � L2ð Þ (9)

qI2 1� ðe0aÞ2 @2

@x2

� �@2h2

@t2þ jGA2 h2 þ

@w2

@x

� �

� EI2

@2h2

@x2¼ 0 0 � x � L2ð Þ (10)

qA3 1� ðe0aÞ2 @2

@x2

� �@2w3

@t2� jGA3

@h3

@xþ @

2w3

@x2

� �¼ 0

L2 < x � L1ð Þ (11)

qI3 1� ðe0aÞ2 @2

@x2

� �@2h3

@t2þ jGA3 h3 þ

@w3

@x

� �� EI3

@2h3

@x2¼ 0

L2 < x � L1ð Þ (12)

where the subscripts 1 and 2 denote the quantities associated with theinner and outer nanotubes, respectively. It is particularly noted thatfor convenience we have introduced subscript 3 that specifies thecorresponding quantities of the inner tube lying in L2 < x � L1,viz. w3 ¼ w1; h3 ¼ h1, I1 ¼ I3; A1 ¼ A3 as L2 < x � L1. This isequivalent to say that for 0 � x � L2, the DWCNT is essentiallycomposed of two tubes, while L2 < x � L1 it is in fact a single tube.

For a cantilever DWCNT carrying a micro mass where micromass specifies those such as a buckyball, a molecular, a bacteriumor a virus, etc., at the free end, the corresponding boundary condi-tions can be stated as

wjð0; tÞ ¼ 0; hjð0; tÞ ¼ 0 j ¼ 1; 2ð Þ (13)

M2ðL2; tÞ ¼ 0; Q2ðL2; tÞ ¼ 0; M3ðL1; tÞ ¼ 0 (14)

Q3ðL1; tÞ þ m0

@2w3ðL1; tÞ@t2

¼ 0 (15)

where m0 is the attached mass at the free tip of the inner tube. Inaddition, at the position x¼L2, the continuity conditions must besatisfied, namely

w1ðL2Þ ¼ w3ðL2Þ; h1ðL2Þ ¼ h3ðL2Þ (16)

M1ðL2; tÞ ¼ M3ðL2; tÞ; Q1ðL2; tÞ ¼ Q3ðL2; tÞ (17)

Furthermore, initial conditions can be assumed as

wjðx; 0Þ ¼@wjðx; 0Þ

@t¼ 0 j ¼ 1; 2; 3ð Þ (18)

Fig. 1 A cantilever DWCNT-based mass sensor with inner andouter nanotubes of different lengths

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hjðx; 0Þ ¼@hjðx; 0Þ

@t¼ 0 ; j ¼ 1; 2; 3ð Þ (19)

2.2 Nonlocal EBT. The governing equations for the nonlocalEBT can also be analytically derived from the above. EliminatingjGA @h

@x þ @2w@x2

� �from Eqs. (1) and (2) leads to

qI 1� ðe0aÞ2 @2

@x2

� �@3h@x@t2

þ qA 1� ðe0aÞ2 @2

@x2

� �@2w

@t2� EI

@3h@x3

¼ 1� ðe0aÞ2 @2

@x2

� �pðxÞ (20)

In the Euler-Bernoulli hypothesis, the effects of shear deformationand rotary inertia of the beam are neglected. The Euler-Bernoullihypothesis is actually equivalent to sufficiently large shear modulusor shear rigidity, which impedes shear deformation in the beam.Consequently, in the above formulation, let jGA=EI !1 andneglect the influence of the rotary inertia, meaning qI@2h=@t2 ¼ 0.Equation (2) reduces to

h ¼ � @w

@x(21)

Bearing qI@2h=@t2 ¼ 0 in mind, we substitute Eq. (21) intoEq. (20) and find that the governing equation of nonlocal EBT

EI@4w

@x4þ qA 1� ðe0aÞ2 @

2

@x2

� �@2w

@t2¼ 1� ðe0aÞ2 @

2

@x2

� �pðxÞ (22)

is recovered. In this case, instead of Eqs. (3) and (4), the nonlocalbending moment M and the nonlocal shearing force Q are,respectively,

Mðx; tÞ ¼ �EI@2w

@x2þ ðe0aÞ2 qA

@2w

@t2� pðxÞ

� �(23)

Qðx; tÞ ¼ @�Mðx; tÞ@x

¼ �EI@3w

@x3þ ðe0aÞ2 qA

@3w

@x@t2� @pðxÞ

@x

� �(24)

Thus, with the only deflection as an unknown in the nonlocalEBT, the transverse vibration of the DWCNT with different tubelengths (Fig. 1) are governed by the following coupled equations:

EI1

@4w1

@x4þ qA1 1� ðe0aÞ2 @

2

@x2

� �@2w1

@t2

¼ 1� ðe0aÞ2 @2

@x2

� �cðw2 � w1Þ (25)

EI2

@4w2

@x4þ qA2 1� ðe0aÞ2 @

2

@x2

� �@2w2

@t2

¼ � 1� ðe0aÞ2 @2

@x2

� �cðw2 � w1Þ (26)

EI3

@4w3

@x4þ qA3 1� ðe0aÞ2 @

2

@x2

� �@2w3

@t2¼ 0 (27)

The boundary conditions in Eqs. (13)–(15) as well as the continu-ity conditions at the position x¼ L2 in Eqs. (16) and (17) and ini-tial conditions in Eqs. (18) and (19) are the same, where M and Qin those equations should be replaced by Eqs. (23) and (24).

3 Solution Method

3.1 Nonlocal TBT. In solving free vibration of Timoshenkobeams, as a powerful semi-analytical and seminumerical method,the TFM is frequently applied to treat relevant problems. For

example, Adhikari et al. [30] presented a closed form solution fora beam with nonlocal damping using the TFM for the distributedparameter system. In this section, the TFM is employed to investi-gate the free vibration of the DWCNT-based mass sensor.

With the aid of the initial conditions, we can perform Laplacetransform for the governing Eqs. (7)–(12). Furthermore, by intro-ducing the following nondimensional parameters:

X ¼ x

L1

; Wi ¼wi

L1

; ai ¼Ii

AiL21

; wi ¼cL2

1

kGAi; b ¼ E

jG;

C ¼ qL21s2

E; k ¼ e0a

L1

i ¼ 1; 2; 3ð Þ

the governing Eqs. (7)–(12) may be rewritten as

ð1þ Cbk2 þ w1k2Þ @

2 ~W1

@X2� w1k

2 @2 ~W2

@X2

¼ ðCbþ w1Þ ~W1 � w1~W2 �

@ ~h1

@X(28)

ð1þ Cbk2 þ w2k2Þ @

2 ~W2

@X2� w2k

2 @2 ~W1

@X2

¼ ðCbþ w2Þ ~W2 � w2~W1 �

@ ~h2

@X(29)

@2 ~W3

@X2¼ Cb

1þ Cbk2~W3 �

1

1þ Cbk2

@ ~h3

@X(30)

@2 ~hi

@X2¼ 1

aibð1þ Ck2Þ@ ~Wi

@Xþ 1þ aibC

aibð1þ Ck2Þ~hi i ¼ 1; 2; 3ð Þ (31)

Combining Eq. (28) with Eq. (29), @2 ~W1

@X2 and @2 ~W2

@X2 may be expressedas

@2 ~W1

@X2¼ Cb

1þ Cbk2þ w1

D

� �~W1 �

1

1þ Cbk2� w1k

2

D

� �@ ~h1

@X

� w1

D~W2 �

w1k2

D@ ~h2

@X(32)

@2 ~W2

@X2¼ Cb

1þ Cbk2þ w2

D

� �~W2 �

1

1þ Cbk2� w2k

2

D

� �@ ~h2

@X

� w2

D~W1 �

w2k2

D@ ~h1

@X(33)

where D ¼ ð1þ Cbk2Þð1þ Cbk2 þ w1k2 þ w2k

2Þ.Next, if we introduce a state vector as

gðX;sÞ¼ ~W1ðX;sÞ;@ ~W1ðX;sÞ

@X; ~h1ðX;sÞ;

@ ~h1ðX;sÞ@X

; :::;@ ~h3ðX;sÞ@X

" #T

12�1

where the superscript T denotes matrix transpose, Eqs. (30)–(33)can be rewritten in a compact form in state space

dgðX; sÞdX

¼ UðsÞgðX; sÞ þ gðX; sÞ (34)

where UðsÞ ¼ ½Uij�12�12 is a 12� 12 matrix with variable s, thedetails of which are given in the Appendix A, and gðX; sÞ is relatedto distributed loading and vanishes in the present study.

Next, after performing the Laplace transform to two sides ofthe boundary conditions (13)–(15) and the continuity conditions(16) and (17), with the vanishing initial conditions, we get

~Wjð0; sÞ ¼ 0; ~hjð0; sÞ ¼ 0 j ¼ 1; 2ð Þ (35)

�w2k2

a2b~W1ðd; sÞþ

k2

a2

Cþw2

b

� �~W2ðd; sÞþ 1þCk2

� @ ~h2ðd; sÞ@X

¼ 0

(36)

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� w2k2 @ ~W1ðd; sÞ

@X� ð1þ Cbk2 þ w2k

2Þ @~W2ðd; sÞ@X

þ ~h2ðd; sÞ ¼ 0

(37)

Ca3

k2 ~W3ð1; sÞ þ ð1þ Ck2Þ @~h3ð1; sÞ@X

¼ 0 (38)

mbC ~W3ð1; sÞ þ ð1þ Cbk2Þ @~W3ð1; sÞ@X

þ ~h3ð1; sÞ ¼ 0 (39)

~W1ðd; sÞ ¼ ~W3ðd; sÞ; ~h1ðd; sÞ ¼ ~h3ðd; sÞ (40)

k2

a1

Cþ w1

b

� �~W1ðd; sÞ þ 1þ Ck2

� @ ~h1ðd; sÞ@X

� w1k2

a1b~W2ðd; sÞ

� Ck2

a1

~W3ðd; sÞ � ð1þ Ck2Þ @~h3ðd; sÞ@X

¼ 0 (41)

ð1þ bCk2 þ w1k2Þ @

~W1ðd; sÞ@X

þ ~h1ðd; sÞ � w1k2 @ ~W2ðd; sÞ

@X

� ð1þ bCk2Þ @~W3ðd; sÞ@X

� ~h3ðd; sÞ ¼ 0 (42)

where m ¼ m0=qAL1 is the ratio of the attached micromass to theinner tube mass, and d ¼ L2=L1 is the outer-to-inner tube lengthratio, respectively. Furthermore, using the introduced vector g,Eqs. (35)–(42) become

Mbgð0; sÞ þ Nbgð1; sÞ þ RcðsÞgðd; sÞ ¼ cðsÞ (43)

where Mb ¼ ½Mi;j�12�12, Nb ¼ ½Ni;j�12�12 and Rc ¼ ½Ri;j�12�12 are12� 12 matrixes, the details of them are given in the AppendixB. cðsÞ in Eq. (43) is a vector and cðsÞ ¼ 0 in the present study.

According to the TFM, the solution of Eq. (34) can beexpressed as follows [31]:

Mb þ NbeUðsÞ þ RcðsÞeUðsÞdh i

gðX; sÞ

¼ðL

0

GðX; n; sÞgðn; sÞdnþ eUðsÞXcðsÞ (44)

where

GðX; n; sÞ ¼ Mbe�UðsÞn X � n�NbeUðsÞð1�nÞ X < n

For the free vibration of the DWCNT-based nanomechanical sen-sor, keeping gðX; sÞ ¼ 0 and cðsÞ ¼ 0 in mind, meaning that theright-hand side of Eq. (44) clearly vanishes, the existence of anontrivial solution of the corresponding homogenous Eq. (44)allows us to get that the determinant of the coefficient matrix mustvanish

det Mb þ NbeUðsÞ þ RcðsÞeUðsÞdh i

¼ 0 (45)

This is the characteristic equation we want to look for.Considering the nondimensional natural frequency

X ¼ffiffiffiffiffiffiffi�Cp

¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi� qL2

1s2

E

r(46)

the circular frequency x of free vibration is related by the follow-ing relation:

x ¼ ImðsÞ ¼ X

ffiffiffiffiffiffiffiffiE

qL21

s(47)

where ImðsÞ denotes the imaginary part of a complex s. After get-ting the solution of Eq. (45), the natural frequencies

f ¼ x2p¼ X

2p

ffiffiffiffiffiffiffiffiE

qL21

s(48)

can be directly obtained from Eq. (47). Also, the correspondingmodal shapes can be evaluated by

gðx; skÞ ¼ eUðskÞXfk (49)

where sk denotes the imaginary frequency of the nanomechanicalsensor for mode k, and fk is a nontrivial vector obtained by substi-tuting sk into the corresponding homogeneous Eq. (44).

3.2 Nonlocal EBT. In a similar manner, the nondimensionalgoverning equations, boundary conditions and continuity condi-tions for the DWCNT sensor based on the nonlocal EBT can beexpressed as

@4 ~W1

@X4¼ Ck2

a1

þ w1k2

a1b

� �@2 ~W1

@X2� C

a1

þ w1

a1b

� �~W1

� w1k2

a1b@2 ~W2

@X2þ w1

a1b~W2 (50)

@4 ~W2

@X4¼ Ck2

a2

þ w1k2

a2b

� �@2 ~W2

@X2� C

a2

þ w2

a2b

� �~W2

� w2k2

a2b@2 ~W1

@X2þ w2

a2b~W1 (51)

@4 ~W3

@X4¼ Ck2

a3

@2 ~W3

@X2� C

a1

~W3 (52)

~Wjð0; sÞ ¼ 0;@ ~Wj

@Xð0; sÞ ¼ 0 j ¼ 1; 2ð Þ (53)

Ca1

k2 ~W3ð1; sÞ �@2 ~W3ð1; sÞ

@X2¼ 0 (54)

mCa3

~W3ð1; sÞ þCk2

a3

@ ~W3ð1; sÞ@X

� @3 ~W3ð1; sÞ@X3

¼ 0 (55)

�w2k2

a2b~W1ðd; sÞ þ

Ck2

a2

þ w2k2

a2b

� �~W2ðd; sÞ �

@2 ~W2ðd; sÞ@X2

¼ 0

(56)

� w2k2

a2b@ ~W1ðd; sÞ

@Xþ Ck2

a2

þ w2k2

a2b

� �@ ~W2ðd; sÞ

@X� @

3 ~W2ðd; sÞ@X3

¼ 0

(57)

~W1ðd; sÞ ¼ ~W3ðd; sÞ;@ ~W1ðd; sÞ

@X¼ @

~W3ðd; sÞ@X

(58)

Ck2

a1

þ w1k2

a1b

� �~W1ðd; sÞ �

@2 ~W1ðd; sÞ@X2

� w1k2

a1b~W2ðd; sÞ

� Ck2

a1

~W3ðd; sÞ þ@2 ~W3ðd; sÞ

@X2¼ 0 (59)

Ck2

a1

þ w1k2

a1b

� �@ ~W1ðd; sÞ

@X� @

3 ~W1ðd; sÞ@X3

� w1k2

a1b@ ~W2ðd; sÞ

@X

� Ck2

a1

@ ~W3ðd; sÞ@X

þ @3 ~W3ðd; sÞ@X3

¼ 0 (60)

A completely analogous procedure can deal with the followingequations using the TFM with the introduction of the followingstate vector:

�gðX; sÞ ¼ ~W1;@ ~W1

@X;@2 ~W1

@X2;@3 ~W1

@X3; :::;

@3 ~W3

@X3

� �T

12�1

:

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Note that in this case, U sð Þ ¼ ½�Ui;j�12�12;Mb ¼ ½ �Mi;j�12�12;Nb

¼ ½ �Ni;j�12�12; Rc ¼ ½ �Ri;j�12�12 are still 12� 12 matrixes, the detailsof which are given in the Appendix C.

4 Results and Discussion

In this section, the effects of the nonlocal parameter, tip micro-mass, and outer-to-inner tube length ratio on the natural frequen-cies of the cantilever DWCNT are analyzed. Consider a DWCNTwith inner diameter d1¼ 2R1¼ 0.7 nm and outer diameter d2

¼ 2R2¼ 1.4 nm, where R1 is the radius of the inner tube center-line, while R2 is the radius of the outer tube centerline. It isassumed that the inner and outer tubes have the same Young’smodulus E¼ 1 TPa, shear modulus G¼ 0.4 Tpa (with Poissonratio � ¼ 0:25), and the effective thickness of SWCNT t¼ 0.3 nm.In accordance with the definition of the effective thicknessand the Young’s modulus mentioned above, a mass densityq ¼ 2:3g=cm3 is adopted. For a CNT with hollow circularcross-section, the shear correction factor j is taken as 0.8 [32].

4.1 Result Validation. Prior to the presentation of numericalresults of the natural frequencies, let us examine the validity andaccuracy of the method suggested here. The DWCNT-basedmicromass sensor is modeled as a microcantilever with a concen-trated mass at the free end of the inner tube. The natural frequen-cies of the DWCNT-based micromass sensor can be computedand numerical results of a DWCNT with L1¼ 14 nm for differentvalues of the tip micromass and the outer-to-inner tube length ra-tio are listed in Table 1 when the nonlocal parameter e0a/L van-ishes, i.e., neglecting the size effects. For comparison, we alsotabulate the corresponding numerical results in Table 1 using acommercial FEM software MSC.NASTRAN. Note that when adoptingMSC.NASTRAN, the vdW force between two tubes is modeled withlinear spring, and the concentrated mass is joined with the innernanotube by interpolation constraint element RBE3. The diagramusing 3D solid finite element model for a DWCNT with inner andouter tubes of different lengths and carrying a micromass at theinner tube tip is shown in Fig. 2. From Table 1, it is seen that thetheoretical results and the FEM simulation results are in goodagreement. For the fundamental frequencies, the maximum rela-tive error is 3.51%, occurring in the absence of the tip mass withL2/L1¼ 0.5. This comparison provides high confidence for use ofthe TBT model in further investigation as a mass sensor. Obtainedresults indicate that with the tip mass rising, the correspondingfrequencies decrease. Moreover, as the ratio L2/L1 varies, the natu-ral frequency of the DWCNT is strongly affected, no matter

whether a tip micromass is attached or not. The fundamental fre-quencies are seen to be very sensitive to the change in the attachedtip mass, in particular for larger tip mass. This is the basic princi-ple of the DWCNT as micromass detection [7].

4.2 Effect of Tip Mass on the SWCNT Based MicromassSensor. Due to the significance of the fundamental frequencies ofDWCNT based mass sensor, in what follows we focus our attentionon the fundamental frequency shift induced by an attached micro-mass. For this purpose, we denote the frequency shift Df as the dif-ference between the fundamental frequencies of the cantileverDWCNT with and without tip micromass, which serves as an indexto assess quantitatively the mechanical behavior of the mass sensor.

Figure 3 shows the fundamental frequency shift of a cantileverDWCNT as a function of the attached mass with different outertube lengths and L1¼ 14 nm. As pointed before, an increase in theattached tip mass decreases the fundamental frequencies. How-ever, from Fig. 3, the frequency shift is viewed to rise as the tipmass is raised, which agrees qualitatively with the resultsdescribed by Lee et al. [26] and Li et al. [7]. Moreover, the varia-tions of frequency shift are apparent when the attached mass islarger than 10�21 g. In other words, the mass sensitivity of thisnanomechanical sensor can reach at least 10�21 g, which has thesame order as mentioned in Refs. [7] and [33]. Furthermore, theeffect of nonlocal parameter on the frequency shift is significant.Note that the impact of the small scale on the frequency shift isdifferent depending on different length ratios. For example, forL2/L1¼ 0.5, the nonlocal parameter makes the frequency shiftbecome smaller. This trend is, however, reversed for L2/L1¼ 1.0.From Fig. 3, it is clear that there is a critical mass mc related to thenonlocal parameter e0a and the length of DWCNT. The frequencyshift of the DWCNT-based mass sensor is more sensitive for thetip micromass m0 less than mc if L2/L1¼ 0.5, and for m0 largerthan mc if L2/L1¼ 1.0.

Table 1 Comparison of the natural frequencies in GHz obtained using the TFM with those usingthe FEM software for SWCNT sensor with L1 5 14 nm for different attached masses

m L2/L1 f1 f2 f3 f4 f5

0.0 0.0 TFM 15.9730 98.0645 266.1967 500.4702 788.5085FEM 15.8037 96.8190 262.1844 491.5864 772.3475

%error �1.0599 �1.2701 �1.5073 �1.7751 �2.04960.5 TFM 32.8253 110.5275 292.1970 545.5307 765.8425

FEM 33.9794 110.4688 304.7367 528.3862 820.3664%error 3.5159 �0.0531 4.2915 �3.1427 7.1195

1.0 TFM 26.0066 153.4298 394.6100 695.4094 1024.0282FEM 25.6823 149.6798 381.7038 671.3197 994.6709

%error �1.2470 �2.4441 �3.2706 �3.4641 �2.86681.0 0.0 TFM 7.0853 72.7607 221.4480 440.1099 715.6924

FEM 6.9354 71.8114 218.2323 432.6588 701.6287%error �2.1156 �1.3047 �1.4521 �1.6930 �1.9650

0.5 TFM 13.0307 92.0735 232.3664 505.1855 683.4943FEM 13.3516 90.7892 243.4285 490.4210 722.3286

%error 2.4626 �1.3949 4.7606 �2.9226 5.68171.0 TFM 16.9716 120.5118 314.2986 532.4463 782.2438

FEM 16.6581 119.4134 324.6293 586.6382 877.5734%error �1.8472 �0.9114 3.2869 10.1779 12.1867

Fig. 2 A FEM model for cantilever DWCNT-based mass sensorwith different tube lengths

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In addition, the length of the DWCNT, L1, can influence thechanges in the fundamental frequency shift of the DWCNT-basedmass sensor. Figure 4 illustrates the effect of length ratio, L2/L1,for two different DWCNT lengths, L1¼ 7 nm and 14 nm, on the

frequency shift of the mass sensor. It is seen that the frequencyshift is significantly affected by the length ratio. The frequencyshift becomes larger when the length of DWCNT is shorter, espe-cially for larger attached mass. Thus, the mass sensitivity

Fig. 3 The fundamental frequency shift of a DWCNT sensor versus attached mass withL1 5 14 nm

Fig. 4 The effect of length ratio and DWCNT length on the fundamental frequency shift of theSWCNT sensor versus attached mass with e0a/L 5 0.3

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increases when smaller size DWCNT resonators are used in masssensors. This finding agrees with previous results for CNT-basednanomechanical resonators [7,33,34]. Another interestingobservation is that a DWCNT-based mass sensor with lengthratio (0< L2/L1� 1.0) is more sensitive than a SWCNT-based one(L2/L1¼ 0), and it with shorter outer tube is more sensitive than itwith equal outer tube length for an atomic-resolution mass around10�21g, which implies that it is better to utilize a DWCNT withshorter outer tube length as a micromass sensor, rather than aSWCNT or a DWCNT with equal tube length.

4.3 Vibration of DWCNT Based Mass Sensor. To furtherstudy the vibration properties of DWCNT-based mass sensor withdifferent inner and outer nanotube lengths, the effects of the non-local parameter, the attached tip micromass, the length ratio, androtary inertia on the fundamental frequencies are examined.

The influence of the nonlocal parameter on the frequency ratiofNT/fCT is displayed in Fig. 5(a) for a DWCNT with L1¼ 14 nm in

the absence of tip mass, where the frequency with the subscriptsNT and CT stands for nonlocal and classical TBT, respectively. Itis clearly seen from Fig. 5(a) that the small scale strongly affectsthe fundamental frequencies. As the length ratio increases, the fre-quency ratio decreases initially and then increases. Moreover, thefrequency ratio is larger than unity when L2/L1 is close to zero orunity, which means that the nonlocal parameter causes a slightrise for fundamental frequencies when L2/L1 is small or gettingclose to unity. This conclusion is in consistency with that for thenonlocal Timoshenko beam without attached mass [21]. However,in most ranges of L2/L1, the fundamental frequencies become lessthan the corresponding classical values. In the presence of tipmass, the variation of the frequency ratio fNT/fCT against L2/L1 ispresented in Fig. 5(b) with e0a/L¼ 0.3. In this case, the trend isstill similar to that observed in Fig. 5(a). Furthermore, wheneither e0a/L or tip mass varies, there are always two fundamentalfrequencies remaining unchanged. For example, fNT/fCT¼ 1.0at about L2/L1¼ 0.06 and 0.76 for any e0a/L (Fig. 5(a)), while

Fig. 5 The effect of small scale on the fundamental frequency of the DWCNTsensor versus length ratio (a) different nonlocal parameters with m 5 0 (b) dif-ferent attached mass with e0a/L 5 0.3

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fNT/fCT¼ 0.87 and 0.71 at about L2/L1¼ 0.1 and 0.48, respec-tively, for variable tip mass. The greatest drop among the fun-damental frequencies occurs around L2/L1¼ 0.2 regardless of tipmass. The location at which the greatest drop among thefundamental frequencies occurs in Figs. 5(a) and 5(b) is clearlydependent on the nonlocal parameter.

Next, let examine the merit of a DWCNT with different walllengths as mass sensor compared with a SWCNT. Figure 6 depictsthe effect of attached mass on the fundamental frequency ratiofDT/fST, where the subscripts DT and ST stand for the DWCNT(0 < L2=L1 � 1) and SWCNT (L2=L1 ¼ 0), respectively. It can beseen from Fig. 6(a) that the frequency fDT> fST always holds ife0a/L¼ 0, whereas it is true for L2/L1> 0.35 if e0a/L¼ 0.3. This isto say that in a general case, the response of the fundamentalfrequencies for DWCNT due to an attached tip mass is strongerthan that for SWCNT. Therefore, a design for nanomechanical

resonators that operate at various frequencies can be realized bycontrolling the length ratio of DWCNTs. Furthermore, due to theirlarger diameter, DWCNTs are easily produced and can preventundesirable kinking and bucking. So DWCNT may be the morepromising materials for mass sensor than SWCNT.

Finally, we compare the difference between the results of thenatural frequencies when adopting the nonlocal EBT and TBT.The former neglects shear deformation and rotary inertia. A com-parison of the frequency ratio fNT/fNE for a DWCNT with e0a/L¼ 0.1 and m¼ 10 is demonstrated in Fig. 7. Here, the sub-scripts NT and NE represent the nonlocal TBT and EBT,respectively. From Fig. 7, the values of fNT/fNE are alwaysless than unity. This means that the frequencies based on thenonlocal EBT are overestimated, as those in the classicaltheory. In particular, we find that frequencies using the nonlo-cal EBT overestimated in the largest extent for L2=L1 ¼ 0:5

Fig. 6 The effect of small scale and attached mass on the fundamental fre-quency of the DWCNT sensor versus length ratio with L1 5 14 nm (a) e0a/L 5 0.0(b) e0a/L 5 0.3

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than those for L2=L1 ¼ 0; 1. In other words, when shear defor-mation and rotary inertia are taken into account, the nonlocalTBT is more adequate than the nonlocal EBT, especially forDWCNT with inner and outer tubes of different lengths.This mainly results from the essential drawback of the Euler-Bernoulli hypothesis, irrespective of the nonlocal and classicalmodels. The difference between TBT and EBT is especiallyevident for short DWCNT-based micromass sensors. However,for slender SWCNT-based micromass sensors, the frequencyfrom the nonlocal EBT still gives enough accurate results. Asa consequence, for a short DWCNT-based micromass sensor,it is accurate to employ the TBT instead of the EBT.

5 Conclusions

In this paper, the frequency response of DWCNT-based micro-mass sensor was investigated. It was modeled as a microcantilevercarrying a concentrated tip mass at the free end of the inner tube.The interaction of two tubes is described by the vdW force. Usingthe nonlocal TBT, the governing equations were derived and thefundamental frequencies were determined by the TFM. Ourresults were confirmed by the results via using the FEM. The con-clusions are drawn as follows

• Increasing the tip micromass decrease the natural frequenciesbut increases the frequency shift.

• The mass sensitivity of DWCNT-based mass sensor can beenhanced when short DWCNTs are used in mass sensors,especially for larger attached mass.

• The mass sensor based on a DWCNT with different walllengths has a noticeable advantage over that based on aSWCNT.

• The nonlocal TBT is more adequate than the nonlocal EBTsince shear deformation and rotary inertia are taken intoaccount.

Appendix A

In Eq. (34), the corresponding elements for the nonlocal TBT

are U1;2 ¼U3;4 ¼ U5;6 ¼U7;8 ¼ U9;10 ¼U11;12 ¼ 1, U2;1 ¼ Cb1þCbk2

þ w1

D , U2;4 ¼ �11þbCk2þ w1k

2

D , U2;5 ¼�w1

D , U2;8 ¼�w1k2

D , U6;1 ¼�w2

D ,

U6;4 ¼�w2k2

D , U6;5 ¼ Cb1þCbk2þ w2

D , U6;8 ¼ �11þbCk2þ w2k

2

D , U4i;4i�2

¼ 1aibð1þCk2Þ , U4i;4i�1 ¼ 1þCaib

aibð1þCk2Þ (i¼ 1,2,3), and other elements in

UðsÞ vanish.

Appendix B

In Eq. (43), the corresponding elements for the nonlocal TBT

are M1;1 ¼ M2;3 ¼ M3;5 ¼ M4;7 ¼ 1, N6;11 ¼ 1, N5;9 ¼ Ck2

a1, N5;12

¼ 1þ Ck2, N6;9 ¼ mbC, N6;10 ¼ 1þ Cbk2, R8;7 ¼ R9;1 ¼ R10;3

¼ R12;3 ¼ �R9;9 ¼ �R10;11 ¼ �R12;11 ¼ 1, R7;1¼�w2k2

a2b, R7;5¼ k2

a2

ðCþw2

b Þ, R7;8 ¼ R11;4 ¼ �R11;12 ¼ 1þ Ck2, R8;2 ¼ �w2k2, R8;6

¼ 1þ bCk2 þ w2k2, R11;1 ¼ k2

a1ðCþ w1

b Þ, R11;5 ¼ � w1k2

a1b, R11;9 ¼

� Ck2

a1, R12;2 ¼ 1þ bCk2 þ w1k

2, R12;6 ¼ �w1k2, R12;10 ¼ �1

�Cbk2 and other elements in Mb, Nb, and Rc vanish.

Appendix C

In Eqs. (34) and (43), the corresponding elements for the nonlo-cal EBT in place of those in the Appendixes A and B are�Ui;iþ1 ¼ 1 ði ¼ 1; 2; 3; 5; 6; 7; 9; 10; 11Þ, �U4;1 ¼ � C

a1� w1

a1b, �U4;3

¼ Ck2

a1þ w1k

2

a1b, �U4;5 ¼ w1

a1b, �U4;7 ¼ � w1k

2

a1b, �U8;1 ¼ w2

a2b, �U8;3 ¼ � w2k

2

a2b,

�U8;5 ¼ � Ca2� w2

a2b, �U8;7 ¼ Ck2

a2þ w2k

2

a2b, �U12;9 ¼ � C

a1, �U12;11 ¼ Ck2

a1,

�M1;1 ¼ �M2;2 ¼ �M3;5 ¼ �M4;6 ¼ 1, �N5;11 ¼ �N6;12 ¼ �1, �N5;9 ¼ �N6;10

¼ Ck2

a3, �N6;9 ¼ mC

a3, �R7;7 ¼ �R8:8 ¼ �R9;9 ¼ �R10;10 ¼ �R11;3 ¼ �R12;4 ¼�1,

�R9;1 ¼ �R10;2 ¼ �R11;11 ¼ �R12;12 ¼ 1, �R7;1 ¼ �R8;2 ¼ � w2k2

a2b, �R7;5

¼ �R8;6 ¼ Ck2

a2þ w2k

2

a2b, �R11;1 ¼ �R12;2 ¼ Ck2

a1þ w1k

2

a1b, �R11;5 ¼ �R12;6

¼ � w1k2

a1band �R11;9 ¼ �R12;10 ¼ � Ck2

a3, other elements in UðsÞ, Mb,

Nb, and Rc vanish.

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