application of nonlocal beam models to double-walled

Acta Mech 216, 165–195 (2011)DOI 10.1007/s00707-010-0362-1

Keivan Kiani

Application of nonlocal beam models to double-walledcarbon nanotubes under a moving nanoparticle.Part I: theoretical formulations

Received: 2 August 2009 / Revised: 1 April 2010 / Published online: 10 July 2010© Springer-Verlag 2010

Abstract The current work suggests mathematical models for the vibration of double-walled carbonnanotubes (DWCNTs) subjected to a moving nanoparticle by using nonlocal classical and shear deform-able beam theories. The van der Waals interaction forces between atoms of the innermost and outermost tubesare modeled by an elastic layer. The equations of motion are derived for the nonlocal double body Euler–Bernoulli, Timoshenko and higher-order beams connected by a flexible layer under excitation of a movingnanoparticle. Analytical solutions of the problem are provided for the aforementioned nonlocal beam modelswith simply supported boundary conditions. The dynamical deflections and nonlocal bending moments of theinnermost and outermost tubes are then obtained during the courses of excitation and free vibration. Finally,the critical velocities of the moving nanoparticle associated with the nonlocal beam theories are expressed interms of small-scale effect parameter, geometry, and material properties of DWCNTs.

1 Introduction

The particular nature of carbon atoms and the molecular perfection of carbon nanotubes (CNTs) provide themwith remarkably high levels of material property such as electrical and thermal conductivity, stiffness, andstrength. In most materials, the electromechanical properties are degraded by initiation of damage or defectin their structures. Nevertheless, CNTs attain values close to those predicted by theoretical models becauseof their inherent molecular perfection. These extraordinary characteristics promise a wide range of potentialapplications for CNTs such as energy storage, conductive adhesives and connectors, molecular electronics, ther-mal materials, structural composites, biomedical purposes [1,2] and nanoscale mass transport systems [3–5].Recently, Regan et al. [6] showed that local temperature gradients cause directional movement of nanoparticlesalong multiwalled carbon nanotubes (MWCNTs). In another work, Regan et al. [7] proposed CNTs as a tool fortransporting indium nanoparticles by applying a voltage gradient. It was monitored that the voltage gradient,rather than the thermal gradient, would determine the direction of mass transport. Moreover, it was shown thatmass transport along CNTs would be controllable by providing particular conditions. The CNTs can also beexploited as nanoconveyers or nanofluidic devices to deliver the extent of attoliter (10−18) volumes of liquid toparticular locations for applications such as probing cells or delivering drugs and proteins inside cells [8]. Fullrecognition of mass transport mechanisms and behavior of CNTs under such loadings would be very usefulin many applications such as more accurate design of complex electromechanical nanostructures. Therefore,

This article is lovingly dedicated to my father and mother, Amrullah Kiani and Kobra Ahmadi, whose love and encouragementsI feel every day of my life.

K. Kiani (B)Department of Civil Engineering, Sharif University of Technology, P.O. Box 11365-9313, Tehran, IranE-mail: [email protected].: +98-21-66164264Fax: +98-21-66014828

166 K. Kiani

it is very important to be aware of the effects of moving molecules or nanoparticles on the dynamic behaviorof CNTs.

Although a few experimental works have been conducted on studying fluid flow inside CNTs [5,8–10],simulations of these nanostructures under fluid flow have been carried out by many researchers [11–16] ofvarious scientific disciplines by using different methods. The commonly used techniques in analyzing thenanostructures are classified into two major classes; one class is based on atomistic simulation and the otherone is the continuum-based modeling. The popular methods in the atomistic simulations are classical moleculardynamics (CMD), molecular static (MS), tight binding molecular dynamics (TBMD), Monte Carlo method(MCM) and density functional theory (DFT). The capabilities of these methods are restricted to the numberof atoms due to their complexities in atomistic calculations. For example, the use of DFT is restricted to about100 atoms in most cases. On the contrary, MCM would be an efficient approach for analyzing a continuumwith higher than 108 atoms. However, its accuracy may be lower than that of DFT. Additionally, most of theatomistic methods are complex and involve much computational effort and time. To overcome the complexi-ties arising in such methods, the appropriate continuum mechanics models are successfully employed in themathematical modeling of CNTs [17–22]. Conducting experimental tests on CNTs to determine their mechan-ical properties is so difficult. Therefore, the mechanical properties and the geometry of the continuum-basedmodels are obtained from the results of the atomistic models. In this regard, Gupta and Batra [23] proposedan equivalent continuum structure (ECS) whose frequencies in axial, torsional and radial breathing modesare equal to those of the single-walled carbon nanotubes (SWCNTs). It is found that the ECS made of linearelastic homogeneous material is a cylindrical tube of mean radii and length equal to those of SWCNTs. Thesimulation results showed that the Young’s modulus of the material of the ECS would be 1 TPa for a wallthickness of the ECS equal to 3.4 A◦. The continuum-based models are also classified into two major groups;one is the classical continuum theory and the other is the nonlocal one. The main difference of these two the-ories is related to considering or not considering the stress effects of other points in evaluating the stress in anarbitrary point of the continuum. When the dimensions of the continuum or the wavelengths of the propagatedsound waves are comparable with the small-scale effect parameter, the classical continuum models could notusually predict the real dynamic feedback of the nanostructures since for calculating the stress in a point, theinteraction effects of that point with its surrounding points due to the interatomic bonds are not taken intoaccount. As a result, the models based on the nonlocal continuum theory have gained much popularity amongmany researchers to examine the effect of small-scale effect parameter on the static and dynamic behavior ofnanostructures [24–27].

In a pioneering work on the movement of nanoparticles due to temperature gradients, Schoen et al. [28]examined motion of solid gold nanoparticles inside CNTs subjected to a range of wall temperature gradientsusing MD technique. For temperature gradients less than a specific value, it was found that the particles move‘on tracks’ in a predictable path as they follow unique helical orbits depending on the geometry of the CNTs.Furthermore, Schoen et al. [29] investigated the thermally driven mass transport of gold nanoparticles acrossCNTs using MD. The influence of the coherent lattice vibrations on the nano-tribological effects of the slidinggold nanoparticles confined inside CNTs is also examined. The obtained results reveal that nanoparticles moveonly under the support of the breathing mode of the CNT, which is responsible for the release of gold-carboncontact points. Moreover, an increase of the nanoparticle size when confined inside armchair CNTs, could pro-duce an unpredicted reduction of the static frictional drag. In another work, Kiani and Mehri [30] investigatednanotube structures under excitation of a moving nanoparticle using nonlocal beam theories. The normalizedequations of motion of nonlocal Euler–Bernoulli, Timoshenko and higher-order beams were solved analyti-cally under simply supported conditions. The role of the small-scale effect parameter, the slenderness ratioof the nanotube and moving nanoparticle velocity on the time history of deflection as well as the dynamicamplitude factor of the nonlocal beams is scrutinized. The obtained results reveal that an appropriate nonlocalshear deformable beam theory should be used, particularly for very stocky nanotube structures acted upon bya moving nanoparticle with low levels of velocity.

A brief survey of the literature indicates that no detailed study has been reported on the effect of a movingnanoparticle on the vibration of MWCNTs until now. Addressing the need to bridge to this scientific gap, thiswork is devoted to investigate the vibration of double-walled carbon nanotubes (DWCNTs) under excitationof a moving nanoparticle employing nonlocal continuum theory. For this purpose, the van der Waals (vdW)interaction force between two adjacent CNTs is assumed to be a linear function of difference of deflectionfields of the innermost and outermost CNTs. Subsequently, DWCNTs are simulated according to nonlocaldouble beams connected by a flexible layer. Application of the nonlocal continuum theory to the classicalgoverning equations of the double Euler–Bernoulli, Timoshenko and higher-order beams under a moving

Vibration of DWCNTs under a moving nanoparticle 167

(a) (b)

Fig. 1 a Schematic representation of a finite length DWCNT under excitation of a moving nanoparticle; b Cross section of theinnermost and outermost tubes connected by transversely continuous springs of total constant Cv

nanoparticle leads to development of the nonlocal equations of motion. The assumed mode method is utilizedfor discretization of the unknown fields of the simply supported DWCNT in the spatial domain and then, theLaplace transform method is employed for solving the resulting ordinary differential equations in the timedomain. The analytical expressions of the deflection and bending moment of the innermost and outermostnanotubes are gained during the phases of excitation and free vibration. In the remainder, static response ofDWCNTs under the mass weight of the nanoparticle as well as the dynamic response of DWCNTs when themoving nanoparticle passes DWCNTs with the critical velocity are formulated for the suggested nonlocalbeam theories.

In part II of this study, a comprehensive parametric study is carried out to investigate the role of importantparameters on dynamic responses of the innermost and outermost tubes of DWCNTs under a moving nano-particle. Additionally, the capabilities of the proposed nonlocal beam models in capturing the critical velocityof the moving nanoparticle as well as the dynamic response of DWCNTs under a moving nanoparticle areexamined in some detail.

2 The vdW interaction force between two adjacent CNTs

The vdW interaction force between two atoms at a distance λ could be simulated by a Lennard-Jones potentialfunction as follows [31,32]:

Φ(λ) = 4ε

[(σλ

)12 −(σλ

)6], (1)

fi j = −dΦ

dλ= 24ε

σ

[2(σλ

)13 −(σλ

)7], (2)

or in vector form

f i j = 24ε

σ 2

[2(σλ

)14 −(σλ

)8]

λ, (3)

where the vector λ describes the position of the atom j with respect to the atom i . In the Cartesian coordinatesystem xyz with the base vectors ex , ey and ez (see Fig. 1), λ is outlined as

λ = (x2 − x1) ex + (rm2 cos(ϕ2)− rm1 cos(ϕ1)) ey + (rm2 sin(ϕ2)− rm1 sin(ϕ1)− �w) ez, (4)

where w1(x, t) and w2(x, t) denote the deflection fields of the innermost and outermost CNTs, respectively,and �w = w1(x, t) − w2(x, t). Moreover, (x1, rm1, ϕ1) and (x2, rm2 , ϕ2) are in turn the cylindrical coordi-nates of two points located in the midplan of the innermost and outermost CNTs such that 0 < xi < lb and0 < ϕi < 2π; i = 1, 2. The components of the total resultant vdW force exerted on the finite length CNTs

168 K. Kiani

are calculated from the following expressions:

Fx = 1

lb

lb∫0

lb∫0

2π∫0

2π∫0

24 ε σ 2C N T

σ 2

[2(σλ

)14 −(σλ

)8](x2 − x1) dϕ1 dϕ2 dx1 dx2,

Fy = 1

lb

lb∫0

lb∫0

2π∫0

2π∫0

24 ε σ 2C N T

σ 2

[2(σλ

)14 −(σλ

)8](rm2 cos(ϕ2)− rm1 cos(ϕ1)) dϕ1 dϕ2 dx1 dx2,

Fz = 1

lb

lb∫0

lb∫0

2π∫0

2π∫0

24 ε σ 2C N T

σ 2

[2(σλ

)14 −(σλ

)8](rm2 sin(ϕ2)− rm1 sin(ϕ1)− �w) dϕ1 dϕ2 dx1 dx2,

(5)

in which σC N T represents the surface density of the carbon atoms. According to the work of Saito et al. [33],σC N T = 9a2/(4

√3) where a is the length of the C–C bond. Evaluation of the integrals in Eq. (5) reveals that

Fx = Fy = 0. Since the small deformation of DWCNTs due to a moving nanoparticle is of concern, Fz couldbe estimated appropriately by the Taylor expansion up to the first-order around the equilibrium position (i.e,w1 = w2 = 0)

Fz = Cv �w, (6)

where

Cv = −81 rm1 rm2 ε a4

×lb∫

0

lb∫0

2π∫0

2π∫0

{σ 12 [χ−7 − 14χ−8(rm2 sin(ϕ2)− rm1 sin(ϕ1))

2]−12σ

6 [χ−4 − 8χ−5(rm2 sin(ϕ2)− rm1 sin(ϕ1))2]

}dϕ1 dϕ2 dx1 dx2, (7)

χ = r2m1

+ r2m2

− 2 rm1 rm2 cos(ϕ2 − ϕ1)+ (x2 − x1)2,

the integrals in Eq. (7) are evaluated numerically by the Gauss-quadrature integration scheme up to the requiredlevel of accuracy. Equation (6) could be efficiently used in the mathematical modeling of DWCNTs takinginto account the vdW interaction forces. For this purpose, a DWCNT is modeled as a double-beam connectedtransversely by linear continuous springs of constant Cv based on the nonlocal continuum theory.

3 Mathematical modeling and analytical solutions of the problem

Different beam models according to the nonlocal continuum theory are employed to explore vibration ofDWCNTs subjected to a moving nanoparticle. To this end, consider a DWCNT consist of two single cylin-drical tubes of mean radius rmi , i = 1, 2, thickness h, length lb, cross section area Abi , modulus of elasticityEbi and shear modulus of elasticity Gbi . A moving nanoparticle of mass weight mg starts to move at theleft-hand end of the DWCNT with the constant velocity v (see Fig. 1a). The two tubes interact with each otherby the vdW forces between their atoms. This phenomenon has been modeled by a continuous spring system ofconstant Cv for infinitesimal deformations as discussed in the previous section (see Fig. 1b). The only appliedload is the mass weight of the moving nanoparticle on the DWCNT, and vibration of the DWCNT only in thez direction is of concern. Additionally, it is assumed that the moving nanoparticle would be in contact with theinner surface of the innermost tube during traveling inside the DWCNT and the mass weight of the movingnanoparticle would be negligible in comparing with the mass weight of the DWCNT; hence, the inertial effectsof the moving nanoparticle would be insignificant during passing the DWCNT.

In the following parts, the equations of motion of DWCNTs under excitation of a moving nanoparticle arederived according to the nonlocal Euler–Bernoulli beam theory (NEBT), nonlocal Timoshenko beam theory(NTBT) and nonlocal higher-order beam theory (NHOBT). Subsequently, analytical solutions to the governingequations are proposed for nonlocal beam models with simply supported conditions.


3.1 Modeling DWCNTs acted upon by a moving nanoparticle based on the NEBT

3.1.1 The equations of motion

For the classical double Euler–Bernoulli beams which are connected continuously by lateral springs under amoving nanoparticle, the transverse equations of motion are provided by [34]

ρb1

(Ab1w

E1 − Ib1w

E1,xx

)+ Cv

(wE

1 − wE2

)− M E

b1,xx = mg δ(x − xm)H(lb − xm),(8)

ρb2

(Ab2w

E2 − Ib2w

E2,xx

)− Cv

(wE

1 − wE2

)− M E

b2,xx = 0,

where δ and H denote the delta and Heaviside step functions, respectively. According to the nonlocal con-tinuum theory of Eringen [35–37], the nonlocal bending moments within the tubes modeled based on theEuler–Bernoulli beam theory are expressed as [24]

M Eb1

− (e0a)2 M Eb1,xx = −Eb1 Ib1w

E1,xx , (9)

M Eb2

− (e0a)2 M Eb2,xx = −Eb2 Ib2w

E2,xx ,

in which e0 is a constant associated with the material of the nanostructure. The value of e0a, small-scale effectparameter, could be determined from experimentally observed data or by matching dispersion curves withthose of atomic models [37]. By substituting M E

b1and M E

b2from Eq. (8) into Eq. (9), the nonlocal bending

moments in terms of displacement fields are given by:

M Eb1

=−Eb1 Ib1wE1,xx + (e0a)2

[ρb1

(Ab1w

E1 − Ib1w

E1,xx

)+ Cv

(wE

1 −wE2

)−mg δ(x−xm)H(lb−xm)

],

M Eb2

=−Eb2 Ib2wE2,xx + (e0a)2

[ρb2

(Ab2w

E2 − Ib2w

E2,xx

)−Cv

(wE

1 −wE2

)], (10)

substituting of Eq. (10) into Eq. (8) leads to the nonlocal equations of motion of the DWCNT under a movingnanoparticle based on the NEBT:

ρb1 Ab1

[wE

1 −(e0a)2wE1,xx

]−ρb1 Ib1

[wE

1,xx −(e0a)2wE1,xxxx

]+ Cv

[(wE

1 −wE2

)−(e0a)2

(wE

1,xx −wE2,xx

)]

+Eb1 Ib1wE1,xxxx = mg

[δ(x−xm)−(e0a)2δ,xx (x−xm)

]H(lb−xm),

ρb2 Ab2

[wE

2 −(e0a)2wE2,xx

]−ρb2 Ib2

[wE

2,xx −(e0a)2wE2,xxxx

]−Cv

[(wE

1 −wE2

)−(e0a)2

(wE

1,xx −wE2,xx

)]

+Eb2 Ib2wE2,xxxx = 0. (11)

The following dimensionless parameters are introduced for analyzing of the problem in a general framework:

ξ = x

lb, wE

1 = wE1

lb, wE

2 = wE2

lb, τ = 1

l2b

√Eb1 Ib1

ρb1 Ab1

t, μ = e0a

lb, λ1 = lb

rb1

, (12)

where rb1 represents the gyration radius of the innermost tube and μ is named normalized small-scale effectparameter. Therefore, the nonlocal non-dimensional equations of motion are obtained as

wE1,ττ−

1

λ21

wE1,ττξξ−μ2wE

1,ττξξ +(μ

λ1

)2

wE1,ττξξξξ + κE

[(wE

1 −wE2

)−μ2

(wE

1,ξξ−wE2,ξξ

)]+ wE

1,ξξξξ

= fE [δ(ξ−ξm)−μ2 δ,ξξ (ξ−ξm)] H(1−ξm), (13)

�21 w

E2,ττ−

�22

λ21

wE2,ττξξ−�2

1 μ2wE

2,ττξξ +(�2 μ

λ1

)2

wE2,ττξξξξ−κE

[(wE

1 −wE2

)−μ2

(wE

1,ξξ−wE2,ξξ

)]

+�23 w

E2,ξξξξ = 0,

where

λ1 = lbrb1

, �21 = ρb2 Ab2

ρb1 Ab1

, �22 = ρb2 Ib2

ρb1 Ib1

, �23 = Eb2 Ib2

Eb1 Ib1

, κE = Cv l4b

Eb1 Ib1

, fE = mgl2

b

Eb1 Ib1

. (14)

170 K. Kiani

By introducing the dimensionless parameters in Eqs. (12) and (14) to Eq. (10), the nonlocal bending momentswithin the innermost and outermost CNTs are readily derived:

M Eb1

= Eb1 Ib1

lb

[−wE

1,ξξ + μ2

(wE

1,ττ − 1

λ21

wE1,ττξξ + κE

(wE

1 − wE2

)− f

Eδ(ξ − ξm)H(1 − ξm)

)],

(15)

M Eb2

= Eb1 Ib1

lb

[−�2

3wE2,ξξ + μ2

(�2

1wE2,ττ − �2

2

λ21

wE2,ττξξ − κE

(wE

1 − wE2

))],

3.1.2 Solution to the governing equations

In order to examine vibration of the DWCNT acted upon by a moving nanoparticle, the assumed mode methodis employed for spatial discretization of the unknown fields; for this purpose, the dimensionless deflectionfields of the innermost and outermost CNTs are approximated by

wE1 (ξ, τ ) =

∞∑n=1

aEn (τ ) φ

w1n (ξ),

wE2 (ξ, τ ) =

∞∑n=1

bEn (τ ) φ

w2n (ξ), (16)

where aEn and bE

n are the unknown coefficients associated with the mode shapes of the innermost and outermostCNTs, namely φw1

n and φw2n , correspondingly. The mode shapes are constructed such that meet the boundary

conditions of the problem exactly. In the case of simply supported boundary conditions for both CNTs

wEi (0, τ ) = wE

i (1, τ ) = 0; i = 1, 2,(17)

M Ebi(0, τ ) = M E

bi(1, τ ) = 0,

the mode shapes φw1n (ξ) = sin(nπξ) and φw2

n (ξ) = sin(nπξ) are presumed as the nth deflection mode shapesof the DWCNT. Moreover,

δ(ξ − ξm)− μ2δ,ξξ (ξ − ξm) = 2∞∑

n=1

(1 + (nπμ)2) sin(nπξm) sin(nπξ). (18)

Substituting Eq. (16) into Eq. (13) and using Eq. (18) yields the following set of ordinary differential equations(ODEs):

{aE

n,ττbE

n,ττ

}+[Γ1n Γ2n

Γ3n Γ4n

]{aE

nbE

n

}={βE

n sin(gEn τ)

0

}, (19)

with the initial conditions {aE

n (0), bEn (0)

}={

aEn,τ (0), bE

n,τ (0)}

= {0, 0} , (20)

where

Γ1n = κE (1+(nπμ)2)+(nπ)4(1+(nπμ)2)

(1+(

nπλ1

)2) , Γ2n = − κE

1+(

nπλ1

)2 ,

Γ3n = − κE

�21+(

nπ�2λ1

)2 , Γ4n = κE (1+(nπμ)2)+�23(nπ)

4

(1+(nπμ)2)(�2

1+(

nπ�2λ1

)2) ,

βEn = 2 f

E

1+(

nπλ1

)2 , gEn = nπvlb

√ρb1 Ab1Eb1 Ib1

.

(21)


Employing Laplace transform to solve the set of ODEs in Eq. (19) leads to

L(

aEn

)= βE

n gEn

(s2 + Γ4n

)ΔE

n (s)(

s2 + (gEn

)2) ,

L(

bEn

)= −βE

n gEn Γ3n

ΔEn (s)

(s2 + (gE

n

)2) , (22)

where

ΔEn (s) = (s2 + Γ1n )(s

2 + Γ4n )− Γ2nΓ3n . (23)

It can be shown that ΔEn (s) = (s2 + (r E

1n)2)(s2 + (r E

2n)2), in which r E

1nand r E

2nare

r E1n

=√(Γ1n + Γ4n )/2 −

√(Γ1n − Γ4n )

2/4 + Γ2nΓ3n ,

(24)

r E2n

=√(Γ1n + Γ4n )/2 +

√(Γ1n − Γ4n )

2/4 + Γ2nΓ3n .

The splitting up of the ratios in Eq. (22) is necessary to produce simpler ratios from which inverse Laplacetransforms can be obtained more conveniently. Therefore,

L(

aEn

)= AE

1n

s2 +(

r E1n

)2 + AE2n

s2 +(

r E2n

)2 + AE3n

s2 + (gEn

)2 ,(25)

L(

bEn

)= B E

1n

s2 +(

r E1n

)2 + B E2n

s2 +(

r E2n

)2 + B E3n

s2 + (gEn

)2 ,

where

AE1n

=βE

n gEn

(Γ4n − (r E

1n)2)

((r E

2n

)2 −(

r E1n

)2)((

gEn

)2 −(

r E1n

)2) , B E

1n= − βE

n gEn Γ3n((

r E2n

)2 −(

r E1n

)2)((

gEn

)2 −(

r E1n

)2) ,

AE2n

= −βE

n gEn

(Γ4n −

(r E

2n

)2)

((r E

2n

)2 −(

r E1n

)2)((

gEn

)2 −(

r E2n

)2) , B E

2n= βE

n gEn Γ3n((

r E2n

)2 −(

r E1n

)2)((

gEn

)2 −(

r E2n

)2) ,

AE3n

=βE

n gEn

(Γ4n − (gE

n

)2)((

r E1n

)2 − (gEn

)2)((r E

2n

)2 − (gEn

)2) , B E3n

= − βEn gE

n Γ3n((r E

1n

)2 − (gEn

)2)((r E

2n

)2 − (gEn

)2) .

(26)

applying the inverse Laplace transform to Eq. (25), the dynamic deflections of the DWCNT during the courseof excitation (ξm < 1) are easily obtained as

wE1 (ξ, τ ) =

∞∑n=1

[AE

1n

r E1n

sin(

r E1nτ)

+ AE2n

r E2n

sin(

r E2nτ)

+ AE3n

gEn

sin(

gEn τ)]

sin(nπξ),

(27)

wE2 (ξ, τ ) =

∞∑n=1

[B E

1n

r E1n

sin(

r E1nτ)

+ B E2n

r E2n

sin(

r E2nτ)

+ B E3n

gEn

sin(

gEn τ)]

sin(nπξ).

172 K. Kiani

In order to analyze the dynamic deflection of the DWCNT based on the NEBT during the course of freevibration (i.e., the second phase of vibration), the following set of ODEs should be solved:{

aEn,ττ

bEn,ττ

}+[Γ1n Γ2n

Γ3n Γ4n

]{aE

nbE

n

}={

00

}, τ > τ E

f , (28)

where τ Ef = 1

vlb

√Eb1 Ib1ρb1 Ab1

. The requirement of the continuity of the deflection and deflection angle and their

velocities for each tube at the end of the excitation phase (i.e., ξm = 1) leads to the following initial boundaryconditions:

W E1n

= aEn

(τ E

f

)= AE

1n

r E1n

sin(

r E1nτ E

f

)+ AE

2n

r E2n

sin(

r E2nτ E

f

)+ AE

3n

gEn

sin(

gEn τ

Ef

),

(29)

W E2n

= bEn

(τ E

f

)= B E

1n

r E1n

sin(

r E1nτ E

f

)+ B E

2n

r E2n

sin(

r E2nτ E

f

)+ B E

3n

gEn

sin(

gEn τ

Ef

),

and

W E1n

= aEn,τ

(τ E

f

)= AE

1ncos(

r E1nτ E

f

)+ AE

2ncos(

r E2nτ E

f

)+ AE

3ncos(

gEn τ

Ef

),

(30)W E

2n= bE

n,τ

(τ E

f

)= B E

1ncos(

r E1nτ E

f

)+ B E

2ncos(

r E2nτ E

f

)+ B E

3ncos(

gEn τ

Ef

).

Utilizing Laplace transform to solve the set of ODEs in Eq. (28), one may arrive at

L(aEn ) =

(sW E

1n+ W E

1n

) (s2 + Γ4n

)− Γ2n

(sW E

2n+ W E

2n

)(

s2 + (r E1n)2) (

s2 + (r E2n)2) ,

L(bEn ) =

(sW E

2n+ W E

2n

) (s2 + Γ1n

)− Γ3n

(sW E

1n+ W E

1n

)(

s2 + (r E1n)2) (

s2 + (r E2n)2) ,

(31)

by decomposing Eq. (31) into simpler ratios and taking the inverse Laplace transform of the resulting expres-sions, the dynamic response of the DWCNT during the course of free vibration (i.e., ξm > 1) could be derivedas follows:

wE1 (ξ, τ

′) =∞∑

n=1

[A′E

1n

r E1n

sin(

r E1nτ ′)+ A′E

2n

r E2n

sin(

r E2nτ ′)+ A′E

3ncos(

r E1nτ ′)+ A′E

4ncos(

r E2nτ ′)]

sin(nπξ),

wE2 (ξ, τ

′) =∞∑

n=1

[B ′E

1n

r E1n

sin(

r E1nτ ′)+ B ′E

2n

r E2n

sin(


3ncos(


4ncos(

r E2nτ ′)]

sin(nπξ),

(32)

in which τ ′ = τ − τ Ef ,

A′E1n

=(

W E1nΓ4n − W E

1nΓ2n − W E

1n

(r E

1n

)2)/((

r E2n

)2 −(

r E1n

)2),

A′E2n

= −(

W E1nΓ4n − W E

1nΓ2n − W E

1n

(r E

2n

)2)/((

r E2n

)2 −(

r E1n

)2),

(33)

A′E3n

=(

W E1nΓ4n − W E

1nΓ2n − W E

1n

(r E

1n

)2)/((

r E2n

)2 −(

r E1n

)2),

A′E4n

= −(

W E1nΓ4n − W E

1nΓ2n − W E

1n

(r E

2n

)2)/((

r E2n

)2 −(

r E1n

)2),


and

B ′E1n

=(

W E2nΓ1n − W E

1nΓ3n − W E

2n

(r E

1n

)2)/((

r E2n

)2 −(

r E1n

)2),

B ′E2n

= −(

W E2nΓ1n − W E

1nΓ3n − W E

2n

(r E

2n

)2)/((

r E2n

)2 −(

r E1n

)2),

(34)

B ′E3n

=(

W E2nΓ1n − W E

1nΓ3n − W E

2n

(r E

1n

)2)/((

r E2n

)2 −(

r E1n

)2),

B ′E4n

= −(

W E2nΓ1n − W E

1nΓ3n − W E

2n

(r E

2n

)2)/((

r E2n

)2 −(

r E1n

)2).

3.2 Modeling DWCNTs acted upon by a moving nanoparticle based on the NTBT


For a classical Timoshenko beam under a moving load which is continuously connected to another Timoshenkobeam by springs of constant Cv , the transverse equations of motion are written as [38]

ρb1 Ab1wT1 − QT

b1,x + Cv(wT

1 − wT2

)= mg δ(x − xm) H(lb − xm),

ρb1 Ib1 θT1 − QT

b1+ MT

b1,x = 0,(35)

ρb2 Ab2wT2 − QT

b2,x − Cv(wT

1 − wT2

)= 0,

ρb2 Ib2 θT2 − QT

b2+ MT

b2,x = 0,

where θTi , QT

biand MT

biare the deflection angle, resultant shear force and bending moment of the i th tube,

correspondingly. In the context of the classical (or local) Timoshenko beam theory

(Ql

bi

)T = ksi Gbi Abi

(wT

i,x − θT)

; i = 1, 2,(Ml

bi

)T = −Ebi Ibi θTi,x ,

(36)

in which the parameter with the superscript l denotes that the parameter is defined in the framework of localcontinuum theory. According to the nonlocal continuum theory of Eringen [35–37]

QTbi

− (e0a)2 QTbi ,xx =

(Ql

bi

)T ; i = 1, 2,(37)

MTbi

− (e0a)2 MTbi ,xx =

(Ml

bi

)T,

from Eqs. (35)–(37), the nonlocal versions of the resultant shear force and bending moment within each CNTcould be outlined as follows:

QTb1

= ks1 Gb1 Ab1

(wT

1,x − θT1

)+ (e0a)2

[ρb1 Ab1w

T1,x − mg δ,x (x − xm) H(lb − xm)

],

MTb1

= −Eb1 Ib1θT1,x +(e0a)2

[ρb1 Ab1w

T1 −ρb1 Ib1 θ

T1,x +Cv (w

E1 − wE

2 )− mg δ(x − xm) H(lb − xm)],

QTb2

= ks2 Gb2 Ab2(wT2,x − θT

2 )+ (e0a)2 ρb2 Ab2wT2,x ,

MTb2

= −Eb2 Ib2θT2,x + (e0a)2

[ρb2 Ab2w

T2 − ρb2 Ib2 θ

T2,x − Cv (w

E1 − wE

2 )].

(38)

174 K. Kiani

By substituting Eq. (38) into Eq. (35), the nonlocal equations of motion of the DWCNT under excitation of amoving nanoparticle based on the NTBT are obtained:

ρb1 Ab1

[wT

1 −(e0a)2wT1,xx

]−ks1 Gb1 Ab1

(wT

1,xx −θT1,x

)+ Cv

[(wT

1 −wT2

)−(e0a)2

(wT

1,xx −wT2,xx

)]

= mg[δ(x−xm)−(e0a)2δ,xx (x−xm)

]H(lb−xm),

ρb1 Ib1

[θT

1 −(e0a)2θT1,xx

]−ks1 Gb1 Ab1

(wT

1,x −θT1

)−Eb1 Ib1θ

T1,xx = 0, (39)

ρb2 Ab2

[wT

2 −(e0a)2wT2,xx

]−ks2 Gb2 Ab2

(wT

2,xx −θT2,x

)−Cv

[(wT

1 −wT2

)−(e0a)2

(wT

1,xx −wT2,xx

)]= 0,

ρb2 Ib2

[θT

2 −(e0a)2θT2,xx

]−ks2 Gb2 Ab2

(wT

2,x −θT2

)−Eb2 Ib2θ

T2,xx = 0.

By defining the nondimensional quantities

wT1 = wT

1

lb, wT

2 = wT2

lb, μ = e0a

lb, τ = 1

lb

√ks1 Gb1

ρb1

t, η = Eb1 Ib1

ks1 Gb1 Ab1l2b

, (40)

the governing equations of the problem could be expressed in terms of the dimensionless deflection anddeflection angle as

wT1,ττ − μ2wT

1,ττξξ −(wT

1,ξξ − θT1,ξ

)+ κT

[(wT

1 − wT2

)− μ2

(wT

1,ξξ − wT2,ξξ

)]

= fT [δ(ξ − ξm)− μ2δ,ξξ (ξ − ξm)

]H(1 − ξm),

1

λ21

(θ

T1,ττ − μ2θ

T1,ττξξ

)−(wT

1,ξ − θT1

)− η θT

1,ξξ = 0, (41)

�21

(wT

2,ττ − μ2wT2,ττξξ

)− �2

4

(wT

2,ξξ − θT2,ξ

)− κT

[(wT

1 − wT2

)− μ2

(wT

1,ξξ − wT2,ξξ

)]= 0,

�22

λ21

(θ

T2,ττ − μ2θ

T2,ττξξ

)− �2

4

(wT

2,ξ − θT2

)− η�2

3 θT2,ξξ = 0,

where

�24 = ks2 Gb2 Ab2

ks1 Gb1 Ab1

, κT = Cv l2b

ks1 Gb1 Ab1

, fT = mg

ks1 Gb1 Ab1

. (42)

Moreover, by utilizing Eqs. (38), (40) and (42), the nonlocal bending moments within the CNTs based on theNTBT are rewritten as

MTb1

= ks1 Gb1 Ab1 lb

{−η θT

1,ξ + μ2

[wT

1,ττ−1

λ21

θT1,ττξ + κT

(wT

1 −wT2

)− f

Tδ(ξ−ξm) H(1−ξm)

]},

(43)

MTb2

= ks1 Gb1 Ab1 lb

{−η �2

3θT2,ξ + μ2

[�2

4wT2,ττ−

�24

λ22

θ2,ττξ−κT(wT

1 −wT2

)]},

in which λ2 = lbrb2

and the parameter rb2 denotes the gyration radius of the outermost CNT.


The assumed mode method is employed for discretization of the unknown fields of the problem in the spatialdomain. Hence,

wT1 (ξ, τ ) =

∞∑n=1

aTn (τ )φ

w1n (ξ), θ

T1 (ξ, τ ) =

∞∑n=1

bTn (τ )φ

θ1n (ξ),

(44)

wT2 (ξ, τ ) =

∞∑n=1

cTn (τ )φ

w2n (ξ), θ

T2 (ξ, τ ) =

∞∑n=1

dTn (τ )φ

θ2n (ξ),


where aTn , bT

n , cTn , and dT

n are the unknown coefficients and the functions φwin and φθin represent the nth mode

shapes associated with the deflection and deflection angle of the i th CNT, correspondingly. In the case ofsimply supported boundary conditions for both innermost and outermost CNTs,

wTi (0, τ ) = wT

i (1, τ ) = 0; i = 1, 2,(45)

θTi,ξ (0, τ ) = θ

Ti,ξ (1, τ ) = 0.

Using Eq. (43), the boundary conditions in Eq. (45) could be rewritten in a simpler form:

wTi (0, τ ) = wT

i (1, τ ) = 0; i = 1, 2,(46)

θTi,ξ (0, τ ) = θ

Ti,ξ (1, τ ) = 0.

The parameters φwin (ξ) = sin(nπξ) and φθin (ξ) = cos(nπξ), i = 1, 2, are assumed as the appropriate mode

shapes of the simply supported DWCNTs, which satisfy the conditions in Eq. (46). Hence, substitution ofEq. (44) into Eq. (41) and utilizing Eq. (18) lead to the following set of ODEs:⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

aTn,ττ

bTn,ττ

cTn,ττ

dTn,ττ

⎫⎪⎪⎪⎪⎬⎪⎪⎪⎪⎭

+

⎡⎢⎢⎢⎣ν1n ν2n ν3n 0

ν4n ν5n 0 0

ν6n 0 ν7n ν8n

0 0 ν9n ν10n

⎤⎥⎥⎥⎦

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

aTn

bTn

cTn

dTn

⎫⎪⎪⎪⎪⎬⎪⎪⎪⎪⎭

=

⎧⎪⎪⎪⎨⎪⎪⎪⎩

βTn sin(gT

n τ)

0

0

0

⎫⎪⎪⎪⎬⎪⎪⎪⎭, (47)

with the initial conditions{aT

n (0), bTn (0), cT

n (0), dTn (0)

}={

aTn,τ (0), bT

n,τ (0), cTn,τ (0), dT

n,τ (0)}

= {0, 0, 0, 0} , (48)

where

ν1n = κT (1 + (nπμ)2)+ (nπ)21 + (nπμ)2 , ν2n = − nπ

1 + (nπμ)2 ,

ν3n = −κT , ν4n = − nπλ21

1 + (nπμ)2 ,

ν5n = λ21(1 + η(nπ)2)1 + (nπμ)2 , ν6n = −κ

T

�21

,

ν7n = κT (1 + (nπμ)2)+ (nπ�3)2

1 + (nπμ�1)2, ν8n = − nπ�2

4

1 + (nπμ�1)2,

ν9n = − nπλ21�

24

1 + (nπμ�2)2, ν10n = λ2

1(�24 + η(nπ�3)

2)

1 + (nπμ�2)2,

βTn = 2 f

T, gT

n = nπv√

ρb1

ks1 Gb1

.

(49)

The Laplace transform is used for solving Eq. (47) in the time domain:

L(

aTn

)= βT

n gTn

ΔTn (s)

(s2 + (gT

n

)2) × [s6 + (ν5n + ν7n + ν10n

)s4 + (ν5nν10n + ν5nν7n − ν8nν9n + ν7nν10n

)s2

+ν5n

(ν7nν10n − ν8nν9n

)],

L(

bTn

)= −β

Tn gT

n

[ν4n s4 + ν4n (ν7n + ν10n )s

2 + ν4n


)]ΔT

n (s)(

s2 + (gTn

)2) , (50)

L(

cTn

)= −β

Tn gT

n

[ν6n s4 + ν6n

(ν5n + ν10n

)s2 + ν5nν6nν10n

]ΔT

n (s)(

s2 + (gTn

)2) ,

L(

dTn

)= βT

n gTn

(ν6nν9n s2 + ν5nν6nν9n

)ΔT

n (s)(

s2 + (gTn

)2) ,

176 K. Kiani

where

ΔTn (s) = s8 + (ν1n + ν5n + ν7n + ν10n )s

6 + (−ν2nν4n + ν1nν5n − ν3nν6n + ν1nν7n + ν5nν7n − ν8nν9n

+ν1nν10n + ν5nν10n + ν7nν10n )s4 + (−ν3nν5nν6n − ν2nν4nν7n + ν1nν5nν7n − ν1nν8nν9n

−ν5nν8nν9n − ν2nν4nν10n + ν1nν5nν10n − ν3nν6nν10n + ν1nν7nν10n + ν5nν7nν10n )s2

+(ν2nν4nν8nν9n − ν1nν5nν8nν9n − ν3nν5nν6nν10n − ν2nν4nν7nν10n + ν1nν5nν7nν10n ). (51)

Taking z = s2,ΔTn (s) is reduced to a quartic equation in terms of z whose roots could be expressed ana-

lytically [39]. The resulting quartic equation has four real negative roots. Therefore, all the roots associatedwith ΔT

n (s) would be imaginary numbers in the form ±ir T1n,±ir T

2n,±ir T

3n, and ±ir T

4n. Hence, Eq. (51) can be

rewritten as

ΔTn (s) =

(s2 +

(r T

1n

)2)(

s2 +(

r T2n

)2)(

s2 +(

r T3n

)2)(

s2 +(

r T4n

)2). (52)

The ratios in Eq. (50) could be splitted up to produce simpler ratios from which inverse Laplace transformscould be readily calculated:

L(

aTn

)=

5∑I=1

ATIn

s2 +(

r TIn

)2 ,

L(

bTn

)=

5∑I=1

BTIn

s2 +(

r TIn

)2 ,

(53)

L(

cTn

)=

5∑I=1

CTIn

s2 +(

r TIn

)2 ,

L(

dTn

)=

5∑I=1

DTIn

s2 +(

r TIn

)2 ,

in which r T5n

= gTn . The Heaviside cover up method is used to obtain the partial fraction terms of

L(aTn ),L(bT

n ),L(cTn ) and L(dT

n ) in Eq. (53):

ATIn

= lims→ir T

In

(s2 +

(r T

In

)2)

L(

aTn

), BT

In= lim

s→ir TIn

(s2 +

(r T

In

)2)

L(

bTn

),

(54)

CTIn

= lims→ir T

In

(s2 +

(r T

In

)2)

L(

cTn

), DT

In= lim

s→ir TIn

(s2 +

(r T

In

)2)

L(

dTn

).

Subsequently, the coefficients ATIn, BT

In, CT

Inand DT

In, I = 1, 2, . . . , 5, are calculated from

ATIn

= ΥIn

(−(

r TIn

)6 + (ν5n + ν7n + ν10n

) (r T

In

)4 − (ν5nν10n + ν5nν7n − ν8nν9n + ν7nν10n

) (r T

In

)2

+ν5n


) ),

BTIn

= −ΥInν4n

((r T

In

)4 − (ν7n + ν10n

) (r T

In

)2 + (ν8nν9n − ν7nν10n

)), (55)

CTIn

= −ΥInν6n

((r T

In

)4 − (ν5n + ν10n

) (r T

In

)2 + ν5nν10n

),

DTIn

= ΥInν6nν9n

((r T

In

)2 − ν5n

),


where

Υ1n = βTn gT

n((r T

1n

)2 −(

r T2n

)2) ((

r T1n

)2 −(

r T3n

)2)((

r T1n

)2 −(

r T4n

)2) ((

r T1n

)2 − (gTn

)2) ,

Υ2n = βTn gT

n((r T

2n

)2 −(

r T1n

)2) ((

r T2n

)2 −(

r T3n

)2) ((

r T2n

)2 −(

r T4n

)2) ((

r T2n

)2 − (gTn

)2) ,

Υ3n = βTn gT

n((r T

3n

)2 −(

r T1n

)2) ((

r T3n

)2 −(

r T2n

)2) ((

r T3n

)2 −(

r T4n

)2) ((

r T3n

)2 − (gTn

)2) , (56)

Υ4n = βTn gT

n((r T

4n

)2 −(

r T1n

)2) ((

r T4n

)2 −(

r T2n

)2) ((

r T4n

)2 −(

r T3n

)2) ((

r T4n

)2 − (gTn

)2) ,

Υ5n = βTn gT

n((gT

n

)2 −(

r T1n

)2) ((

gTn

)2 −(

r T2n

)2) ((

gTn

)2 −(

r T3n

)2) ((

gTn

)2 −(

r T4n

)2) .

By applying the inverse Laplace transform to Eq. (53), the dynamic response of the DWCNT subjected to amoving nanoparticle would be readily determined during the course of excitation:

wT1 (ξ, τ ) =

∞∑n=1

5∑I=1

ATIn

r TIn

sin(

r TInτ)

sin(nπξ), θT1 (ξ, τ ) =

∞∑n=1

5∑I=1

BTIn

r TIn

sin(

r TInτ)

cos(nπξ),

(57)

wT2 (ξ, τ ) =

∞∑n=1

5∑I=1

CTIn

r TIn

sin(

r TInτ)

sin(nπξ), θT2 (ξ, τ ) =

∞∑n=1

5∑I=1

DTIn

r TIn

sin(

r TInτ)

cos(nπξ).

Concerning the free vibration of the DWCNT based on the NTBT, the following set of ODEs should besolved appropriately in the time domain:⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

aTn,ττ

bTn,ττ

cTn,ττ

dTn,ττ

⎫⎪⎪⎪⎪⎬⎪⎪⎪⎪⎭

+

⎡⎢⎢⎢⎣ν1n ν2n ν3n 0

ν4n ν5n 0 0

ν6n 0 ν7n ν8n

0 0 ν9n ν10n

⎤⎥⎥⎥⎦

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

aTn

bTn

cTn

dTn

⎫⎪⎪⎪⎪⎬⎪⎪⎪⎪⎭

=

⎧⎪⎪⎪⎨⎪⎪⎪⎩

0

0

0

0

⎫⎪⎪⎪⎬⎪⎪⎪⎭, τ > τ T

f (58)

with the following initial conditions

W T1n

= aTn

(τ T

f

)=

5∑I=1

ATIn

r TIn

sin(

r TInτ T

f

), ΘT

1n= bT

n

(τ T

f

)=

5∑I=1

BTIn

r TIn

sin(

r TInτ T

f

),

(59)

W T2n

= cTn

(τ T

f

)=

5∑I=1

CTIn

r TIn

sin(

r TInτ T

f

), ΘT

2n= dT

n

(τ T

f

)=

5∑I=1

DTIn

r TIn

sin(

r TInτ T

f

),

and

W T1n

= aTn,τ

(τ T

f

)=

5∑I=1

ATIn

cos(

r TInτ T

f

), ΘT

1n= bT

n,τ

(τ T

f

)=

5∑I=1

BTIn

cos(

r TInτ T

f

),

(60)

W T2n

= cTn,τ

(τ T

f

)=

5∑I=1

CTIn

cos(

r TInτ T

f

), ΘT

2n= dT

n,τ

(τ T

f

)=

5∑I=1

DTIn

cos(

r TInτ T

f

),

178 K. Kiani

where τ Tf = 1

v

√ks1 Gb1ρb1

. After application of the Laplace transform to Eq. (58) with the initial conditions in

Eqs. (59) and (60), the resulting algebraic equations take the following form:

⎡⎢⎢⎢⎢⎣

s2 + ν1n ν2n ν3n 0

ν4n s2 + ν5n 0 0

ν6n 0 s2 + ν7n ν8n

0 0 ν9n s2 + ν10n

⎤⎥⎥⎥⎥⎦

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

L (aTn

)L (bT

n

)L (cT

n

)L (dT

n

)

⎫⎪⎪⎪⎪⎬⎪⎪⎪⎪⎭

=

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

sW T1n

+ W T1n

sΘT1n

+ ΘT1n

sW T2n

+ W T2n

sΘT2n

+ ΘT2n

⎫⎪⎪⎪⎪⎬⎪⎪⎪⎪⎭. (61)

The computed expressions of L(aTn ),L(bT

n ),L(cTn ) and L(dT

n ) have been given in Appendix A.1. By splittingup the ratios of the aforementioned expressions to simpler ratios, Eqs. (108)–(111) could be reexpressed in theform of partial fraction expansion as below:

L(

aTn

)=

4∑I=1

⎛⎜⎝ s A′T

In

s2 +(

r TIn

)2 + A′′TIn

s2 +(

r TIn

)2

⎞⎟⎠ ,

L(

bTn

)=

4∑I=1

⎛⎜⎝ s B ′T

In

s2 +(

r TIn

)2 + B ′′TIn

s2 +(

r TIn

)2

⎞⎟⎠ ,

(62)

L(

cTn

)=

4∑I=1

⎛⎜⎝ sC ′T

In

s2 +(

r TIn

)2 + C ′′TIn

s2 +(

r TIn

)2

⎞⎟⎠ ,

L(

dTn

)=

4∑I=1

⎛⎜⎝ s D′T

In

s2 +(

r TIn

)2 + D′′TIn

s2 +(

r TIn

)2

⎞⎟⎠ ,

where A′TIn, A′′T

In, B ′T

In, B ′′T

In, C ′T

In, C ′′T

In, D′T

Inand D′′T

In, I = 1, . . . , 4, are calculated via the Heaviside

cover up method. The statements of these parameters have been presented in A.2. After employing the inverseLaplace transform to Eq. (62), the free vibration response of the DWCNT according to the NTBT would bereadily gained:

wT1 (ξ, τ

′) =∞∑

n=1

4∑I=1

[A′T

Incos(

r TInτ ′)+ A′′T

In

r TIn

sin(

r TInτ ′)]

sin(nπξ),

θT1 (ξ, τ

′) =∞∑

n=1

4∑I=1

[B ′T

Incos(

r TInτ ′)+ B ′′T

In

r TIn

sin(

r TInτ ′)]

cos(nπξ),

wT2 (ξ, τ

′) =∞∑

n=1

4∑I=1

[C ′T

Incos(

r TInτ ′)+ C ′′T

In

r TIn

sin(

r TInτ ′)]

sin(nπξ),

θT2 (ξ, τ

′) =∞∑

n=1

4∑I=1

[D′T

Incos(

r TInτ ′)+ D′′T

In

r TIn

sin(

r TInτ ′)]

cos(nπξ),

(63)

where τ ′ = τ − τ Tf .


3.3 Modeling DWCNTs acted upon by a moving nanoparticle based on the NHOBT


Based on the classical continuum theory for double deformable higher-order beams of Reddy [40] which areconnected by an elastic layer of constant Cv , the transverse equations of motion are expressed as

I0(1) wH1 − (α2

1 I6(1) − α1 I4(1)

)ψH

1,x − α21 I6(1) w

H1,xx − Q H

b1,x − α1 P Hb1,xx + Cv

(wH

1 − wH2

)

= mg δ(x − xm) H(lb − xm),(I2(1) − 2α1 I4(1) + α2

1 I6(1)

)ψH

1 + (α21 I6(1) − α1 I4(1)

)wH

1,x + Q Hb1

+ α1 P Hb1,x − M H

b1,x = 0,

I0(2) wH2 − (α2

2 I6(2) − α2 I4(2)

)ψH

2,x − α22 I6(2) w

H2,xx − Q H

b2,x − α2 P Hb2,xx − Cv

(wH

1 − wH2

)= 0,

(I2(2) − 2α2 I4(2) + α2

2 I6(2)

)ψH

2 + (α22 I6(2) − α2 I4(2)

)wH

2,x + Q Hb2

+ α2 P Hb2,x − M H

b2,x = 0,

(64)

where the parameters with the subscript (i), i = 1, 2, are associated with the i th CNT. The local expressionsof Q H

bi, P H

biand M H

biare

(Ql

bi

)H = κi

(ψH

i + wHi,x

), i = 1, 2,

(Pl

bi

)H = J4(i)ψHi,x − αi J6(i)

(ψH

i,x + wHi,xx

), (65)

(Ml

bi

)H = J2(i)ψHi,x − αi J4(i)

(ψH

i,x + wHi,xx

),

in which

αi = 1/(3r2oi), i = 1, 2,

κi =∫

Abi

Gbi (1 − 3αi z2) d A,

(66)In(i) =

∫Abi

ρbi zn d A, n = 0, 2, 4, 6,

Jn(i) =∫

Abi

Ebi zn d A, n = 2, 4, 6,

where roi denotes the outer radius of the i th CNT. By using the nonlocal continuum theory of Eringen [35–37]

M Hbi

− (e0a)2 M Hbi ,xx =

(Ml

bi

)H, i = 1, 2,

(67)(Q H

bi+ αi P H

bi ,x

)− (e0a)2

(Q H

bi+ αi P H

bi ,x

),xx

=(

Qlbi

)H + αi

(Pl

bi ,x

)H.

According to Eqs. (64) and (67), the nonlocal M Hbi

and Q Hbi

+ αi P Hbi ,x, i = 1, 2, in terms of the elastic

deformation fields of the double higher-order beams can be obtained as

180 K. Kiani

M Hb1

= J2(1)ψH1,x −α1 J4(1)

(ψH

1,x +wH1,xx

)+(e0a)2

[(I2(1)−α1 I4(1)

)ψH

1,x + I0(1) wH1 − α1 I4(1) w

H1,xx

+Cv(wH

1 − wH2

)− mg δ(x − xm) H(lb − xm)

],

Q Hb1

+ α1 P Hb1,x = κ1

(ψH

1 + wH1,x

)+ (α1 J4(1) − α2

1 J6(1)

)ψH

1,xx − α21 J6(1)w

H1,xxx + (e0a)2

[I0(1) w

H1,x

+ (α1 I4(1)−α21 I6(1)

)ψH

1,xx −α21 I6(1) w

H1,xxx + Cv

(wH

1,x −wH2,x

)− mg δ,x (x − xm) H(lb − xm)

],

M Hb2

= J2(2)ψH2,x −α2 J4(2)

(ψH

2,x +wH2,xx

)+(e0a)2

[(I2(2)−α2 I4(2)

)ψH

2,x + I0(2) wH2 − α2 I4(2) w

H2,xx

−Cv(wH

1 − wH2

)],

Q Hb2

+ α2 P Hb2,x = κ2

(ψH

2 + wH2,x

)+ (α2 J4(2) − α2

2 J6(2)

)ψH

2,xx − α22 J6(2)w

H2,xxx + (e0a)2

[I0(2) w

H2,x

+ (α2 I4(2) − α22 I6(2)

)ψH

2,xx − α22 I6(2) w

H2,xxx − Cv

(wH

1,x − wH2,x

)].

(68)

Through substitution of Eq. (68) into Eq. (64), the nonlocal equations of motion of the DWCNT under excitationof a moving nanoparticle based on the NHOBT are derived:

I0(1)

[wH

1 −(e0a)2wH1,xx

]−(α2

1 I6(1)−α1 I4(1)

) [ψH

1,x −(e0a)2ψH1,xxx

]−α2

1 I6(1)

[wH

1,xx −(e0a)2wH1,xxxx

]

−κ1

(ψH

1,x + wH1,xx

)+(α2

1 J6(1)−α1 J4(1)

)ψH

1,xxx + α21 J6(1)w

H1,xxxx + Cv

[(wH

1 −wH2

)−(e0a)2

(wH

1,xx −wH2,xx

)]

= mg[δ(x−xm)−(e0a)2δ,xx (x−xm)

]H(lb−xm),(

I2(1)−2α1 I4(1) + α21 I6(1)

) [ψH

1 −(e0a)2ψH1,xx

]+(α2

1 I6(1)−α1 I4(1)

) [wH

1,x −(e0a)2wH1,xxx

]

−(

J2(1)−2α1 J4(1) + α21 J6(1)

)ψH

1,xx −(α2

1 J6(1)−α1 J4(1)

)wH

1,xxx = 0,

(69)I0(2)

[wH

2 −(e0a)2wH2,xx

]−(α2

2 I6(2)−α2 I4(2)

) [ψH

2,x −(e0a)2ψH2,xxx

]−α2

2 I6(2)

[wH

2,xx −(e0a)2wH2,xxxx

]

−κ2

(ψH

2,x +wH2,xx

)+(α2

2 J6(2)−α2 J4(2)

)ψH

2,xxx +α22 J6(2)w

H2,xxxx −Cv

[(wH

1 −wH2 )−(e0a)2

(wH

1,xx −wH2,xx

)]=0,(

I2(2)−2α2 I4(2) + α22 I6(2)

) [ψH

2 −(e0a)2ψH2,xx

]+(α2

2 I6(2)−α2 I4(2)

) [wH

2,x −(e0a)2wH2,xxx

]

−(

J2(2)−2α2 J4(2) + α22 J6(2)

)ψH

2,xx −(α2

2 J6(2)−α2 J4(2)

)wH

2,xxx = 0.

The following dimensionless quantities are introduced for general analyzing of the problem regardless of thedimensions of the DWCNT:

wH1 = wH

1

lb, wH

2 = wH2

lb, ψ

H1 = ψH

1 , ψH2 = ψH

2 , τ = α1

l2b

√J6(1)

I0(1)t, (70)

and by introducing Eq. (70) into Eq. (69)

wH1,ττ − μ2wH

1,ττξξ + γ 21

(ψ

H1,ττξ − μ2ψ

H1,ττξξξ )− γ 2

2 (wH1,ττξξ − μ2wH

1,ττξξξξ

)− γ 2

3

(ψ

H1,ξ + wH

1,ξξ

)

−γ 24 ψ

H1,ξξξ+wH

1,ξξξξ+γ 25

[(wH

1 −wH2

)−μ2

(wH

1,ξξ−wH2,ξξ

)]= f

H [δ(ξ−ξm)−δ,ξξ (ξ−ξm)]H(1−ξm),ψ

H1,ττ − μ2ψ

H1,ξξττ − γ 2

6

(wH

1,ττξ − μ2wH1,ττξξξ

)+ γ 2

7

(ψ

H1 + wH

1,ξ

)− γ 2

8 ψH1,ξξ + γ 2

9wH1,ξξξ = 0,

(71)ϑ2

1

(wH

2,ττ − μ2wH2,ττξξ

)+ ϑ2

2γ21

(ψ

H2,ττξ − μ2ψ

H2,ττξξξ

)− ϑ2

3γ22

(wH

2,ττξξ − μ2wH2,ττξξξξ

)

−ϑ24γ

23

(ψ

H2,ξ + wH

2,ξξ

)− ϑ2

5γ24 ψ

H2,ξξξ + ϑ2

6wH2,ξξξξ − γ 2

5

[(wH

1 − wH2

)− μ2

(wH

1,ξξ − wH2,ξξ

)]= 0,

ϑ27ψ

H2,ττ−μ2ψ

H2,ξξττ−ϑ2

2γ26

(wH

2,ττξ−μ2wH2,ττξξξ

)+ϑ2

4γ27

(ψ

H2 +wH

2,ξ

)−ϑ2

8γ28 ψ

H2,ξξ+ϑ2

5γ29w

H2,ξξξ = 0,


where

ϑ21 = I0(2)

I0(1), ϑ2

2 = α2 I4(2) − α22 I6(2)

α1 I4(1) − α21 I6(1)

, ϑ23 = α2

2 I6(2)

α21 I6(1)

, ϑ24 = κ2

κ1, ϑ2

5 = α2 J4(2) − α22 J6(2)

α1 J4(1) − α21 J6(1)

,

ϑ26 = α2

2 J6(2)

α21 J6(1)

, ϑ27 = I2(2) − 2α2 I4(2) + α2

2 I6(2)

I2(1) − 2α1 I4(1) + α21 I6(1)

, ϑ28 = J2(2) − 2α2 J4(2) + α2

2 J6(2)

J2(1) − 2α1 J4(1) + α21 J6(1)

,

γ 21 = α1 I4(1) − α2

1 I6(1)

I0(1)l2b

, γ 22 = α2

1 I6(1)

I0(1)l2b

, γ 23 = κ1l2

b

α21 J6(1)

, γ 24 = α1 J4(1) − α2

1 J6(1)

α21 J6(1)

, (72)

γ 25 = Cv l4

b

α21 J6(1)

, γ 26 = α1 I4(1) − α2

1 I6(1)

I2(1) − 2α1 I4(1) + α21 I6(1)

, γ 27 = κ1 I0(1)l

4b

α21 J6(1) (I2(1) − 2α1 I4(1) + α2

1 I6(1) ),

γ 28 = (J2(1) − 2α1 J4(1) + α2

1 J6(1) )I0(1)l2b

α21 J6(1) (I2(1) − 2α1 I4(1) + α2

1 I6(1) ), γ 2

9 = (α1 J4(1) − α21 J6(1) )I0(1)l

2b

α21 J6(1) (I2(1) − 2α1 I4(1) + α2

1 I6(1) ), f

H = mgl2b

α21 J6(1)

.

Using Eqs. (68) and (70), the nonlocal bending moments within the innermost and outermost CNTs are rewrittenas:

M Hb1

= J2(1)

lbψ

H1,ξ − α1 J4(1)

lb

(ψ

H1,ξ + wH

1,ξξ

)+ μ2

[(I2(1) − α1 I4(1) )α

21 J6(1)

I0(1)l3b

ψH1,ξττ

+α21 J6(1)

lbwH

1,ττ − α31 I4(1) J6(1)

I0(1)l3b

wH1,ττξξ + Cv l3

b

(wH

1 − wH2

)− mg lb δ(ξ − ξm) H(1 − ξm)

],

M Hb2

= J2(2)

lbψ

H2,ξ − α2 J4(2)

lb

(ψ

H2,ξ + wH

2,ξξ

)+ μ2

[(I2(2) − α2 I4(2) )α

21 J6(1)

I0(1)l3b

ψH2,ξττ

+α21ϑ

21 J6(1)

lbwH

2,ττ − α21α2 I4(2) J6(1)

I0(1)l3b

wH2,ττξξ − Cv l3

b(wH1 − wH

2 )

].

(73)


To discretize the unknown fields of the problem, the assumed mode method is exploited as follows:

wH1 (ξ, τ ) =

∞∑n=1

aHn (τ )φ

w1n (ξ), ψ

H1 (ξ, τ ) =

∞∑n=1

bHn (τ )φ

ψ1n (ξ),

wH2 (ξ, τ ) =

∞∑n=1

cHn (τ )φ

w2n (ξ), ψ

H2 (ξ, τ ) =

∞∑n=1

d Hn (τ )φ

ψ2n (ξ),

(74)

where aHn , bH

n , cHn , and d H

n denote the unknown coefficients and the functions φwin and φψi

n represent the nthmode shapes associated with the deflection and rotation of the i th CNT, respectively. In the case of simplysupported boundary conditions for both innermost and outermost CNTs,

wHi (0, τ ) = wH

i (1, τ ) = 0, i = 1, 2,

M Hbi(0, τ ) = M H

bi(1, τ ) = 0,

(75)

φwin (ξ) = sin(nπξ) and φψi

n (ξ) = cos(nπξ), i = 1, 2, are assumed as the corresponding mode shapes ofthe simply supported DWCNTs such that they satisfy the conditions in Eq. (75). Substituting Eq. (74) intoEq. (71) and using Eq. (18) leads to the following set of ODEs:

⎡⎢⎢⎢⎣ζ1n ζ2n 0 0

ζ3n ζ4n 0 0

0 0 ζ5n ζ6n

0 0 ζ7n ζ8n

⎤⎥⎥⎥⎦

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

aHn,ττ

bHn,ττ

cHn,ττ

d Hn,ττ

⎫⎪⎪⎪⎪⎬⎪⎪⎪⎪⎭

+

⎡⎢⎢⎢⎣η1n η2n η3n 0

η4n η5n 0 0

η3n 0 η6n η7n

0 0 η8n η9n

⎤⎥⎥⎥⎦

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

aHn

bHn

cHn

d Hn

⎫⎪⎪⎪⎪⎬⎪⎪⎪⎪⎭

=

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

βHn sin(gH

n τ)

0

0

0

⎫⎪⎪⎪⎪⎬⎪⎪⎪⎪⎭, (76)

182 K. Kiani

with the initial conditions{aH

n (0), bHn (0), cH

n (0), d Hn (0)

}={

aHn,τ (0), bH

n,τ (0), cHn,τ (0), d H

n,τ (0)}

= {0, 0, 0, 0} , (77)

where

ζ1n = (1 + (nπμ)2)(1 + (nπγ2)2), ζ2n = −γ 2

1 ((nπ)+ μ2(nπ)3),

ζ3n = −γ 26 ((nπ)+ μ2(nπ)3), ζ4n = 1 + (nπμ)2,

ζ5n = (1 + (nπμ)2)(ϑ21 + (nπγ2ϑ3)

2), ζ6n = −ϑ22γ

21 ((nπ)+ μ2(nπ)3),

ζ7n = −ϑ22γ

26 ((nπ)+ μ2(nπ)3), ζ8n = ϑ2

7 (1 + (nπμ)2),η1n = (1 + (nπμ)2)γ 2

5 + (nπ)2γ 23 + (nπ)4, η2n = (nπ)γ 2

3 − (nπ)3γ 24 ,

η3n = −(1 + (nπμ)2)γ 25 , η4n = (nπ)γ 2

7 − (nπ)3γ 29 ,

η5n = γ 27 + (nπ)2γ 2

8 , η6n = (1 + (nπμ)2)γ 25 + (nπ)2ϑ2

4γ23 + (nπ)4ϑ2

6 ,

η7n = (nπ)ϑ24γ

23 − (nπ)3ϑ2

5γ24 , η8n = (nπ)ϑ2

4γ27 − (nπ)3ϑ2

5γ29 ,

η9n = ϑ24γ

27 + (nπ)2ϑ2

8γ28 , βH

n = 2 fH(1 + (nπμ)2),

gHn = nπvlb

α1

√I0(1)J6(1)

.

(78)

By applying the Laplace transform to the set of ODEs in Eq. (76), the coupled time domain equations areconverted to the uncoupled equations expressed in terms of the Laplace operator s as follows:

L(aHn ) = βH

n gHn

(s2 + (gHn )

2)ΔHn (s)

×{ζ4n (ζ5nζ8n − ζ6nζ7n )s

6 + [η5n (ζ5nζ8n − ζ6nζ7n )+ ζ4n (ζ5nη9n + η6nζ8n − ζ6nη8n − η7nζ7n )]s4

+ [η5n (ζ5nη9n + η6nζ8n − ζ6nη8n − η7nζ7n )+ ζ4n (η6nη9n − η7nη8n )]

s2 + η5n (η6nη9n − η7nη8n )

},

L(bHn ) = βH

n gHn

(s2 + (gHn )

2)ΔHn (s)

×{−ζ3n (ζ5nζ8n − ζ6nζ7n )s

6 − [η4n (ζ5nζ8n − ζ6nζ7n )+ ζ3n (ζ5nη9n + η6nζ8n − ζ6nη8n − η7nζ7n )]s4

+ [−η4n (ζ5nη9n + η6nζ8n − ζ6nη8n − η7nζ7n )− ζ3n (η6nη9n − η7nη8n )]

s2 − η4n (η6nη9n − η7nη8n )

},

L(cHn ) = βH

n gHn

(s2 + (gHn )

2)ΔHn (s)

[−η3nζ4nζ8n s4 − η3n (η5nζ8n + ζ4nη9n )s2 − η3nη5nη9n

],

L(d Hn ) = βH

n gHn

(s2 + (gHn )

2)ΔHn (s)

[η3nζ4nζ7n s4 + η3n (η5nζ7n + ζ4nη8n )s

2 + η3nη5nη8n

], (79)

where the expression of ΔHn (s) has been presented in A.3. Assessing the quartic equation associated with

ΔHn (s) reveals that ΔH

n (s) has eight imaginary roots in the form ±ir H1n,±ir H

2n,±ir H

3n, and ±ir H

4n. Hence,

ΔHn (s) =P8n (s

2 + (r H1n)2)(s2 + (r H

2n)2)(s2 + (r H

3n)2)(s2 + (r H

4n)2),

P8n =ζ1nζ4n (ζ5nζ8n − ζ6nζ7n )+ ζ3nζ2n (ζ6nζ7n − ζ5nζ8n ).(80)

Splitting up of the ratios in Eq. (79) is necessary to produce simpler ratios from which the inverse Laplacetransforms could be more conveniently calculated. Therefore,

L(

aHn

)=

5∑I=1

AHIn

s2 +(

r HIn

)2 ,

L(

bHn

)=

5∑I=1

B HIn

s2 +(

r HIn

)2 ,


L(

cHn

)=

5∑I=1

C HIn

s2 +(

r HIn

)2 ,

L(

d Hn

)=

5∑I=1

DHIn

s2 +(

r HIn

)2 ,

(81)

where r H5n

= gHn . Using the Heaviside cover up method to derive the coefficients AH

In, B H

In, C H

Inand DH

In,

I = 1, 2, . . . , 5:

AHIn

= ΛIn ×

⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩

ζ4n

(ζ6nζ7n − ζ5nζ8n

) (r H

In

)6 + [η5n (ζ5nζ8n − ζ6nζ7n )+ ζ4n

(ζ5nη9n

+ η6nζ8n − ζ6nη8n − η7nζ7n

)] (r H

In

)4 − [η5n

(ζ5nη9n + η6nζ8n − ζ6nη8n − η7nζ7n

)

+ ζ4n

(η6nη9n − η7nη8n

)] (r H

In

)2 + η5n


)

⎫⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎭,

(82)

B HIn

= ΛIn ×

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

ζ3n


) (r H

In

)6 − [η4n


)+ ζ3n

(ζ5nη9n + η6nζ8n

−ζ6nη8n − η7nζ7n

)] (r H

In

)4 + [η4n

(ζ5nη9n + η6nζ8n − ζ6nη8n − η7nζ7n

)+ζ3n


)] (r H

In

)2 − η4n


)

⎫⎪⎪⎪⎪⎬⎪⎪⎪⎪⎭, (83)

C HIn

= −ΛInη3n

(ζ4nζ8n

(r H

In

)4 − (η5nζ8n + ζ4nη9n

) (r H

In

)2 + η5nη9n

), (84)

DHIn

= ΛInη3n

(ζ4nζ7n

(r H

In

)4 − (η5nζ7n + ζ4nη8n

) (r H

In

)2 + η5nη8n

), (85)

where

Λ1n = βHn gH

n

P8n

((r H

1n

)2 −(

r H2n

)2)((

r H1n

)2 −(

r H3n

)2) ((

r H1n

)2 −(

r H4n

)2)((

r H1n

)2 − (gHn

)2) ,

Λ2n = βHn gH

n

P8n

((r H

2n

)2 −(

r H1n

)2)((

r H2n

)2 −(

r H3n

)2) ((

r H2n

)2 −(

r H4n

)2) ((

r H2n

)2 − (gHn

)2) ,

Λ3n = βHn gH

n

P8n

((r H

3n

)2 −(

r H1n

)2)((

r H3n

)2 −(

r H2n

)2) ((

r H3n

)2 −(

r H4n

)2) ((

r H3n

)2 − (gHn

)2) ,

Λ4n = βHn gH

n

P8n

((r H

4n

)2 −(

r H1n

)2)((

r H4n

)2 −(

r H2n

)2) ((

r H4n

)2 −(

r H3n

)2) ((

r H4n

)2 − (gHn

)2) ,

Λ5n = βHn gH

n

P8n

((gH

n

)2 −(

r H1n

)2)((

gHn

)2 −(

r H2n

)2) ((

gHn

)2 −(

r H3n

)2) ((

gHn

)2 −(

r H4n

)2) .

(86)

184 K. Kiani

By application of the inverse Laplace transform to Eq. (81), the deflection and rotation fields of the DWCNTaccording to the NHOBT could be determined during the course of excitation as

wH1 (ξ, τ ) =

∞∑n=1

5∑I=1

AHIn

r HIn

sin(

r HInτ)

sin(nπξ), ψH1 (ξ, τ ) =

∞∑n=1

5∑I=1

B HIn

r HIn

sin(

r HInτ)

cos(nπξ),

wH2 (ξ, τ ) =

∞∑n=1

5∑I=1

C HIn

r HIn

sin(

r HInτ)

sin(nπξ), ψH2 (ξ, τ ) =

∞∑n=1

5∑I=1

DHIn

r HIn

sin(

r HInτ)

cos(nπξ).

(87)

In order to investigate the dynamic response of the DWCNT based on the NHOBT during the course offree vibration, the following set of ODEs should be solved in the time domain:

⎡⎢⎢⎢⎣ζ1n ζ2n 0 0

ζ3n ζ4n 0 0

0 0 ζ5n ζ6n

0 0 ζ7n ζ8n

⎤⎥⎥⎥⎦

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

aHn,ττ

bHn,ττ

cHn,ττ

d Hn,ττ

⎫⎪⎪⎪⎪⎬⎪⎪⎪⎪⎭

+

⎡⎢⎢⎢⎣η1n η2n η3n 0

η4n η5n 0 0

η3n 0 η6n η7n

0 0 η8n η9n

⎤⎥⎥⎥⎦

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

aHn

bHn

cHn

d Hn

⎫⎪⎪⎪⎪⎬⎪⎪⎪⎪⎭

=

⎧⎪⎪⎪⎨⎪⎪⎪⎩

0

0

0

0

⎫⎪⎪⎪⎬⎪⎪⎪⎭, τ > τ H

f (88)

with the following initial conditions

W H1n

= aHn

(τ H

f

)=

5∑I=1

AHIn

r HIn

sin(

r HInτ H

f

), Ψ H

1n= bH

n

(τ H

f

)=

5∑I=1

B HIn

r HIn

sin(

r HInτ H

f

),

W H2n

= cHn

(τ H

f

)=

5∑I=1

C HIn

r HIn

sin(

r HInτ H

f

), Ψ H

2n= d H

n

(τ H

f

)=

5∑I=1

DHIn

r HIn

sin(

r HInτ H

f

),

(89)

and

W H1n

= aHn,τ

(τ H

f

)=

5∑I=1

AHIn

cos(

r HInτ H

f

), Ψ H

1n= bH

n,τ

(τ H

f

)=

5∑I=1

B HIn

cos(

r HInτ H

f

),

W H2n

= cHn,τ

(τ H

f

)=

5∑I=1

C HIn

cos(

r HInτ H

f

), Ψ H

2n= d H

n,τ

(τ H

f

)=

5∑I=1

DHIn

cos(

r HInτ H

f

),

(90)

in which τ Hf = α1

v lb

√J6(1)I0(1)

. After application of the Laplace transform to Eq. (88) with the initial conditions in

Eqs. (89) and (90), the following algebraic equations could be obtained:⎡⎢⎢⎢⎢⎣

ζ1n s2 + η1n ζ2n s2 + η2n η3n 0

ζ3n s2 + η4n ζ4n s2 + η5n 0 0

η3n 0 ζ5n s2 + η6n ζ6n s2 + η7n

0 0 ζ7n s2 + η8n ζ8n s2 + η9n

⎤⎥⎥⎥⎥⎦

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

L (aHn

)L (bH

n

)L (cH

n

)L (d H

n

)

⎫⎪⎪⎪⎪⎬⎪⎪⎪⎪⎭

=

⎧⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎩

(ζ1n W H

1n + ζ2n ΨH

1n

)+ s

(ζ1n W H

1n+ ζ2nΨ

H1n

)(ζ3n W H

1n+ ζ4n Ψ

H1n

)+ s

(ζ3n W H

1n+ ζ4nΨ

H1n

)(ζ5n W H

2n+ ζ6n Ψ

H2n

)+ s

(ζ5n W H

2n+ ζ6nΨ

H2n

)(ζ7n W H

2n+ ζ8n Ψ

H2n

)+ s

(ζ7n W H

2n+ ζ8nΨ

H2n

)

⎫⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎭, (91)

the explicit expressions of L(aHn ),L(bH

n ),L(cHn ) and L(d H

n ) have been provided in A.4. Following the sameprocedure mentioned in the previous part, the Laplace transforms of aH

n , bHn , c

Hn and d H

n in Eqs. (117)–(120)


could be rewritten in the form of partial fraction expansion by using the Heaviside cover up method; after that,the free vibration response of the DWCNT based on the NHOBT could be outlined as

wH1 (ξ, τ

′) =∞∑

n=1

4∑I=1

[A′H

Incos(

r HInτ ′)+ A′′H

In

r HIn

sin(

r HInτ ′)]

sin(nπξ),

ψH1 (ξ, τ

′) =∞∑

n=1

4∑I=1

[B ′H

Incos(

r HInτ ′)+ B ′′H

In

r HIn

sin(

r HInτ ′)]

cos(nπξ),

wH2 (ξ, τ

′) =∞∑

n=1

4∑I=1

[C ′H

Incos(

r HInτ ′)+ C ′′H

In

r HIn

sin(

r HInτ ′)]

sin(nπξ),

ψH2 (ξ, τ

′) =∞∑

n=1

4∑I=1

[D′H

Incos(

r HInτ ′)+ D′′H

In

r HIn

sin(

r HInτ ′)]

cos(nπξ),

(92)

in which τ ′ = τ − τ Hf ; moreover, the parameters A′H

In, A′′H

In, B ′H

In, B ′′H

In,C ′H

In,C ′′H

In, D′H

Inand D′′H

Inare given

in A.5.

4 Particular case studies

4.1 v = 0: Static loading

In this part, it is shown that the dynamic response of various nonlocal beam models could also include the staticresponse of them in the special case of the zero velocity of the nanoparticle. To this end, it is just necessaryto set coincidentally g[ ]

n = 0 and g[ ]n τ = 0 into the dynamic response of the DWCNT during the excitation

phase; moreover, the sign [ ] denotes the kind of nonlocal beam theory, E, T or H .

4.1.1 Static response of the DWCNT under the weight of nanoparticle based on the NEBT

By substituting gEn = 0 and gE

n τ = 0 into Eq. (27), the dimensionless static deflection fields of the DWCNTbased on the NEBT due to the point load mg at ξm are readily obtained as

wE1 (ξ) =

∞∑n=1

βEn Γ4n

Γ1nΓ4n − Γ2nΓ3n

sin(nπξm) sin(nπξ),

wE2 (ξ) = −

∞∑n=1

βEn Γ3n

Γ1nΓ4n − Γ2nΓ3n

sin(nπξm) sin(nπξ).

(93)

Moreover, by substituting Eq. (93) into Eq. (15), the nonlocal bending moments within each CNT which ismodeled based on the NEBT, could be explicitly stated as follows:

M Eb1(ξ) = Eb1 Ib1

lb

∞∑n=1

βEn

[(nπ)2Γ4n + μ2κE

(Γ4n + Γ3n

)]Γ1nΓ4n − Γ2nΓ3n


M Eb2(ξ) = − Eb1 Ib1

lb

∞∑n=1

βEn

[(nπ�3)

2Γ3n + μ2κE(Γ4n + Γ3n

)]Γ1nΓ4n − Γ2nΓ3n


(94)

4.1.2 Static response of the DWCNT under the weight of a nanoparticle based on the NTBT

Through replacing gTn = 0 and gT

n τ = 0 into Eq. (57), the static deformation fields of the DWCNT basedon the NTBT due to the applied weight load of the nanoparticle at ξm are derived after some manipulation asbelow:

186 K. Kiani

wT1 (ξ) = −

∞∑n=1

βTn ν5n

ν2nν4n − ν1nν5n


θT1 (ξ) = −

∞∑n=1

βTn ν4n


sin(nπξm) cos(nπξ),

wT2 (ξ) = −

∞∑n=1

βTn ν5nν6nν10n(


) (ν8nν9n − ν7nν10n

) sin(nπξm) sin(nπξ),

θT2 (ξ) = −

∞∑n=1

βTn ν5nν6nν10n(



) sin(nπξm) cos(nπξ).

(95)

By substitution Eq. (95) into Eq. (43), the nonlocal bending moment within each CNT according to the NTBTcould be explicitly expressed as:

MTb1(ξ) = −ks1 Gb1 Ab1 lb

∞∑n=1

βTn

[nπη ν4n

(ν8nν9n −ν7nν10n

)+ μ2κT ν5n

(ν8nν9n −ν6nν10n

)](ν2nν4n −ν1nν5n

) (ν8nν9n −ν7nν10n


MTb2(ξ) = −ks1 Gb1 Ab1 lb

∞∑n=1

βTn ν5n

[nπη�2

3 ν6nν10n − μ2κT (ν8nν9n − ν6nν10n

)](ν2nν4n − ν1nν5n



(96)

4.1.3 Static response of the DWCNT under the weight of nanoparticle based on the NHOBT

By setting gHn = 0 and gH

n τ = 0 into Eq. (87), the dimensionless static deflection and rotation field of theDWCNT based on the NHOBT due to the applied weight load of the nanoparticle at ξm are provided aftersome manipulation:

wH1 (ξ) =

∞∑n=1

βHn η5n


)(η1nη5n − η2nη4n

) (η6nη9n − η7nη8n

)− η23nη5nη9n


ψH1 (ξ) = −

∞∑n=1

βHn η4n




)− η23nη5nη9n

sin(nπξm) cos(nπξ),

wH2 (ξ) = −

∞∑n=1

βHn η3nη5nη9n(

η1nη5n − η2nη4n


)− η23nη5nη9n


ψH2 (ξ) =

∞∑n=1

βHn η3nη5nη8n(

η1nη5n − η2nη4n


)− η23nη5nη9n

sin(nπξm) cos(nπξ).

(97)

Additionally, substitution of Eq. (97) into Eq. (73) leads to the nonlocal bending moment within each CNTaccording to the NHOBT:

M Hb1(ξ) =

∞∑n=1

βHn

lb

([(nπ)2η5nα1 J4(1)−nπη4n

(J2(1)−α1 J4(1)

)] (η6nη9n −η7nη8n

)+μ2η5n

(η6nη9n −η7nη8n + η3nη9n

)Cv l4

b

)(η1nη5n −η2nη4n

) (η6nη9n −η7nη8n

)−η23nη5nη9n


M Hb2(ξ) = −

∞∑n=1

βHn

lb

(η3nη5n

[(nπ)2η9nα2 J4(2) − nπη8n

(J2(2) − α2 J4(2)

)]+μ2η5n

(η6nη9n − η7nη8n + η3nη9n

)Cv l4

b



)− η23nη5nη9n


(98)

When the parameters ξm and ξ are exchanged, the expressions of the deformations in Eqs. (93), (95)and (97) do not change; it means that the Betti–Maxwell’s law is even valid for the nonlocal elastic beams.Moreover, by setting μ = 0 into these equations, one may arrive at the common static response of the simplysupported DWCNTs under the weight of the nanoparticle at an arbitrary location ξm based on the classicalcontinuum theory.


4.2 g[ ]m = r [ ]

Im([ ] = E or T or H )

In this case, Eqs. (27), (57) and (87) could not be utilized explicitly to examine the dynamic response of theDWCNT because the denominators of these equations take zero values.To overcome this difficulty, the above-mentioned equations should be reconstructed from the initial expressions before application of the Laplacetransform. In the following parts without loss of generality, it is presumed that g[ ]

m = r [ ]1m

and the deformationfields of the nonlocal classical and shear deformable beams are totally reproduced once more.

4.2.1 gEm = r E

1m

By setting r E1m

= gEm in Eq. (27) and applying the inverse Laplace transform to the resulting equations, the

deflection fields of the DWCNT based on the NEBT are obtained as

wE1 (ξ, τ ) =

⎧⎨⎩

βEm

(Γ4m − (gE

m

)2)

2(gE

m

)2 ((r E

2m

)2 − (gEm

)2)[sin(

gEmτ)

−(

gEmτ)

cos(

gEmτ)]

+βE

m gEm

(Γ4m − (r2m

)2)

r E2m

((gE

m

)2 − (r E2m

)2) sin(

r E2mτ)⎫⎬⎭ sin(nπξ)

+∞∑

n=1, n �=m

[AE

1n

r E1n

sin(

r E1nτ)

+ AE2n

r E2n

sin(

r E2nτ)

+ AE3n

gEn

sin(

gEn τ)]

sin(nπξ),

wE2 (ξ, τ ) = −

⎧⎪⎪⎨⎪⎪⎩

βEmΓ3m

2(gE

m

)2 ((r E

2m

)2 − (gEm

)2)[sin(

gEmτ)

−(

gEmτ)

cos(

gEmτ)]

+ βEm gE

mΓ3m

r E2m

((gE

m

)2 −(

r E2m

)2) sin

(r E

2mτ)⎫⎪⎪⎬⎪⎪⎭

sin(nπξ)

+∞∑

n=1, n �=m

[B E

1n

r E1n

sin(

r E1nτ)

+ B E2n

r E2n

sin(

r E2nτ)

+ B E3n

gEn

sin(

gEn τ)]

sin(nπξ). (99)

4.2.2 gTm = r T

1m

Substituting r T1m

= gTm into Eq. (57) and taking the inverse Laplace transform of the resulting equations, the

deformation fields of the DWCNT based on the NTBT are provided as follows:

wT1 (ξ, τ ) = AT

1m

2(gT

m

)3[sin(

gTmτ)

−(

gTmτ)

cos(

gTmτ)]

sin(mπξ)+∞∑

n=1, n �=m

5∑I=1

ATIn

r TIn

sin(

r TInτ)

sin(nπξ),

θT1 (ξ, τ ) = BT

1m

2(gT

m

)3[sin(

gTmτ)

−(

gTmτ)

cos(

gTmτ)]

cos(mπξ)+∞∑

n=1, n �=m

5∑I=1

BTIn

r TIn

sin(

r TInτ)

cos(nπξ),

188 K. Kiani

wT2 (ξ, τ ) = CT

1m

2(gT

m

)3[sin(

gTmτ)

−(

gTmτ)

cos(

gTmτ)]

sin(mπξ)+∞∑

n=1, n �=m

5∑I=1

CTIn

r TIn

sin(

r TInτ)

sin(nπξ),

θT2 (ξ, τ ) = DT

1m

2(gT

m

)3[sin(

gTmτ)

−(

gTmτ)

cos(

gTmτ)]

cos(mπξ)+∞∑

n=1, n �=m

5∑I=1

DTIn

r TIn

sin(

r TInτ)

cos(nπξ),

(100)

where

AT1m

=βT

m gTm

⎛⎝(gT

m

)6 − (ν5m + ν7m + ν10m

) (gT

m

)4 + (ν5mν10m + ν5mν7m

−ν8mν9m + ν7mν10m

) (gT

m

)2 − ν5m

(ν7mν10m − ν8mν9m

)⎞⎠

((r T

2m

)2 − (gTm

)2) ((r T

3m

)2 − (gTm

)2) ((r T

4m

)2 − (gTm

)2) ,

BT1m

=βT

m gTmν4m

((gT

m

)4 − (ν7m + ν10m

) (gT

m

)2 + ν8mν9m − ν7mν10m

)((

r T2m

)2 − (gTm

)2) ((r T

3m

)2 − (gTm

)2) ((r T

4m

)2 − (gTm

)2) ,

CT1m

=βT

m gTmν6m

((gT

m

)4 − (ν5m + ν10m

) (gT

m

)2 + ν5mν10m

)((

r T2m

)2 − (gTm

)2) ((r T

3m

)2 − (gTm

)2) ((r T

4m

)2 − (gTm

)2) ,

DT1m

=βT

m gTmν6mν9m

(ν5m − (gT

m

)2)((

r T2m

)2 − (gTm

)2) ((r T

3m

)2 − (gTm

)2) ((r T

4m

)2 − (gTm

)2) . (101)

4.2.3 gHm = r H

1m

Substituting r H1m

= gHm into Eq. (87) and taking the inverse Laplace transform of the resulting equations lead

to the following deformation fields of the DWCNT based on the NHOBT:

wH1 (ξ, τ ) = AH

1m

2(gH

m

)3[sin(

gHm τ)−(

gHm τ)

cos(

gHm τ)]

sin(mπξ)+∞∑

n=1, n �=m

5∑I=1

AHIn

r HIn

sin(

r HInτ)

sin(nπξ),

ψH1 (ξ, τ ) = B H

1m

2(gH

m

)3[sin(

gHm τ)−(

gHm τ)

cos(

gHm τ)]

cos(mπξ)+∞∑

n=1, n �=m

5∑I=1

B HIn

r HIn

sin(

r HInτ)

cos(nπξ),

wH2 (ξ, τ ) = C H

1m

2(gH

m

)3[sin(

gHm τ)−(

gHm τ)

cos(

gHm τ)]

sin(mπξ)+∞∑

n=1, n �=m

5∑I=1

C HIn

r HIn

sin(

r HInτ)

sin(nπξ),

ψH2 (ξ, τ ) = DH

1m

2(gH

m

)3[sin(

gHm τ)−(

gHm τ)

cos(

gHm τ)]

cos(mπξ)+∞∑

n=1, n �=m

5∑I=1

DHIn

r HIn

sin(

r HInτ)

cos(nπξ),

(102)


where

AH1m

=Λ1m

×

⎧⎪⎨⎪⎩ζ4m

(ζ6m ζ7m −ζ5m ζ8m

) (gH

m

)6+[η5m


)+ζ4m

(ζ5m η9m +η6m ζ8m −ζ6m η8m −η7m ζ7m

)] (gH

m

)4

−[η5m


)+ζ4m

(η6m η9m −η7m η8m

)] (gH

m

)2+η5m


)⎫⎪⎬⎪⎭,

B H1m

=Λ1m

×

⎧⎪⎨⎪⎩ζ3m


) (gH

m

)6−[η4m


)+ζ3m


)] (gH

m

)4

+[η4m


)+ζ3m


)] (gH

m

)2−η4m


)⎫⎪⎬⎪⎭,

C H1m

= − Λ1m η3m

(ζ4m ζ8m

(gH

m

)4 − (η5m ζ8m + ζ4m η9m

) (gH

m

)2 + η5m η9m

),

DH1m

=Λ1m η3m

(ζ4m ζ7m

(gH

m

)4 − (η5m ζ7m + ζ4m η8m

) (gH

m

)2 + η5m η8m

),

(103)

in which

Λ1m = βHm gH

m

P8m

((gH

m

)2 − (r H2m

)2) ((gH

m

)2 − (r H3m

)2) ((gH

m

)2 − (r H4m

)2) . (104)

4.2.4 The concept of the critical velocity of the moving nanoparticle

As is obvious from Eqs. (99), (100) and (102), the deflections and rotations of each point of the DWCNT’smedium dramatically grow with time during excitation. Moreover, the corporation of the first mode of vibra-tion in increasing deflections and rotations would be most influential comparing with other terms. As a result,finding the velocity levels in which the quantities of the deformation fields magnify with time would be ofgreat importance.

The lowest velocity of the nanoparticle associated with the relation g[ ]m = min(r [ ]

1m, . . . , r [ ]

4m) is defined as

the critical velocity of the nanoparticle pertinent to the nonlocal [ ] beam theory, v[ ]cr . It could be shown that

the critical velocity could be sought from g[ ]1 = min(r [ ]

11, . . . , r [ ]

41). For instant, the explicit statement of the

critical velocity associated with the NEBT could be obtained as

vEcr = v′E

π2

√(Γ11 + Γ41)/2 −

√(Γ11 − Γ41)

2/4 + Γ21Γ31, (105)

where v′E denotes the critical velocity associated with the classical Euler–Bernoulli beam, v′E =πlb

√Eb1 Ib1ρb1 Ab1

[41]. Furthermore, analytical expressions of the critical velocity for the nanoparticle acted upon the

DWCNT based on the NTBT and NHOBT would not be stated in simple forms since the analytical expressionsof the roots of ΔT

1 (s) or ΔH1 (s) are more complex. Therefore,

vTcr = 1

π2 min(

r T11, r T

21, r T

31, r T

41

)v′T ,

vHcr = 1

π2 min(

r H11, r H

21, r H

31, r H

41

)v′H ,

(106)

where

v′T = π

√ks1 Gb1

ρb1

,

v′H = πα1

lb

√J6(1)

I0(1).

(107)

According to Eq. (105), it can be concluded that vEcr → v′E when λ → ∞, κE → 0 and μ → 0. It means

that by disappearing vdW interaction force and the small-scale effect parameter from a slender DWCNT, the

190 K. Kiani

predicted results of critical velocity approach to those of the innermost carbon nanotube based on the classicalcontinuum theory. In the next paper, the effects of different parameters on the critical velocities of a movingnanoparticle associated with various nonlocal beam theories are also investigated in some detail.

5 Conclusions

Vibration of DWCNTs under excitation of a moving nanoparticle is carried out by using nonlocal double bodyEuler–Bernoulli, Timoshenko and higher-order beams acted together due to the vdW interaction forces. Theexisting vdW interaction force between two CNTs is simulated by a flexible layer. The constant of the flexiblelayer is evaluated from the expansion of the Lennard-Jones potential function about the equilibrium positionfor infinitesimal deformation of DWCNTs with of finite length. The equations of motion are derived for thedouble body classical and shear deformable beams connected by a flexible layer under a moving nanoparticlewithin the framework of nonlocal continuum theory of Eringen. The analytical solutions of the problem arethen provided for the nonlocal beams with fully simply supported boundary conditions employing assumedmode method and Laplace transform approach. The critical velocities of the moving nanoparticle for differentnonlocal beam theories are introduced and then, the dynamic displacements of the innermost and outermosttubes associated with the critical velocities are obtained.

A Appendix

A.1 The expressions of L(aTn ), L(bT

n ), L(cTn ) and L(dT

n ) during the course of free vibration

L(

aTn

)= 1

ΔTn (s)

{s7W T

1n+ s6W T

1n+(ν7n W T

1n− ν2nΘ

T1n

+ ν5n W T1n

+ ν10n W T1n

− ν3n W T2n

)s5

+(−ν2n Θ

T1n

+ ν7n W T1n

− ν3n W T2n

+ ν10n W T1n

+ ν5n W T1n

)s4 +

(−ν2n ν7n Θ

T1n

+ ν5n ν7n W T1n

− ν3n ν10n W T2n

+ν3n ν8n ΘT2n

− ν3n ν5n W T2n

− ν2n ν10n ΘT1n

+ ν5n ν10n W T1n

+ ν7n ν10n W T1n

− ν8n ν9n W T1n

)s3 +

(−ν2n ν10n Θ

T1n

−ν2n ν7n ΘT1n

−ν3n ν5n W T2n

+ ν7n ν10n W T1n

−ν8n ν9n W T1n

+ ν5nν7n W T1n

−ν3n ν10n W T2n

+ ν3n ν8n ΘT2n

+ ν5n ν10n W T1n

)s2

+(ν5n ν7n ν10n W T

1n− ν5n ν8n ν9n W T

1n− ν2n ν7n ν10n Θ

T1n

+ ν2n ν8n ν9n ΘT1n


− ν3n ν5n ν10n W T2n

)s

−ν2n ν7n ν10n ΘT1n

+ ν2nν8n ν9n ΘT1n

−ν3n ν5n ν10n W T2n

+ ν5n ν7n ν10n W T1n



},

(108)

L(

bTn

)= 1

ΔTn (s)

{s7ΘT

1n+ s6ΘT

1n+(ν10n Θ

T1n

+ ν7nΘT1n

− ν4n W T1n

+ ν1n ΘT1n

)s5 +

(−ν4n W T

1n+ ν1n Θ

T1n

+ ν7n ΘT1n

+ν10n ΘT1n

)s4 +

(−ν6n ν3nΘ

T1n

+ ν1n ν10n ΘT1n

+ ν1nν7n ΘT1n

− ν4n ν7n W T1n

− ν8n ν9n ΘT1n


+ν7n ν10n ΘT1n

+ ν3n ν4n W T2n

)s3 +

(−ν4n ν10n W T

1n− ν8n ν9n Θ

T1n

+ ν1n ν10n ΘT1n

+ ν3n ν4n W T2n

− ν4n ν7n W T1n

+ν1n ν7n ΘT1n

+ ν7nν10n ΘT1n

− ν6n ν3n ΘT1n

)s2 +

(ν1n ν7n ν10nΘ

T1n

− ν4n ν3n ν8n ΘT2n


+ν4n ν8n ν9n W T1n

+ ν3nν4n ν10n W T2n

− ν1n ν8nν9n ΘT1n

− ν6n ν3n ν10nΘT1n

)s + ν1n ν7n ν10n Θ

T1n



+ ν4nν8n ν9n W T1n




}, (109)

L(

cTn

)= 1

ΔTn (s)

{s7W T

2n+ s6W T

2n+(−ν6n W T

1n+ ν1n W T

2n− ν8n Θ

T2n

+ ν5n W T2n

+ ν10n W T2n

)s5 +

(−ν6n W T

1n

+ν10n W T2n

− ν8n ΘT2n

+ ν5n W T2n

+ ν1n W T2n

)s4 +

(−ν8n ν5n Θ

T2n

− ν6n ν5n W T1n

+ ν2n ν6n ΘT1n

+ ν1n ν10n W T2n

+ν1n ν5n W T2n

+ ν5n ν10n W T2n


− ν4n ν2n W T2n

− ν8n ν1n ΘT2n

)s3 +

(−ν6n ν10n W T

1n− ν6n ν5n W T

1n

+ν1n ν10n W T2n

+ν1n ν5n W T2n

+ν2n ν6n ΘT1n

−ν8n ν1n ΘT2n

+ν5n ν10n W T2n

−ν4n ν2n W T2n

−ν8nν5n ΘT2n

)s2

+(−ν8n ν1n ν5n Θ

T2n

+ν8nν4n ν2n ΘT2n

+ν1n ν5nν10n W T2n



+ν2nν6n ν10n ΘT1n

)s

+ν2n ν6n ν10n ΘT1n


−ν8nν1n ν5n ΘT2n




}, (110)


L(

dTn

)= 1

ΔTn (s)

{s7ΘT

2n+ s6ΘT

2n+(−ν9n W T

2n+ ν1nΘ

T2n

+ ν5n ΘT2n

+ ν7nΘT2n

)s5 +

(ν1n Θ

T2n

+ ν5n ΘT2n

+ ν7n ΘT2n

−ν9n W T2n

)s4+

(ν6n ν9n W T

1n−ν6n ν3n Θ

T2n

+ν5nν7n ΘT2n

−ν9n ν5n W T2n

−ν4n ν2n ΘT2n

−ν9n ν1n W T2n

+ν1n ν7n ΘT2n

+ν1n ν5n ΘT2n

)s3+

(ν6nν9n W T

1n+ν5nν7n Θ

T2n

+ν1n ν5n ΘT2n

−ν6n ν3n ΘT2n

−ν9n ν5n W T2n

−ν4n ν2n ΘT2n

+ ν1n ν7n ΘT2n

−ν9n ν1n W T2n

s2 +(−ν6nν3n ν5n Θ

T2n

+ ν9n ν4nν2n W T2n

− ν4n ν2n ν7nΘT2n





)s − ν9n ν1n ν5n W T

2n+ν1n ν5n ν7n Θ

T2n




−ν6nν3n ν5n ΘT2n

−ν4n ν2nν7n ΘT2n

}, (111)

A.2 The expressions of A′TIn, B ′T

In,C ′T

In, D′T

In, A′′T

In, B ′′T

In,C ′′T

Inand D′′T

In

A′TIn

= ΞIn ×[−W T

1n

(r T

In

)6 +(ν7n W T

1n− ν2n Θ

T1n

+ ν5n W T1n

+ ν10n W T1n

− ν3n W T2n

) (r T

In

)4 −(−ν2n ν7n Θ

T1n

+ν5n ν7n W T1n


+ ν3n ν8n ΘT2n

− ν3nν5n W T2n

− ν2n ν10n ΘT1n

+ ν5n ν10n W T1n

+ ν7n ν10n W T1n

−ν8n ν9n W T1n

) (r T

In

)2 +(ν5nν7n ν10n W T

1n− ν5n ν8nν9n W T

1n− ν2n ν7n ν10nΘ

T1n




) ], (112)

B ′TIn

= ΞIn ×[−ΘT

1n

(r T

In

)6 +(ν10n Θ

T1n

+ ν7n ΘT1n

− ν4n W T1n

+ ν1n ΘT1n

) (r T

In

)4 −(−ν6n ν3n Θ

T1n

+ ν1n ν10n ΘT1n

+ν1n ν7n ΘT1n

− ν4n ν7n W T1n

− ν8n ν9n ΘT1n

− ν4nν10n W T1n

+ ν7n ν10n ΘT1n

+ ν3n ν4n W T2n

) (r T

In

)2 +(ν1n ν7n ν10n Θ

T1n





− ν1nν8n ν9n ΘT1n


) ],

(113)

C ′TIn

= ΞIn ×[−W T

2n

(r T

In

)6 +(−ν6n W T

1n+ ν1n W T

2n− ν8n Θ

T2n

+ ν5n W T2n

+ ν10n W T2n

) (r T

In

)4 −(−ν8n ν5n Θ

T2n

−ν6n ν5n W T1n

+ ν2n ν6nΘT1n

+ ν1n ν10n W T2n

+ ν1nν5n W T2n

+ ν5n ν10n W T2n

− ν6nν10n W T1n

− ν4n ν2n W T2n

−ν8n ν1n ΘT2n

) (r T

In

)2 +(−ν8n ν1n ν5n Θ

T2n


+ ν1n ν5nν10n W T2n




) ], (114)

D′TIn

= ΞIn ×[−ΘT

2n

(r T

In

)6 +(−ν9n W T

2n+ ν1n Θ

T2n

+ ν5nΘT2n

+ ν7n ΘT2n

) (r T

In

)4 −(ν6n ν9n W T

1n− ν6n ν3nΘ

T2n

+ν5n ν7n ΘT2n

− ν9n ν5n W T2n

− ν4n ν2n ΘT2n

− ν9nν1n W T2n

+ ν1n ν7n ΘT2n

+ ν1n ν5n ΘT2n)(

r TIn

)2 +(−ν6n ν3n ν5n Θ

T2n


− ν4nν2n ν7n ΘT2n





) ], (115)

in which In = ΥInβT

n gTn

. Moreover, the explicit expressions of the parameters A′′TIn, B ′′T

In,C ′′T

Inand D′′T

Incould be readily obtained in

order from A′TIn, B ′T

In,C ′T

Inand D′T

Inin Eqs. (112)–(115) if the parameters W T

1n,W T

2n,ΘT

1nandΘT

2nare replaced with W T

1n, W T

2n, ΘT

1n

and ΘT2n

, respectively.

A.3 The expression of ΔHn (s)

ΔHn (s) = [

ζ1n ζ4n

(ζ5n ζ8n − ζ6n ζ7n

)+ ζ3n ζ2n

(ζ6n ζ7n − ζ5n ζ8n

)]s8 + (−ζ1n ζ4n ζ6nη8n + ζ1n ζ4n ζ5nη9n + ζ3n ζ2nη7n ζ7n

−ζ3n ζ2nη6n ζ8n − η1n ζ4n ζ6n ζ7n − ζ1n ζ4nη7n ζ7n − ζ3n ζ2n ζ5nη9n + ζ1nη5n ζ5n ζ8n + η4n ζ2n ζ6n ζ7n − ζ3nη2n ζ5n ζ8n

+ζ3n ζ2n ζ6nη8n − ζ1nη5n ζ6n ζ7n + ζ3nη2n ζ6n ζ7n + ζ1n ζ4nη6n ζ8n + η1n ζ4n ζ5n ζ8n − η4n ζ2n ζ5n ζ8n

)s6 + (ζ1n ζ4nη6nη9n

192 K. Kiani

−ζ1n ζ4nη7nη8n + ζ1nη5n ζ5nη9n + ζ1nη5nη6n ζ8n − ζ1nη5n ζ6nη8n − ζ1nη5nη7n ζ7n + η1n ζ4n ζ5nη9n + η1n ζ4nη6n ζ8n

−η1n ζ4n ζ6nη8n − η1n ζ4nη7n ζ7n + η1nη5n ζ5n ζ8n − η1nη5n ζ6n ζ7n − ζ3n ζ2nη6nη9n + ζ3n ζ2nη7nη8n − ζ3nη2n ζ5nη9n

−ζ3nη2nη6n ζ8n + ζ3nη2n ζ6nη8n + ζ3nη2nη7n ζ7n − η4n ζ2n ζ5nη9n − η4n ζ2nη6n ζ8n + η4n ζ2n ζ6nη8n + η4n ζ2nη7n ζ7n

−η4nη2n ζ5n ζ8n + η4nη2n ζ6n ζ7n − η23nζ4n ζ8n

)s4 + (−η1n ζ4nη7nη8n − η2

3nη5n ζ8n − ζ3nη2nη6nη9n − η4nη2n ζ5nη9n

−η23nζ4nη9n − η1nη5nη7n ζ7n + ζ3nη2nη7nη8n + η4nη2nη7n ζ7n − ζ1nη5nη7nη8n + η1n ζ4nη6nη9n − η4n ζ2nη6nη9n

+η1nη5nη6n ζ8n + ζ1nη5nη6nη9n + η4n ζ2nη7nη8n + η4nη2n ζ6nη8n − η1nη5n ζ6nη8n − η4nη2nη6n ζ8n + η1nη5n ζ5nη9n

)s2

+η1nη5nη6nη9n − η1nη5nη7nη8n − η4nη2nη6nη9n + η4nη2nη7nη8n − η23nη5nη9n , (116)

A.4 The expressions of L(aHn ), L(bH

n ), L(cHn ) and L(d H

n ) during the course of free vibration

L(

aHn

)= 1

ΔHn (s)

×{(

−ζ4n ζ6n ζ7n W H1n

+ζ4n ζ5n ζ8n W H1n

+ζ2n ζ6n ζ7nΨH

1n−ζ2n ζ5n ζ8nΨ

H1n

)s7+

(−ζ2n ζ5n ζ8n Ψ

H1n

+ζ2n ζ6n ζ7n ΨH

1n


+ζ4n ζ5n ζ8n W H1n

)s6+

(−ζ4n η7n ζ7n W H

1n−ζ2n ζ5n η9nΨ

H1n

−ζ2n η6n ζ8nΨH

1n+η3n ζ4n ζ6nΨ

H2n

+ζ2n ζ6n η8nΨH

1n

+ζ4n η6n ζ8n W H1n

+ζ4n ζ5n η9n W H1n

−η2n ζ5n ζ8nΨH

1n−η5n ζ6n ζ7n W H

1n−ζ4n ζ6n η8n W H


2n+η2n ζ6n ζ7nΨ

H1n

+η5n ζ5n ζ8n W H1n

+ζ2n η7n ζ7nΨH

1n

)s5+

(ζ4n ζ5n η9n W H

1n−ζ4n η7n ζ7n W H

1n+η5n ζ5n ζ8n W H

1n−ζ2n η6n ζ8n Ψ

H1n

−ζ4n ζ6n η8n W H1n

+η2n ζ6n ζ7n ΨH

1n+ζ4n η6n ζ8n W H

1n+ζ2n ζ6n η8n Ψ

H1n

−ζ2n ζ5n η9n ΨH

1n+ζ2n η7n ζ7n Ψ

H1n

−η2n ζ5n ζ8n ΨH


2n

+η3n ζ4n ζ6n ΨH


1n

)s4+

(−η3n ζ4n η9n W H

2n+η5n η6n ζ8n W H

1n−ζ4n η7n η8n W H

1n+ζ4n η6n η9n W H

1n−η2n η6n ζ8nΨ

H1n

−η5n η7n ζ7n W H1n

+ζ2n η7n η8nΨH

1n−ζ2n η6n η9nΨ

H1n

−η5n ζ6n η8n W H1n

−η2n ζ5n η9nΨH

1n+η5n ζ5n η9n W H

1n−η3n η5n ζ8n W H

2n

+η3n η5n ζ6nΨH

2n+η2n η7n ζ7nΨ

H1n

+η3n ζ4n η7nΨH

2n+η2n ζ6n η8nΨ

H1n

)s3+

(−η2n η6n ζ8n Ψ

H1n


+η5n η6n ζ8n W H1n

−ζ2n η6n η9n ΨH

1n−η2n ζ5n η9n Ψ

H1n

+η3n η5n ζ6n ΨH


1n+ζ2n η7n η8n Ψ

H1n


−ζ4n η7n η8n W H1n

+ζ4n η6n η9n W H1n

+η2n η7n ζ7n ΨH


1n+η3n ζ4n η7n Ψ

H2n


+η2n ζ6n η8n ΨH

1n

)s2+

(η3n η5n η7nΨ

H2n

−η5n η7n η8n W H1n

+η5n η6n η9n W H1n


+η2n η7n η8nΨH

1n−η2n η6n η9nΨ

H1n

)s+η3n η5n η7n Ψ

H2n




−η2n η6n η9n ΨH

1n+η2n η7n η8n Ψ

H1n

}, (117)

L(

bHn

)= 1

ΔHn (s)

×{(ζ1n ζ5n ζ8nΨ

H1n


−ζ1n ζ6n ζ7nΨH

1n+ζ3n ζ6n ζ7n W H

1n

)s7+

(ζ1n ζ5n ζ8n Ψ

H1n


−ζ1n ζ6n ζ7n ΨH


1n

)s6+

(−ζ1n η7n ζ7nΨ

H1n


1n+η1n ζ5n ζ8nΨ

H1n

+ζ1n ζ5n η9nΨH

1n+ζ1n η6n ζ8nΨ

H1n


−ζ1n ζ6n η8nΨH

1n+ζ3n η7n ζ7n W H




1n−η3n ζ3n ζ6nΨ

H2n


−ζ3n η6n ζ8n W H1n

)s5+

(−ζ1n ζ6n η8n Ψ

H1n


−ζ3n ζ5n η9n W H1n



+ζ1n η6n ζ8n ΨH


1n+η1n ζ5n ζ8n Ψ

H1n



H1n

+ζ1n ζ5n η9n ΨH

1n−η1n ζ6n ζ7n Ψ

H1n



)s4+

(η3n η4n ζ8n W H

2n−η1n ζ6n η8nΨ

H1n

+ζ1n η6n η9nΨH


H2n

−η1n η7n ζ7nΨH

1n

−η23nζ8nΨ

H1n


−ζ1n η7n η8nΨH

1n+η1n ζ5n η9nΨ

H1n




1n


+η1n η6n ζ8nΨH


1n−η4n ζ5n η9n W H


1n

)s3+

(ζ3n η7n η8n W H


H1n

−η23nζ8n Ψ

H1n

+η1n η6n ζ8n ΨH



H2n

−η1n η7n ζ7n ΨH


H1n



1n


+η1n ζ5n η9n ΨH



1n−η3n η4n ζ6n Ψ

H2n


+η4n ζ6n η8n W H1n

)s2

+(η1n η6n η9nΨ

H1n

−η3n η4n η7nΨH

2n+η3n η4n η9n W H

2n−η2

3nη9nΨ

H1n




1n

)s

+η23nη9n Ψ

H1n



2n−η1n η7n η8n Ψ

H1n



+η1n η6n η9n ΨH

1n

}, (118)

L(

cHn

)= 1

ΔHn (s)

×{(


2n−ζ3n ζ2n ζ8n W H


2n+ζ3n ζ2n ζ6nΨ

H2n

)s7+

(ζ1n ζ4n ζ8n W H

2n+ζ3n ζ2n ζ6n Ψ

H2n

−ζ1n ζ4n ζ6n ΨH

2n−ζ3n ζ2n ζ8n W H

2n

)s6+

(ζ1n η5n ζ8n W H

2n+η3n ζ2n ζ8nΨ

H1n

−ζ1n η5n ζ6nΨH



2n

−η3n ζ4n ζ8n W H1n

+ζ3n η2n ζ6nΨH

2n+ζ1n ζ4n η9n W H

2n−η1n ζ4n ζ6nΨ

H2n



2n+η4n ζ2n ζ6nΨ

H2n


+ζ3n ζ2n η7nΨH


2n

)s5+

(−η1n ζ4n ζ6n Ψ

H2n






+ζ3n η2n ζ6n ΨH



2n−ζ1n ζ4n η7n Ψ

H2n

+η3n ζ2n ζ8n ΨH


H2n

+η4n ζ2n ζ6n ΨH


H2n

)s4+

(ζ1n η5n η9n W H

2n+η4n ζ2n η7nΨ

H2n




H2n



+η3n η2n ζ8nΨH


1n+η3n ζ2n η9nΨ

H1n

+ζ3n η2n η7nΨH


2n


+η4n η2n ζ6nΨH



H2n

)s3+

(−η3n ζ4n η9n W H

1n+η4n ζ2n η7n Ψ

H2n


+η3n ζ2n η9n ΨH




2n+η3n η2n ζ8n Ψ

H1n

−η1n ζ4n η7n ΨH

2n+η4n η2n ζ6n Ψ

H2n




H2n




2n

)s2+

(−η1n η5n η7nΨ

H2n



+η4n η2n η7nΨH

2n−η3n η5n η9n W H

1n+η3n η2n η9nΨ

H1n

)s+η3n η2n η9n Ψ

H1n




+η4n η2n η7n ΨH


H2n

}, (119)

L(

d Hn

)= 1

ΔHn (s)

×{(ζ1n ζ4n ζ5nΨ

H2n




2n

)s7+

(−ζ3n ζ2n ζ5n Ψ

H2n


+ζ1n ζ4n ζ5n ΨH


2n

)s6+

(−ζ1n ζ4n η8n W H

2n+ζ1n η5n ζ5nΨ

H2n






2n−ζ3n η2n ζ5nΨ

H2n

+ζ1n ζ4n η6nΨH



2n+η1n ζ4n ζ5nΨ

H2n


1n−ζ3n ζ2n η6nΨ

H2n

)s5+

(−η1n ζ4n ζ7n W H




1n+η1n ζ4n ζ5n Ψ

H2n



H2n


−ζ3n ζ2n η6n ΨH




H2n

+ζ1n η5n ζ5n ΨH

2n−η4n ζ2n ζ5n Ψ

H2n

)s4+

(−η2

3nζ4nΨ

H2n





H2n





H2n

+η1n η5n ζ5nΨH

2n−η3n ζ2n η8nΨ

H1n





+η1n ζ4n η6nΨH

2n+ζ1n η5n η6nΨ

H2n

)s3+

(η1n η5n ζ5n Ψ

H2n


2n

+ζ1n η5n η6n ΨH


H2n

−η23nζ4n Ψ

H2n



2n−η3n η2n ζ7n Ψ

H1n

+η1n ζ4n η6n ΨH

2n


−η3n ζ2n η8n ΨH






2n


)s2+

(η1n η5n η6nΨ

H2n


2n−η3n η2n η8nΨ

H1n


−η23nη5nΨ

H2n



)s+η4n η2n η8n W H

2n−η2

3nη5n Ψ

H2n

+η1n η5n η6n ΨH


H2n



1n


}, (120)

A.5 The expressions of A′HIn, B ′H

In,C ′H

In, D′H

In, A′′H

In, B ′′H

In,C ′′H

Inand D′′H

In

A′HIn

= ΣIn ×[

−(−ζ4n ζ6n ζ7n W H

1n+ ζ4n ζ5n ζ8n W H

1n+ ζ2n ζ6n ζ7nΨ

H1n

− ζ2n ζ5n ζ8nΨH

1n

) (r H

In

)6 +(−ζ4n η7n ζ7n W H

1n


1n− ζ2n η6n ζ8nΨ

H1n

+ η3n ζ4n ζ6nΨH

2n+ ζ2n ζ6n η8nΨ

H1n

+ ζ4n η6n ζ8n W H1n

+ ζ4n ζ5n η9n W H1n

− η2n ζ5n ζ8nΨH

1n


− ζ4n ζ6n η8n W H1n

− η3n ζ4n ζ8n W H2n

+ η2n ζ6n ζ7nΨH

1n+ η5n ζ5n ζ8n W H

1n+ ζ2n η7n ζ7nΨ

H1n

) (r H

In

)4

−(−η3n ζ4n η9n W H

2n+ η5n η6n ζ8n W H

1n− ζ4n η7n η8n W H

1n+ ζ4n η6n η9n W H

1n− η2n η6n ζ8nΨ

H1n

− η5n η7n ζ7n W H1n

+ ζ2n η7n η8nΨH

1n


1n− η5n ζ6n η8n W H

1n− η2n ζ5n η9nΨ

H1n

+ η5n ζ5n η9n W H1n


+ η3n η5n ζ6nΨH

2n+ η2n η7n ζ7nΨ

H1n

+η3n ζ4n η7nΨH

2n+ η2n ζ6n η8nΨ

H1n

) (r H

In

)2 + η3n η5n η7nΨH

2n− η5n η7n η8n W H

1n+ η5n η6n η9n W H

1n− η3n η5n η9n W H

2n

+η2n η7n η8nΨH

1n− η2n η6n η9nΨ

H1n

], (121)

B′HIn

= ΣIn ×[−(ζ1n ζ5n ζ8nΨ

H1n

− ζ3n ζ5n ζ8n W H1n



1n

) (r H

In

)6 +(−ζ1n η7n ζ7nΨ

H1n


1n+ η1n ζ5n ζ8nΨ

H1n

+ ζ1n ζ5n η9nΨH

1n+ ζ1n η6n ζ8nΨ

H1n

+ η3n ζ3n ζ8n W H2n

− ζ1n ζ6n η8nΨH

1n+ ζ3n η7n ζ7n W H

1n





2n+ ζ3n ζ6n η8n W H

1n− ζ3n η6n ζ8n W H

1n

) (r H

In

)4

−(η3n η4n ζ8n W H

2n− η1n ζ6n η8nΨ

H1n

+ ζ1n η6n η9nΨH

1n− η3n η4n ζ6nΨ

H2n

− η1n η7n ζ7nΨH

1n− η2

3nζ8nΨ

H1n

− ζ3n η6n η9n W H1n

194 K. Kiani



H1n

− η3n ζ3n η7nΨH

2n+ η3n ζ3n η9n W H




H1n


− η4n ζ5n η9n W H1n


) (r H

In

)2 +(η1n η6n η9nΨ

H1n

− η3n η4n η7nΨH

2n+ η3n η4n η9n W H

2n

−η23nη9nΨ

H1n

+ η4n η7n η8n W H1n

− η4n η6n η9n W H1n


1n

)], (122)

C ′HIn

= ΣIn ×[−(−ζ1n ζ4n ζ6nΨ

H2n


+ ζ1n ζ4n ζ8n W H2n

+ ζ3n ζ2n ζ6nΨH

2n

) (r H

In

)6 +(ζ1n η5n ζ8n W H

2n

+η3n ζ2n ζ8nΨH

1n− ζ1n η5n ζ6nΨ

H2n

− ζ3n η2n ζ8n W H2n



+ ζ3n η2n ζ6nΨH


2n


2n− ζ1n ζ4n η7nΨ

H2n

+ η1n ζ4n ζ8n W H2n

+ η4n ζ2n ζ6nΨH

2n+ ζ3n ζ2n η7nΨ

H2n


) (r H

In

)4

−(ζ1n η5n η9n W H


H2n


2n− η3n η5n ζ8n W H

1n− ζ1n η5n η7nΨ

H2n

− ζ3n η2n η9n W H2n

+ η1n η5n ζ8n W H2n

+η3n η2n ζ8nΨH



H1n

+ ζ3n η2n η7nΨH




H2n



2n

) (r H

In

)2 +(−η1n η5n η7nΨ

H2n



+ η4n η2n η7nΨH

2n


+ η3n η2n η9nΨH

1n

)], (123)

D′HIn

= ΣIn ×[−(ζ1n ζ4n ζ5nΨ

H2n




2n

) (r H

In

)6 +(−ζ1n ζ4n η8n W H

2n

+ζ1n η5n ζ5nΨH

2n− ζ1n η5n ζ7n W H

2n+ ζ3n η2n ζ7n W H



1n− η4n ζ2n ζ5nΨ

H2n

− ζ3n η2n ζ5nΨH

2n

+ζ1n ζ4n η6nΨH

2n− η1n ζ4n ζ7n W H


2n+ η1n ζ4n ζ5nΨ

H2n


1n− ζ3n ζ2n η6nΨ

H2n

) (r H

In

)4 −(−η2

3nζ4nΨ

H2n




1n− ζ3n η2n η6nΨ

H2n

− η1n ζ4n η8n W H2n


2n− ζ1n η5n η8n W H

2n



H2n


1n+ ζ3n η2n η8n W H




2n

+η1n ζ4n η6nΨH

2n+ ζ1n η5n η6nΨ

H2n

) (r H

In

)2 +(η1n η5n η6nΨ

H2n


2n− η3n η2n η8nΨ

H1n


− η23nη5nΨ

H2n



)], (124)

where ΣIn = ΛInβH

n gHn

. Moreover, the explicit expressions of the parameters A′′HIn, B ′′H

In,C ′′H

Inand D′′H

Incould be obtained in order

from A′HIn, B ′H

In,C ′H

Inand D′H

Inin Eqs. (121)–(124) if the parameters W H

1n,W H

2n, Ψ H

1nand Ψ H

2nare replaced with W H

1n, W H

2n, Ψ H

1n

and Ψ H2n

, respectively.

References

1. Coleman, J.N., Khan, U., Blau, W.J., Gunko, Y.K.: Small but strong: a review of the mechanical properties of carbonnanotubepolymer composites. Carbon 44, 1624–1652 (2006)

2. Gibson, R.F., Ayorinde, E.O., Wen, Y.F.: Vibrations of carbon nanotubes and their composites: a review. Compos. Sci.Technol. 67, 1–28 (2007)

3. Hummer, G., Rasaiah, J.C., Noworyta, J.P.: Water conduction through the hydrophobic channel of a carbon nanotube. Nature414, 188–190 (2001)

4. Supple, S., Quirke, N.: Rapid imbibition of fluids in carbon nanotubes. Phys. Rev. Lett. 90, 214501 (2003)5. Majumder, M., Chopra, N., Andrews, R., Hinds, B.J.: Nanoscale hydrodynamics enhanced flow in carbon nanotubes.

Nature 438, 444 (2005)6. Regan, B.C., Aloni, S., Huard, B., Fennimore, A., Ritchie, R.O., Zettl, A.: Nanowicks: nanotubes as tracks for mass transfer.

In: Kuzmany, H., Fink, J., Mehring, M., Roth, S. (eds.) Molecular Nanostructures. AIP Conference Proceedings, vol. 685,pp. 612–615 (2003)

7. Regan, B.C., Aloni, S., Ritchie, R.O., Dahmen, U., Zettl, A.: Carbon nanotubes as nanoscale mass conveyors. Nature 428, 924–927 (2004)

8. Babu, S., Ndungu, P., Bradley, J.C., Rossi, M.P., Gogotsi, Y.: Guiding water into carbon nanopipes with the aid of bipolarelectrochemistry. Microfluid. Nanofluid. 1, 284–288 (2005)

9. Gogotsi, Y.: In situ multiphase fluid experiments in hydrothermal CNTs. Appl. Phys. Lett. 79, 1021–1023 (2001)10. Waghe, A., Rasaiah, J.C.: Filling and emptying kinetics of carbon nanotubes in water. J. Chem. Phys. 117, 10790–10795 (2002)11. Tuzun, R.E., Noid, D.W., Sumpter, B.G., Merkle, R.C.: Dynamics of fluid flow inside CNTs. Nanotechnology 7, 241–

246 (1996)12. Mao, Z., Sinnott, S.B.: A computational study of molecular diffusion and dynamics flow through CNTs. J. Phys. Chem.

B 104, 4618–4624 (2000)13. Sokhan, V.P., Nicholson, D., Quirke, N.: Fluid flow in nanopores: an examination of hydrodynamic boundary conditions.

J. Chem. Phys. 115, 3878–3887 (2001)14. Yoon, J., Ru, C.Q., Mioduchowski, A.: Vibration and instability of CNTs conveying fluid. Compos. Sci. Technol. 65, 1326–

1336 (2005)


15. Yoon, J., Ru, C.Q., Mioduchowski, A.: Flow-induced flutter instability of cantilever CNTs. Int. J. Solids. Struct. 43, 3337–3349 (2006)

16. Yana, Y., He, X.Q., Zhang, L.X., Wang, C.M.: Dynamic behavior of triple-walled carbon nanotubes conveying fluid.J. Sound. Vib. 319, 1003–1018 (2009)

17. Ru, C.Q.: Effect of van der Waals forces on axial buckling of a double-wall carbon nanotube. J. Appl. Phys. 87, 7227–7231 (2000)

18. Ru, C.Q.: Axially compressed buckling of a double walled carbon nanotube embedded in an elastic medium. J. Mech. Phys.Solids 49, 1265–1279 (2001)

19. Li, C., Chou, T.W.: Elastic moduli of multi-walled carbon nanotubes and the effect of van der Waals forces. Compos. Sci.Technol. 63, 1517–1524 (2003)

20. Yoon, J., Ru, C.Q., Mioduchowski, A.: Timoshenko-beam effects on transverse wave propagation in carbon nanotubes.Composites B 35, 87–93 (2004)

21. He, X.Q., Kitipornchai, S., Liew, K.M.: Buckling analysis of multi-walled carbon nanotubes: a continuum model accountingfor van der Waals interaction. J. Mech. Phys. Solids 53, 303–326 (2005)

22. Xu, C.L., Wang, X.: Matrix effects on the breathing modes of multiwall carbon nanotubes. Compos. Struct. 80, 73–81 (2007)23. Gupta, S.S., Batra, R.C.: Continuum structures equivalent in normal mode vibrations to single-walled carbon nanotubes.

Comput. Mater. Sci. 43, 715–723 (2008)24. Peddieson, J., Buchanan, G.R., McNitt, R.P.: Application of nonlocal continuum models to nanotechnology. Int. J. Eng.

Sci. 41, 305–312 (2003)25. Sudak, L.J.: Column buckling of multiwalled carbon nanotubes using nonlocal continuum mechanics. J. Appl. Phys. 94, 7281–

7287 (2003)26. Wang, Q., Liew, K.M.: Application of nonlocal continuum mechanics to static analysis of micro- and nano-structures. Phys.

Lett. A 363, 236–242 (2007)27. Ece, M.C., Aydogdu, M.: Nonlocal elasticity effect on vibration of in-plane loaded double-walled carbon nano-tubes. Acta.

Mech. 190, 185–195 (2007)28. Schoen, P.A.E., Walther, J.H., Arcidiacono, S., Poulikakos, D., Koumoutsakos, P.: Nanoparticle traffic on helical tracks:

thermophoretic mass transport through carbon nanotubes. Nano. Lett. 6, 1910–1917 (2006)29. Schoen, P.A.E., Walther, J.H., Poulikakos, D., Koumoutsakos, P.: Phonon assisted thermophoretic motion of gold nanopar-

ticles inside carbon nanotubes. Appl. Phys. Lett. 90, 253116 (2007)30. Kiani, K., Mehri, B.: Assessment of nanotube structures under a moving nanoparticle using nonlocal beam theories. J. Sound.

Vib. 329, 2241–2264 (2010)31. Lennard-Jones, J.E.: The determination of molecular fields: from the variation of the viscosity of a gas with temperature. Proc.

Roy. Soc. Lond. Ser. A 106, 441–462 (1924)32. Girifalco, L.A., Lad, R.A.: Energy of cohesion, compressibility and the potential energy function of graphite system.

J. Chem. Phys. 25, 693–697 (1956)33. Saito, R., Dresselhaus, G., Dresselhaus, M.S.: Physical Properties of Carbon Nanotubes. Imperial College, London (1998)34. Frýba, L.: Vibration of Solids and Structures Under Moving Loads. Thomas Telford, London (1999)35. Eringen, A.C.: Nonlocal polar elastic continua. Int. J. Eng. Sci. 10, 1–16 (1972)36. Eringen, A.C.: On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves. J. Appl.

Phys. 54, 4703–4710 (1983)37. Eringen, A.C.: Nonlocal Continuum Field Theories. Springer, New York (2002)38. Timoshenko, S.: Vibration Problems in Engineering. Van Nostrand, New Jersey (1955)39. Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions, with Formulas, Graphs and Mathematical Tables.

Dover, New York (1972)40. Reddy, J.N.: Mechanics of Laminated Composite Plates. CRC press, Florida (1997)41. Kiani, K., Nikkhoo, A., Mehri, B.: Prediction capabilities of classical and shear deformable beam models excited by a moving

mass. J. Sound. Vib. 320, 632–648 (2009)

application of nonlocal beam models to double-walled

Documents