Natural Frequencies and Mode Shapes of Initially Curved Carbon Nanotube Resonators Under Electric...

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Journal of Sound and Vibration

Journal of Sound and Vibration 330 (2011) 3182–3195

0022-46

doi:10.1

n Corr

E-m

journal homepage: www.elsevier.com/locate/jsvi

Natural frequencies and mode shapes of initially curved carbonnanotube resonators under electric excitation

Hassen M. Ouakad, Mohammad I. Younis n

Department of Mechanical Engineering, State University of New York at Binghamton, Binghamton, NY 13902, USA

a r t i c l e i n f o

Article history:

Received 26 March 2010

Received in revised form

6 December 2010

Accepted 11 December 2010

Handling Editor: M.P. Cartmellstretching of doubly clamped CNTs, the DC electrostatic force, and the initial curvature

Available online 18 February 2011

0X/$ - see front matter & 2011 Elsevier Ltd.

016/j.jsv.2010.12.029

esponding author. Tel.: +1 607 777 4983; fa

ail address: [email protected] (M.I. Y

a b s t r a c t

Estimating accurately the natural frequencies of electrically actuated carbon nanotubes

(CNTs) has been an active research subject over the past few years. Despite the

importance of the topic, robust knowledge is still missing in the understanding of

the role of various physical parameters affecting the natural frequencies, such as the

of slack CNTs. In this investigation, we use a 2D nonlinear curved beam model in the

form of an arch to simulate the coupled in-plane and out-of-plane motions of a CNT

with curvature. We calculate the variation of its natural frequencies and mode shapes

with the level of slackness and the DC electrostatic load. Towards this end, we derive a

reduced-order model using a multimode Galerkin procedure. We show various

scenarios of mode crossing and mode veering as the levels of slackness and DC load

are varied. Finally, we tackle the forced vibration problem of a curved CNT when

actuated by small DC and AC loads. The results show the transfer of energy among the

vibration modes involved in the veering phenomenon.

& 2011 Elsevier Ltd. All rights reserved.

1. Introduction

The interest in nanoelectromechanical systems (NEMS) has grown considerably in the past few years. Since thediscovery of carbon nanotubes (CNTs) [1], they were under extensive research and investigation [2–24] to be able toimplement them as NEMS devices [25,26] and to utilize them as resonant sensors. Hence, vibrations of thesenanostructures as well as estimating their natural frequencies have been under extensive research [2–26].

Several theoretical investigations have been conducted to estimate the in-plane natural frequencies of straightclamped–clamped CNTs [2–6]. Dequesnes et al. [2] calculated the first natural frequency of clamped–clamped CNTs,including mid-plane stretching, using molecular dynamics and continuum models. They concluded that the nonlinearcontinuum model yields good match with molecular dynamics model and is computationally much cheaper. Witkampet al. [3] and Poot et al. [4] used a nonlinear beam model to estimate the variations of the bending natural frequencies ofclamped–clamped CNTs versus the electrostatic DC load. Recently, Elishakoff and Pentaras [5] derived analyticalexpressions for the fundamental natural frequencies of double-walled carbon nanotubes under various boundaryconditions (simply supported and doubly clamped) using two different decritization schemes, the Bubnov–Galerkin andthe Petrov–Galerkin methods. They used a linear beam model and showed a possibility to quickly evaluate the naturalfrequencies of such systems. In another investigation, Elishakoff and Pentaras [6] adopted a simplified Bresse–Timoshenkobeam model to evaluate the natural frequencies of simply supported CNTs taking into account the shear deformation as

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ounis).

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H.M. Ouakad, M.I. Younis / Journal of Sound and Vibration 330 (2011) 3182–3195 3183

well as the rotary inertia effects. They showed that the model yields excellent results compared to the full Bresse–Timoshenko theory as well as the Euler–Bernoulli beam model.

The previous theoretical investigations were motivated especially by the several reported phenomena of previouslyconducted experimental works [7–12]. Many groups were able to measure the natural frequency of CNTs up to THz.Rabieirad et al. [7] tested a clamped–clamped CNT resonator excited by AC and DC loads and were able to measureresonance frequencies up to 70 MHz. San Paulo et al. [8] measured a natural frequency of a clamped–clamped CNT up to3 GHz by using the atomic force microscope (AFM). Amlani et al. [9] measured and predicted a cut-off frequency up to50 GHz of an electrically actuated CNT based resonator. Recently, three groups [10–12] were successful to design nano-devices with CNTs in the terahertz regime, which are an optical polarizer [10] made of highly aligned single-walled CNTand nanotransistors [11,12].

Motivated by the need of more understanding for the dynamical behavior of CNTs, several works [13–17] simulatedtheoretically their vibration under several dynamic loadings. Postma et al. [13] investigated the dynamic behavior of aclamped–clamped CNT using a nonlinear beam model along with a Galerkin procedure. They concluded that CNTs arealmost useless in the linear regime even for small electric load due to the presence of thermal random noise. Lately,Georgantzinos et al. [14] proposed a linear spring–mass model to describe and evaluate the vibration characteristics of asingle walled CNT. They observed that the aspect ratio has significant role on the basic modes of vibration of thenanotubes. Conley et al. [15] proposed a model for a CNT accounting for both in-plane and out-of-plane motions. Theyreported the possibility of non-planar motions in straight CNT resonators, which they attributed to the symmetry of thenanotube as well as to the stretching it undergoes while deforming. Following [15], Rhoads et al. [16] developed aphysically consistent model by introducing linear and nonlinear parametric excitation to the system direct excitations.They studied the nonlinear oscillations of the CNT and presented preliminary data described to be potentially exploitablein practical applications. Recently, Ouakad and Younis [17] investigated using a Galerkin procedure the nonlinear dynamicbehavior of electrically actuated CNT resonators. They showed complex nonlinear dynamics phenomena, such ashysteresis, dynamic pull-in, hardening and softening behaviors, and frequency bands with an inevitable escape from apotential well.

Most of the mentioned literature models clamped–clamped CNTs to be perfectly straight. However, due to theirfabrication process using chemical vapor deposition, fabricating perfectly straight CNTs with controlled geometry andorientation is very difficult. Indeed, many studies have indicated that clamped–clamped CNTs are fabricated with somelevel of curvature (slack) [18–24]. Kang et al. [18] studied the interatomic interaction between a CNT bridge and thesubstrate underneath it used as a NEMS memory device. They showed that the value of the CNT bridge slack is veryimportant for the operation of the NEMS memory device as a nonvolatile memory. Gibson et al. [19] stressed theimportance of modeling the slack of CNTs and indicated that no consistent model has been presented so far to address thisissue despite knowing the strong effect of the curvature of curved beams on modal frequencies. Among those whoreported experimental investigations showing the importance of slack on estimating accurately the natural frequencies ofCNTs are Sazonova et al. [20] and Sazanova [21] who tested slacked clamped–clamped CNT to a DC and AC load andcharacterized experimentally their free vibration response. Comparing their experimental data with the numerical resultsof [22], they reported some unexplained phenomena [21], such as the sub-linear variation of the frequencies, the avoidedcrossings (veering), the abundance of resonances, and the negative variation of the frequencies for certain carbonnanotubes. Ustunel et al. [22] were among the very few who attempted to investigate theoretically the effect of slack on aCNT oscillator. They based their work on a one-dimensional elastic continuum model by assuming the electrostatic force tobe uniformly distributed along the length of the CNT and ignoring the nonlinear elastic effects. Then, they derivedapproximate analytical expressions of the natural frequencies of the CNT for several behavior regimes of the CNTdepending on the gate voltage value. They identified three zones for the response: bending dominated, catenarydominated, and elastic dominated. They related this to the impact of slack on the natural frequencies. They reportedthat the bending regime can be described by a buckled doubly clamped beam model, the catenary regime is modeled as asimple string under variable tension, and the elastic regime is similar to the hanging chain under constant tension. Anothergroup [23] attempted to model the CNT using a finite-element method. They also reported discrepancy among theirtheoretically predicted and experimentally measured resonance frequencies of electrostatically actuated CNTs and citedslack as a possible reason of this. Recently, Mayoof and Hawwa [24] proposed a nonlinear curved beam model to describethe nonlinear dynamics of a slacked single-walled carbon nanotube under linear harmonic excitation. They showed severalscenarios of chaotic behavior of the CNT.

From the aforementioned review, one can note that despite this important topic, a robust knowledge is still missing tounderstand the role of various physical parameters that affect the natural frequencies of CNTs, such as the stretching ofdoubly clamped CNTs, the DC electrostatic force, and the curvature of CNTs with slacks. It is well-known fact that curvedstructures introduce quadratic and cubic nonlinearities. The interaction of these nonlinearities (how they compete,strengthen, or cancel each other) still is not fully understood. The objective of this paper is to investigate in depth the in-plane and out-of-plane natural frequencies and mode shapes of slacked CNTs while varying the slack level and the DCvoltages for a range starting from very small voltage up to the pull-in instability.

The organization of this paper is as follows. First, the applicability of the continuum mechanics theory is discussed.Then, a nonlinear 2D arch beam model is presented to model the in-plane and out-of-plane motions of a CNT with slack.A reduced-order model is derived to solve the coupled in-plane and out-of-plane equations. Then, the free-vibration

H.M. Ouakad, M.I. Younis / Journal of Sound and Vibration 330 (2011) 3182–31953184

problem of the CNT under the actuation of a DC load is solved with and without including the effect of slack. Finally theforced vibration problem of the slacked CNT is simulated for small DC and AC loads.

2. The applicability of continuum mechanics laws

Several investigations have been conducted to simulate the response of CNTs using molecular dynamics and their results werecompared to results obtained using continuum mechanics theories, such as beams and shells. All the investigations concludedthat continuum mechanics serve the purpose of modeling and simulating CNTs both accurately and efficiently from acomputational point of view. For example, Yakobson et al. [27] studied using a molecular dynamics model the large deformationof CNTs. They estimated the buckling of CNTs using continuum theory, a beam model, to the predictions of the molecularmechanics simulations. They concluded that this behavior of CNTs can be well described by a continuum model.

Based on scaling analysis, Harik [28,29] tackled the buckling problem of CNTs and proposed three nondimensional numbersrelated to the CNT geometric parameters to check the validity of the beam assumption for modeling the mechanics of CNTs. Therelation between these ratios and molecular dynamics simulation was discussed. Liu and Chen [30] mentioned that investigatingthe global responses of CNTs such as deformations, effective stiffness, or load transfer, can be done using continuum mechanicsboth effectively and efficiently. Pantano et al. [31,32] used finite element FE approaches to model a single-walled carbon nanotubeSWCNT and a multiwalled carbon nanotube MWCNT. The CNT walls were modeled as thin shells while the inter-wall interactionswere modeled as pressure. Pantano et al. [31,32] validated their model by comparing the FE results with the molecular mechanicssimulations and the experimental data. They found good agreement among all the results. Arroyo and Belytschko [33] andArroyo [34] also developed a FE model for MWNTs. They developed a membrane wall model directly using a Tersoff–Brennerpotential and a modified Cauchy–Born rule. They were able to reproduce local buckling, kinking, and rippling effects, which arenearly identical to the deformed states of the parent molecular simulation, by using fine meshes for a variety of loadings includingcompression, torsion and bending. They concluded that the continuum/FE calculations are surprisingly accurate compared to theatomistic calculations [34]. Dequesnes et al. [35,36] employed a nonlinear beam theory with molecular dynamics simulations toextract the beam material properties (the beam bending (EI) and stretching (EA) constants). They showed good agreement whencomparing the beam model to the molecular dynamic simulations except for slight difference very close to pull-in.

In more recent works, Wang et al. [37] remarked that atomistic and molecular methods are limited to CNTs with smallnumber of atoms, due to the high cost of computation, and are therefore restrained to the study of localized effects onsmall portion of the CNT. In order to simulate the mechanical behavior of large-sized atomic CNTs, Wang et al. [37]proposed the use of continuum models. Sears and Batra [38] showed that continuum models predict both global and localresponses for buckling of SWCNT by comparing their continuum predictions for bending and buckling to atomisticsimulations. They employed a simple Euler–Bernoulli beam to model the cantilever bending. They showed that thecontinuum SWCNT strain energy was found to match that of the molecular simulation very well.

From the aforementioned review, we can see that the continuum mechanics is worthy to be used to model themechanics of CNTs. As will be shown in Section 5.1, we have verified that all the applicability conditions of the continuumtheory as stated by Harik [28] are satisfied for the CNTs under consideration of this work. Hence, we will adopt next theEuler–Bernoulli beam model to solve the nonlinear electrically actuated curved CNT problem.

3. Problem formulation

In this section, we formulate the 2D problem governing the static and dynamic behavior of an electrically actuatedslacked carbon nanotube resonator (Fig. 1). The nanotube is actuated by an electrode underneath it with a gap width d. It ismodeled as a hollow cylindrical Euler–Bernoulli beam of radius ~R, shell thickness h, and length L. It has a cross-sectionalarea A and an area moment of inertia I. The nanotube is assumed to have Young’s modulus E=1.2 TPa and a mass densityr=1.3 g/cm3. The CNT is considered here to be initially curved in the direction of the lower electrode with an initial shapew0ðxÞ ¼ bo sinðpx=LÞ.

The 2D equations describing the in-plane deflection w(x,t) and out-of-plane deflection v(x,t) of the clamped–clampedCNT resonator can be written as [16,39]

EI@4w

@x4þrA

@2w

@t2þ ~c

@w

@t¼

EA

2L

Z L

0

@w

@x

� �2

þ@v

@x

� �2

�2@w

@x

dw0

dx

� �( )dx

" #@2w

@x2�

d2w0

dx2

" #þ Fe, (1)

EI@4v

@x4þrA

@2v

@t2þ ~c

@v

@t¼

EA

2L

Z L

0

@w

@x

� �2

þ@v

@x

� �2

�2@w

@x

dw0

dx

� �( )dx

" #@2v

@x2

, (2)

where

Fe ¼pe0ðVDCþVAC cosð ~OtÞ2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ðd�w�w0Þðd�w�w0þ2 ~RÞq

cosh�1 1þd�w�w0~R

� �� 2, (3)

where ~c is the viscous damping coefficient and e0 is the air permittivity.

L

VDC

d VAC

b0

x

y

z

L

b0

d VAC

V DC

w0 (x) ˆ ˆw (x, t)

ˆ ˆw (x, t)

ˆ ˆv (x, t)

x

z ˆ

Fig. 1. (a) 3D schematic of an electrically actuated slacked carbon nanotube resonator and (b) in-plane view of the nanotube.


The boundary conditions are as follows:

wð0, tÞ ¼ 0,@w

@xð0, tÞ ¼ 0, wðL,tÞ ¼ 0,

@w

@xðL, tÞ ¼ 0,

vð0,tÞ ¼ 0,@v

@xð0, tÞ ¼ 0, vðL, tÞ ¼ 0,

@v

@xðL, tÞ ¼ 0: (4)

For convenience, we introduce the following nondimensional variables:

w¼w

d, v¼

v

d, x¼

x

L, t¼

t

T, (5)

where T is a time constant defined by T ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðrAL4Þ=ðEIÞ

p. By substituting Eq. (5) into Eqs. (1)–(4), the nondimensional

equations of motions and associated boundary conditions of the considered clamped–clamped CNT are written as

@4w

@x4þ@2w

@t2þc

@w

@t¼ a2Feþa1

Z 1

0

@w

@x

� �2

þ@v

@x

� �2

�2@w

@x

dw0

dx

� �( )dx

" #@2w

@x2�

d2w0

dx2

" #, (6)

@4v

@x4þ@2v

@t2þc

@v

@t¼ a1

Z L

0

@w

@x

� �2

þ@v

@x

� �2

�2@w

@x

dw0

dx

� �( )dx

" #@2v

@x2, (7)

wð0,tÞ ¼ 0,@w

@xð0,tÞ ¼ 0, wð1,tÞ ¼ 0,

@w

@xð1,tÞ ¼ 0,

vð0,tÞ ¼ 0,@v

@xð0,tÞ ¼ 0, vð1,tÞ ¼ 0,

@v

@xð1,tÞ ¼ 0, (8)

where

Fe ¼ðVDCþVAC cosðOtÞÞ2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ð1�w�w0Þð1�w�w0þ2RÞp

ðcosh�1ð1þ1�w�w0=RÞÞ2

, w0ðxÞ ¼bo

dsinðpxÞ, (9)

a1 ¼Ad2

2I, a2 ¼

pe0L4

EId2, c¼

~cL4

EIT, O¼

~OffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðEIÞ=ðrAL4Þ

p , R¼~R

d, (10)


4. The reduced-order model

To solve the obtained nondimensional equation of motions of the slacked CNT, Eqs. (6)–(8) are discretized using theGalerkin procedure to yield a Reduced-Order Model (ROM) [17,40]. Hence, the in-plane and out-of-plane responses of theCNT are approximated, respectively, as

wðx,tÞ ¼Xn

i ¼ 1

uiðtÞfiðxÞ,

vðx,tÞ ¼Xm

i ¼ 1

xiðtÞfiðxÞ, (11)

where fiðxÞ are the normalized linear undamped mode shapes of a straight beam and uiðtÞ and xiðtÞ are the nondimensionalmodal coordinates of the in-plane and out-of-plane motions, respectively.

To obtain the ROM, we substitute Eq. (11) into Eqs. (6)–(8), multiply by fiðxÞ, use the orthogonality conditions of themode shapes, and then integrate the outcome from 0 to 1. The results are differential equations in terms of the modalcoordinates uiðtÞ and xiðtÞ. To simulate the dynamic behavior, the obtained ROM can be integrated with time using theRunge–Kutta technique. We should mention here that the mode shapes fiðxÞ will remain embedded inside thedenominator of the electrostatic force term, Eq. (9), in the ROM. To deal with the complicated integral terms due tothat electrostatic force, we evaluate the spatial integrals containing the mode shapes fiðxÞ numerically simultaneouslywhile integrating the differential equations of the modal coordinates uiðtÞ and xiðtÞ.

To calculate the static in-plane and out-of-plane deflections ws and vs of the CNT, all time dependent terms in the ROMdifferential equations are set equal to zero. Then the modal coordinates uiðtÞ and xiðtÞ are replaced by unknown constantcoefficients ai and bi. This results in a system of nonlinear algebraic equations in terms of those coefficients. The system isthen solved numerically using the Newton–Raphson method.

5. Results

5.1. The static response

As a case study, a CNT of L=3000 nm, h=0.34 nm, ~R =1 nm, d=500 nm, and initial rise bo=100 nm is considered. Suchdimensions are typical of CNTs with slack [20,21]. It is worth to mention that using these dimensions, the conditions statedin [28,29] have been checked to assure the validity of the continuum mechanics beam theory, which are

ac=L� 8� 10�551,

dNT=L� 6:6� 10�451,

where dNT its diameter and ac is the width of the hexagonal carbon rings, which is estimated to be near 0.24 nm [41].We have shown in a previous investigation that using one mode only in the ROM is enough to capture the response of a

CNT [17]. Next a one mode is used in the ROM to compare the variation of the maximum in-plane static deflection of theCNT with and without slack (Fig. 4). The figure indicates that for small DC load, the CNT with slack is relatively stiffer dueto the linear stiffness term added from the initial curvature. It can be also seen from the figure that the CNT with slackundergoes a pull-in instability at a lower DC load than the one without slack. This is expected since in the slackedconfiguration, the CNT is nearer to the electrode. Note here that the static response of the out-of-plane motion is zero sincethe in-plane deflection appears as a homogenous term in the out-of-plane equation (Eq. (7)).

The stability of the obtained solutions in Fig. 2 is studied by calculating the eigenvalues of the Jacobian matrix of theROM evaluated at these solutions [40]. The results show that one of the eigenvalues of the Jacobian matrix correspondingto the upper branches is always positive indicating unstable solution (dashed line in Fig. 2). It is also seen that all of theeigenvalues of the lower branches are pure imaginary indicating stable solutions (continuous line in Fig. 2). At pull-in, bothstable and unstable branches collide and destroy each other with one eigenvalue tending to zero corresponding to asaddle-node bifurcation.

5.2. The eigenvalue problem of CNTs with slack

Next, we investigate the eigenvalue problem of the slacked CNT by calculating the variation of the in-plane and out-of-plane natural frequencies and mode shapes with and without slack and under the actuation of the DC voltage. Toward this,we consider the ROM obtained in Section 2, which can be represented in a state-space form as

_X ¼ RðXÞ, (12)

where

X ¼ ½u1,u2,:::,un,x1,x2,:::,xm�, (13)


is the modal amplitudes vector and RðuÞ is a right-hand side vector representing the stiffness coefficients. RðuÞ is anonlinear function of the modal coordinates uiðtÞ and xiðtÞ. Note here that we consider the symmetric and anti-symmetricmode shapes in the ROM to get all the possible natural frequencies and mode shapes of the slacked CNT.

0 10 20 30 40 50 60 70 80 900

50

100

150

200

250

300

350

400

450

500

wm

ax (n

m) b0 = 200 nm

VDC (Volt)

b0 = 100 nmb0 = 0 nm

Fig. 2. Variation of the maximum in-plane static deflection of the CNT with the DC voltage for various levels of initial curvature b0. Solid (––––––) and

dashed (———) lines denote the stable and unstable branches, respectively.

0 10 20 30 40 50 60 700

50

100

150

200

250

300

350

400

450

500

Freq

uenc

y (M

Hz)

0.10.06 0.240.1650.03Slack (%)

0.30.0070

�2

b0 (nm)

�4

�3

�2

�1

�4

�3

�6

�5

�1

Fig. 3. Variation of the first few in-plane and out-of-plane natural frequencies of a CNT with slack for the case study of [19]. Solid line (––––––), dashed line

(———), and circles (o) denote, respectively, the odd in-plane, even in-plane and out-of-plane, and the odd out-of-plane frequencies.


Next, we split X into a static componentXs, representing the equilibrium position due to the DC actuation, and adynamic component ZðtÞ representing the dynamic perturbation around the equilibrium position, that is,

X ¼ XsþZðtÞ, (14)

Then, substituting Eq. (14) into Eq. (12), using the Taylor series expansion assuming small ZðtÞ, eliminating the higher-order terms, and using the fact that RðXsÞ ¼ 0, we get the following equation:

_Z ¼ JðXsÞZ, (15)

where JðXsÞ is the Jacobian matrix calculated at the equilibrium points [42].

0 50 100 150 2000

200

400

600

800

1000

1200

1400

1600

1800

2000

Non

dim

ensi

onal

freq

uenc

y

�6

�5

�4

�3

�2

�1

0 20 40 60 80 1000

200

400

600

800

1000

1200

Non

dim

ensi

onal

freq

uenc

y

b0 (nm)

b0 (nm)

�11

�10

�10

�9

�7

�8

�5

�6

�3

�4�1�2

Fig. 4. (a) Variation of the in-plane and out-of-plane natural frequencies of a CNT with various levels of slack at zero DC load. (b) A zoomed view of Fig. 4a

showing the crossings and veering of the in-plane frequencies (the odd out-of-plane are not shown for clarity). Solid line (––––––), dashed line (———), and

circles (o) denote, respectively, the odd in-plane, the even in-plane and the out-of-plane, and the odd out-of-plane frequencies (for interpretation of the

references to color in this figure, the reader is referred to the web version of this article).


To calculate the natural frequencies of the slacked CNT for a given voltage, we substitute the static stable solution,Xs

into the matrix J and then find its corresponding eigenvalues. The eignvalues are calculated by solving the belowcharacteristic equation, which gives an algebraic equation for the eigenvalue l

detðJðXsÞ�lIÞ ¼ 0, (16)

where I is the identity matrix and ‘‘det’’ refers to the determinant. Then by taking the square-root of each individualeigenvalue, we obtain the natural frequencies of the CNT. Finally, by solving for the eigenfunctions at each eigenvalue, thecorresponding mode shapes are obtained.

First, we consider the case study of Ustunel et al. [22], which is a non-hollow CNT of L=1750 nm, ~R =1 nm, andd=500 nm. For the following results, we will denote the in-plane natural frequencies by oiand the out-of-planefrequencies byli. In Fig. 3, the variation of the in-plane and out-of-plane natural frequencies is calculated at zero DCload for various values of initial rise of the CNT or slack percentages. The slack percentage is defined as %slack=(L� ~L )/L,where~Lis the length of the CNT in the deformed (curved) position. Fig. 3 compares the results with those reported in [22],which indicates good qualitative agreement. However, there is a quantitative discrepancy due to the fact that the modelused in [22] is applicable only when the strain is small enough that the nonlinear elastic effects may be ignored. Unlike themodel in [22], our approach is a continuous model that can describe the nonlinear oscillations of the CNT under its variousparameters without dividing the CNT behavior into different regimes.

In Fig. 4, we investigate in more depth the effect of varying the initial rise of the CNT of Fig. 2, assuming zero DC load, onthe in-plane and out-of-plane natural frequencies. We can see clearly that the even in-plane (dashed lines in Fig. 4a) andout-of-plane frequencies (circles in Fig. 4a) are insensitive to the variation of slack whereas the odd in-plane frequenciesvary with slack (continuous lines in Fig. 4a). These frequencies appear in the large scale to intersect (Fig. 4a). Whenenlarging the apparent intersection zones, one can see that they do not intersect (blue circles in Fig. 4b); they diverge in amanner called curve veering [43,44]. This phenomenon is common and has been cited for the natural frequencies of arectangular membrane when varying the ratio of its lengths’ sides [44] and also for the natural frequencies of cables whenvarying their sagging levels [45,46]. A frequency veering occurs when the loci of two eigenvalues, in an eigenvalueproblem, approach each other when a parameter is varied and then veer away when being too close like two repulsivecharges [47]. In a frequency veering, the eigenfunctions associated with the eigenvalues on each locus before veering isinterchanged during the veering [44]. To further clarify this, we plot the corresponding eigenfunctions of the odd in-planenatural frequencies in Table 1. We can see that the modes are exchanging shapes (bold squares in Table 1) when varyingthe slack level. This table describes all the veering scenarios shown in Fig. 4b, which appear following a straight line. In thisline, the shape of the first mode is transferred into the different odd modes, from the lower to the higher modes, dependingon the slack level. A final note to be mentioned here is that the odd in-plane frequencies intersects the even ones for

Table 1The simulated eigenfunctions of the first five odd in-plane modes for various slack levels.


certain levels of slack offering many possibilities of internal resonances and exchange of energy among higher- and lower-order modes.

Next we investigate the variation of the in-plane natural frequencies of the CNT with the DC load. First, we consider theunslacked case. We can see from Fig. 5 that all the natural frequencies are sensitive to the variation of the DC voltage. All ofthem are increasing dramatically, with a sub-linear variation behavior, except for the first frequency near pull-in where itdrops to zero. This increase in the frequencies suggests a tunable resonator over a wide range of frequencies. Practically,

0 10 20 30 40 50 60 70 80 900

0.5

1

1.5

2

2.5

3x 104

Non

dim

ensi

onal

freq

uenc

y

0 2 4 6 8 100

500

1000

1500

2000

2500

3000

3500

4000

4500

5000

Non

dim

ensi

onal

freq

uenc

y

VDC (Volt)

�10�11

�11

�9

�8

�7

�6

�5

�4

�3

�2

�10

�9

�8

�7

�6

�5

�4

�3

�2

�1

�1

VDC (Volt)

Fig. 5. (a) Variation of the in-plane natural frequencies with the DC load of a CNT without slack. (b) A zoomed view of Fig. 5a near small values of

voltages. Solid line (––––––) and dashed line (———) denote the odd in-plane and the even in-plane frequencies, respectively.


when designing a resonator made with such a CNT, one can easily tune the frequencies from MHz range to GHz and evenTHz range. In addition, one can see from Fig. 5b that there is no possibility of modes veering or modes crossing even forsmall range of voltages. These scenarios however appear when slack is added (Fig. 6). The figure shows the variation of thefirst few in-plane frequencies of the slacked CNT (b0=100 nm) with the DC load. We can see that the odd modes exhibitfrequency veering phenomenon (blue circles in Fig. 6b), but in this case from the higher to the lower modes where the

0 10 20 30 40 50 60 70 80 900

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�11

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�9

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�3

�2

�1

�11

�10

�9

�8

�7

�6

�5

�4

�3

�2

�1

VDC (Volt)

Fig. 6. . (a) Variation of the in-plane natural frequencies of the CNT with the DC load of a CNT with a 100 nm slack. (b) A zoomed view of Fig. 6a near

small values of voltages showing the crossings and veering of the modes. Solid line (––––––) and dashed line (———) denote the odd and even in-plane

frequencies, respectively (for interpretation of the references to color in this figure, the reader is referred to the web version of this article).


energy is transferred to the lowest fundamental frequency, which eventually drops to zero at pull-in. There are alsopossibilities of odd and even in-plane modes crossings (blue arrows in Fig. 6b).

It is worth to mention here that our demonstrated results can justify and predict some of observed experimentallyphenomena in [21], such as the sub-linear variation of the frequencies, the avoided crossings with the DC load (veering),

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�4

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�1

�4

�3

�2

�5

�1

�2

�4

�5

�2

�3

�4

�5

VDC (Volt)

Fig. 7. (a) Variation of the first five in-plane and out-of-plane natural frequencies with the DC load of a CNT with a 100 nm slack. (b) A zoomed view

of Fig. 7a near small values of voltages. Solid line (––––––), dashed line (———), and circles (o) denote, respectively, the odd in-plane, the even in-plane and

the out-of-plane, and the odd out-of-plane frequencies.


the abundance of resonances, and the negative variation of the frequencies for certain nanotubes. Other models of [21,22]could not explain the previously mentioned phenomenon. For example, Fig. 5 shows what was cited as sub-lineardispersion of the frequency in [21]. The avoided crossings for some frequency range in the case of slack cited in [21], can beconsidered to be the same as veering (Fig. 6). Also, our results, based on the coupled in-plane and out-of-plane motions,have shown abundance of resonances as cited in [21].

Next, we investigate the sensitivity of the out-of-plane natural frequencies to the DC load, which although not directlyactuated, they are affected due to the nonlinear coupling between the out-of-plane and in-plane deflections. Fig. 7 showsthe first five in-plane and out-of plane natural frequencies as they vary with the DC load. We can note that, except for thefirst mode, the first few out-of-plane frequencies are larger than those of the in-plane frequencies especially at high DCloads. For the higher order modes, both in-plane and out-of-plane natural frequencies are equal.

5.3. The dynamic response

Next, we use the ROM developed in Section 3 to integrate the differential equations of motion in time to obtain thedynamic response of the slacked CNT under a very small DC and AC harmonic load. The choice of the very small voltageloads is to guarantee linear forced vibration response. The response to small electric loads is important to enable preciseprediction of the resonance frequency in the linear regime. The resonance frequency is the frequency that is commonlymeasured experimentally for CNTs when driven by AC and DC loads. Here we use a nondimensional damping coefficientc=1.196. Figs. 8a and b shows the response of the 100 and the 200 nm slacked CNTs, respectively. We can see from thefigures that the linear dynamic response is significant in the neighborhood of the first natural frequency (o1) and thefrequencies that are located on the veering straight line, as predicted from Fig. 4. Those frequencies are o9 in the case ofthe 100 nm slack and o11 in the case of the 200 nm slack. It is clear from Fig. 8a and b that the higher-order modes locatedon the veering line are sharing the energy of vibration with the fundamental mode. This conclusion might explain one ofthe reasons behind the low quality factor reported experimentally for CNTs [20,21] when driven harmonically at resonancenear their fundamental modes.

To further clarify this point, we calculate in Fig. 9 the participation of each individual odd in-plane mode shape in thedynamic response of the 100 nm slacked CNT. It is clear from the figure that the participations of the first mode and ninth

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� �

� � �

�3

�3

�5�7

�5

�7

�9

�9

�11

�1

�1

Fig. 8. Frequency-response curves of (a) 100 nm slacked CNT and (b) 200 nm slacked CNT showing peaks at the odd in-plane natural frequencies. Results

are shown for VDC=0.01 V, VAC=0.01 V, and 100 quality factor.

2 4 6 8 10 12 14

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�1

(0.5

) u1

(t)

�3

(0.5

) u3

(t)

�5

(0.5

) u5

(t)

�7

(0.5

) u7

(t)

�9

(0.5

) u9

(t)

t

Fig. 9. Time–response curves of the 100 nm slacked CNT at O�o1 showing the participation of the odd in-plane mode shapes. Results are shown for

VDC=0.01 V, VAC=0.01 V, and 100 quality factor.


mode (the one located on the veering line) are the most important ones. This indicates that even when exciting the CNTnear its first natural frequency, significant participation is anticipated of the mode located on the veering line. In summary,the dynamical behavior a slacked CNT of such a case is considered governed by two modes: the lowest mode, which isalways the dominant one, and the mode located on the veering line.

6. Conclusion

In this paper, an investigation into the 2D eigenvalue problem of slacked carbon nanotubes when actuated by a DC loadwas presented. The in-plane and out-of-plane natural frequencies of the CNTs have been calculated numerically for variousvalues of DC voltages and slack levels using the reduced-order model. Our approach has been validated by comparing themto other published results. It has been also demonstrated to be an efficient continuous model, which is computationallycheap compared to molecular dynamics models. We found that the effect of slack (curvature and initial deformation) ofcarbon nanotube has significant influence on the stability, natural frequencies, and pull-in voltages of such structures. Wehave showed possible scenarios of modes crossings and mode veering offering the possibilities of energy transfers betweenmodes as well as possibilities of internal resonances. We showed also the possibilities of activating resonances at higherorder modes other than the fundamental one. We found that those modes depend on the slack level and are located on theveering line obtained when solving the eigenvalue problem for various slacks. The understanding of these phenomena canbe useful to realize smart devices. This will also allow researchers to utilize those nanostructures in useful applications andenable their calibration as sensors and resonators of distinguished characteristics despite their nonlinear behaviors.

Acknowledgment

The authors would like to thank the National Science Foundation, which supported this work through Grant no.0928494, and Prof. Paul L. McEuen and Mr. Arend van der Zande of Cornell University for their useful discussions.

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