Composite Structures 126 (2015) 52–61

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Composite Structures

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

Free vibrations and stability of high-speed rotating carbon nanotubespartially resting on Winkler foundations

http://dx.doi.org/10.1016/j.compstruct.2015.02.0370263-8223/� 2015 Elsevier Ltd. All rights reserved.

⇑ Corresponding author.E-mail address: torkaman-asadi@ae.sharif.edu (M.A. Torkaman-Asadi).

M.A. Torkaman-Asadi ⇑, M. Rahmanian, R.D. Firouz-AbadiDepartment of Aerospace Engineering, Sharif University of Technology, P.O. Box 11155-8639, Tehran, Iran

a r t i c l e i n f o

Article history:Available online 25 February 2015

Keywords:Nonlocal elasticityCarbon nanotubesFree vibrationsSize effectsNanomotor

a b s t r a c t

In the present study, free vibrations and stability of rotating single walled carbon nanotubes (SWCNT) isinvestigated by nonlocal theory of elasticity; while the CNT is partially resting on an elastic foundation.The governing equations of motion are presented by using Love’s shell assumptions. An exact seriesexpansion method of solution is employed and very accurate results are obtained. Some parameter stud-ies including the effects of rotating speed, foundation stiffness, slenderness ratio and nonlocal parameteron the natural frequency and stability margins of the current model are studied. The studies show thatrotation rates and foundation elasticity can contribute significantly in the dynamic characteristics ofrotating MEMS devices.

� 2015 Elsevier Ltd. All rights reserved.

1. Introduction

The advent of new technologies and fabrication processes hasled to many novelties in the field of nano/micro electromechanicalsystems (N/MEMS). The most recent ones are the micro heat engi-nes, nanoturbines, nanoscale molecular bearings, shaft and gear,and multiple gear systems which can revolutionize not only aero-nautical and astronautical industries but also the automotive andelectronic industries [1]. Rotating nanotubes have attracted atten-tions in the recent years due to their promising future to be used innanomachines [2,3] which include shaft of nanomotor [4,5] devicesand carbon nanotube (CNT) gears [6]. These rotating nano struc-tures can be also considered as MEMS gyroscope sensors [7].With dramatically reduced cost, size, and weight, MEMS gyro-scopes potentially have a wide application spectrum in the aero-space industry, military, automotive and consumer electronicsmarkets including advanced automotive safety systems, high per-formance navigation and guidance systems, ride stabilization,roll-over detection and prevention and image stabilization in digi-tal cameras. Miniaturization of gyroscopes also enable higher-endapplications including micro-satellites, microrobotics, and evenimplantable devices to cure vestibular disorders. to be used innavigation systems while their design would surely require a com-prehensive insight over their dynamic behavior/response. Due tothe size effect property in nano/micro scales, CNTs must be treatedby theories which include these effects. One such theory is called

the nonlocal elasticity theory which was first proposed byEringen [4,5] to consider the scale effect in elasticity by assumingthe stress at a reference point to be a function of the strain fieldat every point in the body.

During the past decade, numerous investigations were pub-lished, addressing different aspects of rotating nanotubes. Somestudies in literature are concerned with the possibility of rotationoccurrence in nanotubes while there are few studies aiming at freevibration and stability analysis of rotating CNTs. Kral andSadeghpour [8] and Krim et al. [9] showed the possibility to spinnanotubes with circularly polarized light, further motivating theusage of nanotubes as possible axis of rotation at the nanoscale.However, no systematic analytical and numerical study of rotationhas yet been done, except the work of Zhang et al. [10], wherenonequilibrium molecular dynamics simulations were carried outto calculate the energy dissipation rate during the rotationalmotion. In their work, simulations were performed byHamiltonian dynamics in the microcanonical ensemble whileavoiding additional sources of dissipation. Researchers have alsoreported the feasibility of nanoscale rotating structures.Examples include study of molecular gears [11], fullerene gears[12], and carbon nanotube gears [13,14]. Srivastava [13] hasreported the rotational dynamics of carbon nanotubes and carbonnanotube gears under a single applied laser field. Lohrasebi andTabar [15] carried out computational modeling of rotatingnanomotor using stochastic molecular dynamics (MD) method.Zhang et al. [10] carried out atomistic simulations of doublewalledcarbon nanotubes as rotational bearings. Recently Fennimore et al.[4] reported the feasibility of rotating nanostructures by the

M.A. Torkaman-Asadi et al. / Composite Structures 126 (2015) 52–61 53

construction and successful operation of a nanoscale electrome-chanical actuator. Bailey et al. [16] proposed a new dc drivemechanism for CNT-based motors. They proposed a new forceother than elastic, electrostatic, friction and van der Waals forceswhich has so far been ignored in the NEMS literature. Pradhanand Murmu [17], applied a nonlocal beam model to investigatethe flap wise bending-vibration characteristics of a uniform rotat-ing nanocantilever. The nonlocal natural frequencies wereobtained using the Differential Quadrature Method (DQM). Theyalso discussed the effects of the nonlocal small-scale, angular velo-city and hub radius on vibration characteristics of the nanocan-tilever. Murmu and Adhikari [18], investigated the sameproblem, but considering an initially prestressed single-walled car-bon nanotube to analyze the effect of the initial preload on vibra-tion characteristics. For efficient design of these rotatingcomponents, proper understanding of the mechanical behaviorssuch as natural frequencies and critical rotating speeds arerequired. The development of simplified models for the dynamicsof complex nano-technological systems is thus necessary becausefor many cases, fully atomistic simulations would be computation-ally expensive and sometimes impossible. Feng et al. [19] haveintroduced a novel idea based on rotating nanotubes. They haveactually implemented the molecular dynamics simulations toinvestigate pumping of water by rotating CNTs and found thatthe radius and the angular velocity of the CNT are the most promi-nent parameters effective in pumping water. They have alsoreported other interesting conclusions and have well defined theinfluential parameters. Wang et al. [20] examined the applicabilityof the simplified Flugge and Donnell cylindrical shell models forCNTs to study their static buckling and free vibrations. The validityand accuracy of theories such as Donnell/Love thin shell theory,Sanders thin shell theory and the first order shear deformable the-ory in predicting the critical buckling strains of axially loadedSWCNTs were examined in [21].

The present study aims at investigating the stability character-istics of rotating carbon nanotubes. Since all the novel ideas includ-ing the rotating nanotubes as their primary component, work atvery high rotation speeds, a comprehensive insight over the sta-bility can help the researchers in designing optimum MEMSdevices. To this aim, the Love shell theory is utilized because thatis shown to be accurate in determining the stability boundariesand vibrational behaviors for the cases where thickness to radiusratios are small [22]. The more accurate shear deformable theoriesare not included in this study to avoid any complexities in thestructural modeling; thus the presented formulation is not sug-gested for the case of large thickness to radius ratios (i.e multi-walled carbon nanotubes). Utilizing the aforementioned assump-tions along the nonlocal Eringen’s theory of elasticity and rotationeffects would lead to the governing equations of motion for theCNT in partial elastic foundation. A special attention is also paidto understand the physical significance of different parameterson the stability margins.

2. Governing equations

2.1. Nonlocal elasticity

When the lattice spacing between the individual atoms of ananostructure is not negligible compared to the structural dimen-sions, usage of the continuum theory with the assumption of con-tinuity in the structure is essentially doubtful [23]. Namely, theclassical continuum models will fail to simulate the structuralbehavior unless they are modified to include the size and nonlocaleffects. The nonlocal theory of elasticity [24] proposes a modifica-tion to classical theories to account for the small size effects.

Regarding the nonlocal elasticity theory [25,26,24], the stressstate at a reference point x in the body depends on not only thelocal strain state at that point but also on the strain state at allother points x0 of the body. For a homogeneous isotropic elasticsolid, the nonlocal stress tensor is defined by the following set ofequations [26]:

tkl;k ¼ �qð f l � €ulÞ ð1Þ

tklðxÞ ¼Z

vað x0 � xj j; sÞrklðx0Þdvðx0Þ ð2Þ

rklðx0Þ ¼ k trð�rrðx0ÞÞdkl þ 2g�klðx0Þ ð3Þ

�klðx0Þ ¼12

@ukðx0Þ@x0l

þ @ulðx0Þ@x0k

� �ð4Þ

where tkl; q; f l and ul are the nonlocal stress tensor, mass density,body force density and the displacement vector at a reference pointx, respectively. Furthermore the classical (Hookean) stress tensor atthe point x0 which is shown by rklðx0Þ is related to the linear straintensor �klðx0Þ by means of k and g parameters which are the Lameconstants and are defined as

k ¼ Emð1þ mÞð1� 2mÞ ; g ¼ E

ð1þ mÞ ð5Þ

where E and m are the elastic modulus and Poisson’s ratio,respectively.

The difference between the classical and nonlocal elastic theorylies in the material constitutive relation (Eq. (2)) which incorpo-rates a nonlocal kernel function which describes the relative influ-ences of the strains at various locations on the stress at a givenlocation. The term að x0 � xj j; sÞ is the nonlocal modulus or atten-uation function incorporating the nonlocal effects into constitutiveequations at the reference point x produced by local strain at thesource x0. Various different forms of attenuation functions areintroduced in literature by matching the curves of plane waveswith those due to atomic lattice dynamics [24]. Also s ¼ e0a=lwhere a and l are internal and external characteristic lengths,respectively and e0 is a constant appropriate for each materialand known as the scaling effect. The external characteristic lengthscan be the wavelength or crack length while the internal character-istic length is the length of the bonds which is a ¼ 0:142 nm forcarbon nanotubes. For the scaling factor e0, Zhang and Sharma[27] have shown that the adopted value depends on the crystalstructure in lattice dynamics and the nature of physics underinvestigation.

The differential form of the integro-differential equations isexpressed as

ð1� l2r2Þtkl ¼ rkl ð6Þ

where l ¼ sl; r2 ¼ @2

@x2 þ 1R2

@2

@h2 is the Laplace operator, x is the axialcoordinate, h is circumferential angular coordinate and R is theradius of the nanotube. Eq. (6) will be the same as the classicalHookean relation in case that l ¼ 0.

2.2. Nonlocal shell model

For the present study, Love’s first approximation hypothesis areapplied via the Love’s shell theory. These assumptions are: (a) thethickness of the shell is small compared with the radius; (b)the strains and the displacements are sufficiently small so thatthe quantities of the second and higher order magnitude in thestrain displacement relations maybe neglected in comparison withthe first order terms; (c) the transverse normal stress is smallcompared with the other normal stress components and may beneglected; (d) The normal to the unreformed middle surfaceremains straight and normal to the deformed middle surface and

Fig. 1. A nanomotor attached to electrodes, which act as reservoirs of electrons.

54 M.A. Torkaman-Asadi et al. / Composite Structures 126 (2015) 52–61

it suffers no extension; (e) the rotary inertia and transverse sheardeformation effects maybe neglected. It’s been shown that suchassumptions are fulfilled for the cases were the thickness to radiusratio of the shell is less than 0.05. It must be kept in mind thatexceeding this value may give inaccurate or spurious results[20,22]. For larger thickness to radius values, the shear deforma-tion effects must also be included.

Carbon nanotubes are often modeled as isotropic elastic cylin-drical shells. The anisotropies due to the intrinsic discrete natureof CNTs and imperfections can be neglected because they give amarginal contribution. The cylindrical shell, shown in Fig. 2, repre-senting the nanomotor under investigation (Fig. 1), is considered tobe thin and made of isotropic, homogeneous material. As shown inFig. 2 it has a length of L, a radius of R and a thickness of h. The shellis rotating about the x axis at a constant angular velocity of X. Thelinear elastic material properties are the Young’s modulus E, thePoisson’s ratio m and the mass density q. A rotating cylindricalcoordinate system ðx; h; zÞ is attached to the mid-surface of theshell and the variables x; h and z represent the axial, circumferen-tial, and radial coordinates, respectively.

It is also partially embedded on an elastic foundation which isconsidered to be symmetric about the middle vertical plane ofthe CNT. The nanotube, can be considered as separate sectionswhere the discontinuity at each interface is removed by employingthe compatibility relations. The length of each division is deter-mined by the length of partial elastic foundation. To study the par-tial foundation effects, a dimensionless parameter, say b, isintroduced which is the ratio of the non-embedding length (L1)to the total length (L) of the nanotube. Hence, b ¼ 0 represents afully suspended nanotube on a Winkler foundation.

To obtain the governing equations of a rotating CNT, one canfirst develop the equations in macro scale. The nonlocal counter-parts to the macro scale equations are obtained by applying theð1� l2r2Þ operator to the inertia and external loading tensors.

The displacement field can be expressed in terms of the mid-surface deformations in x; h and z directions which are denotedby u;v and w, respectively. Also, the velocity vector of any genericpoint on the rotating shell is expressed as

h

R

kw

θ L2

Fig. 2. An equivalent shell model for th

V ¼ _u iþ _v jþ _wkþX i�wkð Þ þX i� v jð Þ ð7Þ

where the first three terms are due to contribution of the linearvelocities in the axial, circumferential and lateral directions, respec-tively. The fourth and fifth terms are the contributions due to theCoriolis and centrifugal effects. The temporal derivatives are shownby overdots, while i; j and k are unit vectors in the x; h and z direc-tions, respectively. Following the Hamiltonian principle, the kineticand potential energies are obtained as [28]

T ¼ 12qhZ l

0

Z 2p

0V:V Rdhdx ð8Þ

U ¼ Uh þ Ue ¼12

Z l

0

Z 2p

0

(Nh

R2

@u@h

� �2

þ @v@hþw

� �2

þ v � @w@h

� �" #

þeTre

)Rdhdx ð9Þ

W ¼Z l

0

Z 2p

0qw Rdhdx ð10Þ

where the first term in the potential energy represents the hoop

strain energy and the term Nh ¼ qhR2X2 is the initial hoop forceresultant due to the centrifugal force. e and r are also defined asthe elastic strain and isotropic stress tensors, respectively. Also,qw is the radial external pressure applied on the shell which isqw ¼ kww based on the linear Winkler foundation model. Afterapplying the nonlocal operator to mass and external loading termsof the classical equations and some manipulations, one obtains thefollowing nonlocal equations for a rotating CNT:

L11 L12 L13

L21 L22 L23

L31 L32 L33

264

375

u

vw

8><>:

9>=>;¼ð1�l2r2Þ qh

€u€vþ2X _w€w�2X _v

8><>:

9>=>;þ

00

qw

8><>:

9>=>;

0B@

1CAð11Þ

where the operators Lij are defined as

L11 ¼ A11@2

@x2 þA66

R2 þ qhX2� �

@2

@h2 ð12Þ

L12 ¼A12 þ A66

R@2

@x@hð13Þ

L13 ¼A12

R@

@xð14Þ

L21 ¼A12 þ A66

R@2

@x@hð15Þ

L22 ¼ A66 þ2D66

R2

� �@2

@x2 þ1R2 A22 þ

D22

R2 þ qhR2X2� �

@2

@h2 ð16Þ

L

LL

1

kw

2

e nanomotor mechanism, (L1 ¼ bL).

Table 1Comparison of the natural frequencies of a simply supported CNT with [20,30] atdifferent radii and length ratios for n ¼ m ¼ 1

Radius(nm)

L/R Frequency (Hz)

Wang et al. [20] Fazelzadeh et al.[30]

Present study

12 12 12

M.A. Torkaman-Asadi et al. / Composite Structures 126 (2015) 52–61 55

L23 ¼A22

R2 þ 2qhX2 � D12 þ 2D66

R2

@2

@x2 �D22

R4

@2

@h2

!@

@hð17Þ

L31 ¼ �A12

R@

@xð18Þ

L32 ¼1R2 ðD12 þ 4D66Þ

@2

@x2 � A22 þD22

R2

@2

@h2

!@

@h� 2qhX2 @

@hð19Þ

L33¼ qhX2�D22

R4

@2

@h2

!@2

@h2� D11@2

@x2þ2ðD12þ2D66Þ

R2

@2

@h2

!@2

@x2�A26

R2

ð20Þ

The constant coefficients Aij and Dij are the extensional and bendingstiffness matrices which are defined as

Aij ¼Z h=2

�h=2Qijdz ð21Þ

Dij ¼Z h=2

�h=2z2Q ijdz ð22Þ

where

Q 11 ¼ Q 22 ¼E

1� m2 ; Q12 ¼ Q21 ¼Em

1� m2 ; Q 66 ¼E

2ð1þ mÞ ð23Þ

In the present study, the well-known boundary condition of Simplysupported-Simply supported ðS� SÞ with no axial constraint is con-sidered which is defined as

v ¼ 0; w ¼ 0; Nx ¼ 0; Mx ¼ 0 ð24Þ

where Nx and Mx are the force and moment resultants which arecommonly expressed in terms of the following relations

Nx ¼Z h

2

�h2

rxdz; Mx ¼Z h

2

�h2

zrxdz ð25Þ

0.65 0.5 8.16 � 10 8.2028 � 10 8.2334 � 10

1 4.97 � 1012 4.9656 � 1012 4.9765 � 1012

5 1.04 � 1012 1.0434 � 1012 1.0427 � 1012

10 3.29 � 1011 3.2928 � 1011 3.2901 � 1011

50 1.47 � 1010 1.4704 � 1010 1.4691 � 1010

100 3.68 � 109 3.6905 � 109 3.6873 � 109

5 0.5 6.78 � 1011 6.7789 � 1011 6.7806 � 1011

1 6.11 � 1011 6.1134 � 1011 6.1141 � 1011

5 1.30 � 1011 1.3543 � 1011 1.3544 � 1011

10 4.28 � 1010 4.2749 � 1010 4.2752 � 1010

50 1.91 � 109 1.909 � 109 1.9091 � 109

100 4.79 � 108 4.7914 � 108 4.7918 � 108

Table 2Comparison of the natural frequencies of a simply supported CNT with [30] atdifferent circumferential half-waves and nonlocal parameters m ¼ 1.

n frequencies (cm�1 = Hertz/speed of light in cm/s)

l ¼ 0:0 nm l ¼ 0:10 nm l ¼ 0:20 nm

Ref. [30] Present Ref. [30] Present Ref. [30] Present

2 20.267 21.293 19.016 19.948 16.313 17.0633 57.307 58.037 50.15 50.723 38.434 38.8024 109.865 110.674 88.427 88.975 61.662 61.9765 177.66 178.566 130.638 131.175 84.718 85.0086 260.608 261.621 174.783 175.311 107.357 107.6347 358.676 359.805 219.717 220.241 129.599 129.878 471.849 473.098 264.827 265.352 151.516 151.7859 600.118 601.494 309.81 310.337 173.177 173.44610 743.474 744.982 354.529 355.059 194.637 194.906

3. Solution procedure

Following the solution procedure outlined in [29], let us assumethe solutions of Eq. (11) as the expansion series

uðx; h; tÞ ¼XN

j¼0

ajx j cosðnhþxtÞ ð26Þ

vðx; h; tÞ ¼XN

j¼0

bjx j sinðnhþxtÞ ð27Þ

wðx; h; tÞ ¼XN

j¼0

cjx j cosðnhþxtÞ ð28Þ

where n is an integer representing the number of circumferentialhalf-waves. Also, the natural frequency of the shell is defined byxðrad=sÞ. The unknown constants aj; bj; cj are to be determinedby means of recursive expressions and then applying desiredboundary conditions. Substituting Eqs. (26)–(28) into Eq. (11) andcollecting for the same powers of x, one obtains the following recur-sive expression for m P 0

amþ2 ¼ G11am þ G12bmþ1 þ G13cmþ1 ð29Þ

bmþ2 ¼ G21amþ1 þ G22bm þ G23cm þ G24cmþ2 ð30Þ

cmþ4 ¼ G31amþ1 þ G32bm þ G33bmþ2 þ G34cm þ G35cmþ2 ð31Þ

where Gij are defined as

G11 ¼qh n2X2 �x2 1þ ln

R

� �2� �h i

þ n2

R2 A66

ðA11 � qhl2x2Þðmþ 1Þðmþ 2Þ ð32Þ

G12 ¼ �ðA12 þ A66Þn

RðA11 � qhl2x2Þðmþ 2Þ ð33Þ

G13 ¼ �A12

RðA11 � qhl2x2Þðmþ 2Þ ð34Þ

G21 ¼nðA12 þ A66Þ

RðA66 þ 2R2 D66 � qhl2x2Þðmþ 2Þ

ð35Þ

G22 ¼qh n2X2 �x2 1þ ln

R

� �2� �h i

þ n2

R2 A22 þ D22R2

� �A66 þ 2D66

R2 � l2qhx2� �

ðmþ 1Þðmþ 2Þð36Þ

G23 ¼n

R2 A22 þ n2D22R2

� �þ 2qhX nX�x 1þ ln

R

� �2� �h i

A66 þ 2D66R2 � qhl2x2

� �ðmþ 1Þðmþ 2Þ

ð37Þ

G24 ¼ �n

R2 ðD12 þ 2D66Þ � 2l2qhxX

A66 þ 2D66

R2 � qhl2x2ð38Þ

G31 ¼ �A12

D11Rðmþ 2Þðmþ 3Þðmþ 4Þ ð39Þ

56 M.A. Torkaman-Asadi et al. / Composite Structures 126 (2015) 52–61

G32 ¼2qhX x 1þ ln

R

� �2� �

� nXh i

� nR2 A22 þ n2D22

R2

� �D11ðmþ 1Þðmþ 2Þðmþ 3Þðmþ 4Þ ð40Þ

G33 ¼n

R2 ðD12 þ 4D66Þ � 2qhxXl2

D11ðmþ 3Þðmþ 4Þ ð41Þ

G34 ¼ �n4

R4 D22 þ A22R2 þ n2qhX2 þ 1þ ln

R

� �2� �

ðkw � qhx2ÞD11ðmþ 1Þðmþ 2Þðmþ 3Þðmþ 4Þ ð42Þ

G35 ¼2n2

R2 ðD12 þ 2D66Þ þ l2ðkw � qhx2ÞD11ðmþ 3Þðmþ 4Þ ð43Þ

Fig. 3. Frequency variations of a S–S rotating nanotube f

These recursive expressions offer exact and closed form solu-tions for any desired boundary conditions. In the aforementionedsolution, unlike the weighted residual methods, one does not needtrial functions satisfying special sets of boundary conditions andthis can be named as one of the most fundamental advantages ofthe series solutions compared to semi-analytical methods. Theaccuracy can also be adjusted as needed by justifying the seriestruncation value ðNÞ.

As previously mentioned, the nanotube is partially suspendedon a Winkler foundation. To account for the elastic foundationeffects, separate sections are considered through the length ofthe nanotube. These sections are totally embedded/non-embedded

or different nonlocal parameters (n ¼ m ¼ 1; b ¼ 1).

M.A. Torkaman-Asadi et al. / Composite Structures 126 (2015) 52–61 57

portions which are associated with b ¼ 0 and b ¼ 1, respectively.Such consideration induces some discontinuities in the nanotube.Hence, in addition to the conventional boundary conditions at bothedges of the nanotube, some compatibility relations should be con-sidered as well. Since the partial foundation is considered to besymmetric with respect to the centroid of the nanotube, at leastthree separate sections should to be considered with 8 com-patibility relations at each discontinuity interface, which aredefined as follows

Fig. 4. Frequency variations of a S-S rotating nanotube for differen

Fig. 5. Frequency variations of a S-S rotating nanotube for diffe

ui ¼ uj; v i ¼ v j; wi ¼ wj; w;xi¼ w;xj

Nxi¼ Nxj

; Mxi¼ Mxj

; Nxhi¼ Nxhj

; Q xzi¼ Qxzj

ð44Þ

where the subscripts i; j corresponds to the any of the divisioninterfaces. Also Nxh and Qxz are the shear force resultants whichare defined as follows based on the Love assumptions for a thincylindrical shell:

Nxh ¼Z h

2

�h2

rxhdz; Qxz ¼ Mx;x þ1R

Mx;h ð45Þ

t circumferential waves (m ¼ 1; L=R ¼ 3; l ¼ 0:5 nm; b ¼ 1).

rent nonlocal parameters (n ¼ m ¼ 1; X ¼ 10 GHz; b ¼ 1).

58 M.A. Torkaman-Asadi et al. / Composite Structures 126 (2015) 52–61

4. Results and discussion

The behavior of a carbon nanotubes considered as thin shellscan be studied by a prior consideration of the mechanical andgeometrical properties including Young’s modulus (E), mass densi-ty (q), Poisson’s ratio (m) and thickness (h). Experimental observa-tions have shown that CNTs have variations of the effectiveYoung’s modulus, Poisson’s ratio and wall thickness amongE ¼ 0:97 � 5:5 TPa; m ¼ 0:12 � 0:28 and h ¼ 0:0665 � 0:35 nm,respectively. The large differences between the reported valuesintroduces uncertainties in drawing any general conclusions, soduring the recent years, some researchers have focused on

Fig. 6. Frequency variations of a S–S rotating nanotube

unification of the mechanical/geometrical parameters. To thisaim, it’s been found that some other combinatory factors such asEh can attain a fixed value for different measurement techniques.Such unification technique has narrowed down these variations,for example the commonly adopted Eh parameter varies in therange of 330 � 380 N/m.

The equivalent Young’s modulus and thickness are determinedby using the relations of the tensile rigidity (C ¼ Eh) and bending

rigidity (D ¼ Eh3=12ð1� m2Þ) from the classical shell theory, while

the values of tensile and bending rigidities are obtained from themolecular dynamics (MD) simulations. Following this procedure,the effective values of material properties used in the present

for different rotating speeds at (n ¼ m ¼ 1; b ¼ 1).

Fig. 7. Forward wave fundamental frequency variations versus the foundation stiffness of a S–S CNT for various rotating speed (L=R ¼ 5; b ¼ 0:0; l ¼ 0:5 nm).

M.A. Torkaman-Asadi et al. / Composite Structures 126 (2015) 52–61 59

study are C ¼ 360 N=m; D ¼ 2 eV; m ¼ 0:25 and q ¼ 2:3 g=cm3.These values correspond to the effective thickness h ¼ 0:1 nmand equivalent Young’s modulus E ¼ 3:6 TPa.

Depending on the desired accuracy in numerical calculations,the upper limit of the summation in the series expansions of theobtained recursive relation is truncated at a finite number N, whichmay have large values, indeed. The base solutions are power seriesand their derivatives can be simply obtained in a systematic proce-dure. Also, the base functions and their derivatives are onlyevaluated at the boundaries to obtain the characteristic equation.Accordingly, the proposed algorithm is executed very fast evenfor a large number of terms N. Based on the presented relations,a computer program was developed to obtain numerical results.

In the first step, the obtained results should be verified bymeans of relevant studies. Due to the lack of published papers inthe field of rotating nanotubes, the CNT’s without rotations areconsidered. The natural frequencies are compared with two otherstudies [20,30] which have utilized the Flugge and simplifiedDonnell theories. The comparison is outlined in Table 1 whichshows obvious agreement of the present results with those of theFlugge theory, which is usually referred as the most accurate the-ory in modeling nanotubes [20]. Another comparison of the naturalfrequencies for different circumferential half-waves and nonlocalparameters are also demonstrated in Table 2 which also confirmsthe validity of the present results.

4.1. Rotation effects

In rotating structures the eigenfrequencies depend on the rota-tion rates due to the induced gyroscopic effects. In such mechan-isms there are two distinct natural frequencies for any specificcondition. These are called forward (F.W.) and backward waves(B.W.). As the rotation speed increases, the forward wave decreaseswhile the backward wave experiences an increase. This leads tosome critical conditions corresponding to the case when the for-ward frequency approaches zero (bifurcation buckling).

The variations of the non-dimensional forward and backwardwaves (x� ¼ x=x0) versus the non-dimensional rotation speeds

(X� ¼ X=x0) are shown in Fig. 3 for different nonlocal parametersof a simply supported SWCNT, where x0 is the non-rotating naturalfrequency. As shown, for any specific radius, as the slenderness ratioincreases, the critical rotating speed decreases. This decrease ismuch more significant for large nonlocal parameters. It can also bededuced from Fig. 3 that as the radius of the CNT increases, the effectof nonlocal parameter decreases substantially so that for largeradius and slenderness ratios, the critical rotating speed isapproximately independent of the nonlocal parameter. This impliesthat for long or large-diametered rotating nanotubes (approximate-ly L > 20 nm or R > 5 nm), the local and nonlocal critical rotatingspeeds are the same and one can simply neglect size effects on thecritical rotating speed estimation of the rotor. This shows the sig-nificant effect of gyroscopic terms, included in the analysis.

An intriguing result is that the only circumferential half wavenumber, which may lead to instability conditions is n ¼ 1 basedon the Love/Donnell shell theory. This can be inferred from Fig. 4where the variations of the non-dimensional frequency versusthe non-dimensional rotating speeds for different circumferentialwave numbers are plotted. Also, as the radius of the CNT increasesthe critical speed decreases.

4.2. Size effects and parameter study

As discussed earlier, the gyroscopic effects can counteract thesize effects and for some specific dimensions, the size effects canbe neglected in determining the critical rotating speeds.Moreover, Figs. 5 and 6 are also presented for a better understand-ing of the effect of nonlocal parameter and rotation speeds on thebifurcations characteristics. Based on Fig. 5, for all nonlocal para-meters at a specified rotating speed, the critical condition occursat a specific slenderness ratio which means that the critical condi-tion is independent of the nonlocal parameter. Furthermore, thecritical slenderness ratio increases for larger radii. The same behav-ior is understood from Fig. 6, so the critical slenderness ratiodecreases for larger rotating speeds, as expected. It is also deducedthat for any specific rotating speed, the critical slender ratiodecreases as the radius of the CNT increases.

Fig. 8. Frequency variations of a rotating CNT versus foundation ratios (b) atdifferent rotating speeds (m ¼ n ¼ 1; kw ¼ 1020 N=m3; R ¼ 5 nm, L ¼ 50 nm).

Fig. 9. Frequency variations of a rotating nanotube at different foundationstiffnesses (m ¼ n ¼ 1; b ¼ 0:4; l ¼ 0:5 nm; R ¼ 5 nm, L ¼ 50 nm).

Fig. 10. Frequency variations of a rotating nanotube at different nonlocalparameters (m ¼ n ¼ 1; b ¼ 0:6; kw ¼ 1020 N

m3 ; R ¼ 5 nm, L ¼ 50 nm).

60 M.A. Torkaman-Asadi et al. / Composite Structures 126 (2015) 52–61

4.3. Elastic foundations

Another important issue in the case of rotating MEMS would bethe foundation characteristics. The general case of partial and fullysuspended nanotubes are taken into account. Physically speaking,foundations can alter the natural frequencies and stability bound-aries of any rotating device where the extent of this enhancementwill certainly depend upon the stiffness of the foundation. This canbe interesting to study this effect in the presence of size and gyro-scopic effects. To this aim, the variations of the CNT’s frequency atn ¼ m ¼ 1 for different Winkler parameters and rotating speeds arepresented in Figs. 7 and 8. A fully suspended nanomotor with b ¼ 0is considered in Fig. 7. It can be clearly inferred that the rotationspeed can have significant effects on stabilizing the system at softelastic foundations (low values of kw) while for hard elastic

foundations such stabilization effect is minimized. RegardingFig. 8, it is concluded that as the length of the Winkler foundationdecreases, the frequency also decreases. Fig. 9 investigates the var-iations of the non-dimensional frequency versus the non-dimen-sional rotation speed for different foundation stiffnesses. It’sobvious that the foundation stiffness has approximately nothingto do with the stability boundaries and causes tiny changes inthe critical rotating speed. Fig. 10 shows that for a CNT onWinkler foundation, the nonlocal parameter still can affect the cri-tical condition in a way that any increment in the nonlocal para-meter can make the rotating mechanism more stable.

5. Conclusion

The free vibration and stability of SWCNTs are studied by theLove’s shell assumptions. The governing equations of motion arederived by the Hamiltonian approach and then solved by obtaininga recursive series solution which proposes a very fast algorithmand satisfies any desired boundary conditions. Reasonable resultsare obtained and compared with the available literature. Resultsshow that the nonlocal parameter has no significant effect on thecritical slenderness ratio at a specific rotation speed. It has beenalso shown that as the radius of the CNT increases, the stabilitymargins are narrowed down. Namely, for a specific slendernessratio the critical speed decreases and also for a specific rotationspeed the critical slenderness ratio would decrease. Furthermore,as the length of the nanotube (the slenderness ratio) increases,the critical rotating speed will decrease. Finally, an interesting anpractical outcome is that the rotation speed can increase the fun-damental frequency significantly at soft foundation foundationswhile such effect is minimized at hard foundations.

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