numerical modeling of dynamic and parametric instabilities of
DESCRIPTION
Numerical modeling of dynamic and parametric instabilities ofsingle-walled carbon nanotubes conveying pulsating and viscous fluidTRANSCRIPT
Composite Structures 125 (2015) 127–143
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Composite Structures
journal homepage: www.elsevier .com/locate /compstruct
Numerical modeling of dynamic and parametric instabilities ofsingle-walled carbon nanotubes conveying pulsating and viscous fluid
http://dx.doi.org/10.1016/j.compstruct.2015.01.0440263-8223/� 2015 Elsevier Ltd. All rights reserved.
⇑ Corresponding author at: Mathematical Modeling and Control, Department ofMathematics, Faculty of Sciences and Techniques of Tangier, Abdelmalek EssaâdiUniversity, BP 416, Tangier 90000, Morocco.
E-mail addresses: [email protected], [email protected] (L. Azrar).
A. Azrar a, L. Azrar a,b,⇑, A.A. Aljinaidi b
a Mathematical Modeling and Control, Department of Mathematics, Faculty of Sciences and Techniques of Tangier, Abdelmalek Essaâdi University, BP 416, Tangier 90000, Moroccob Department of Mechanical Engineering, Faculty of Engineering, King Abdulaziz University, Jeddah, Saudi Arabia
a r t i c l e i n f o
Article history:Available online 7 February 2015
Keywords:Carbon nanotubeVibrationComplex modesDynamic and parametric instabilitiesPulsating and viscous fluidDifferential quadrature method
a b s t r a c t
The dynamic and parametric instabilities of single-walled carbon nanotubes (CNTs) conveying pulsatingand viscous fluid embedded in an elastic medium are modeled and numerically investigated. The partialdifferential equation of motion based on the nonlocal elasticity theory, Euler Bernoulli beam’s model andfluid–tube interaction is given. Based on the differential quadrature method, complex eigenmodes andassociated eigenfrequencies are investigated with respect to the flow velocity as well as to the other con-sidered physical parameters. Multimodal formulation based on real and complex eigenmodes are pre-sented in the frequency and time domains. Models are elaborated for dynamic instabilities such asdivergence and flutter as well as for parametric instability behaviors. The influences of the nonlocal para-meter, the fluid pulsation and viscosity, the viscoelastic CNT parameter and the thermal effects on thedynamic behaviors of the CNT-fluid system are analyzed. Instability boundaries and interaction betweenthe dynamic and parametric instabilities are investigated.
� 2015 Elsevier Ltd. All rights reserved.
1. Introduction
Carbon nanotubes (CNT) conveying fluid have become ones ofthe most important structures in nanotechnology. They may beused at micro or nano-levels for fluid storage, fluid transport, drugdelivery, micro-resonator, molecular reactors as well as for manynano-fluidic device applications. In such applications, the dynamiccharacteristics, such as natural frequencies, eigenmodes, stability,critical flow velocity and parametric instability zones are of consid-erable interest. During the last years, a significant amount ofresearch has been elaborated for the dynamic behavior of CNT. Areview paper on vibration of CNT and their composites has beenpublished by Gibson et al. [1]. Lee and Chang [2] studied the vibra-tion analysis of a viscous fluid conveying single walled carbon nan-otube embedded in an elastic medium. Wang [3] proposed thevibration analysis of fluid-conveying nanotubes with considerationof surface effects. Eringen’s nonlocal elasticity theory [4,5] allowsone to account for the small scale effect that is very significantwhen dealing with micro and nanostructures. Reddy [6] studied
the nonlocal theories for bending, buckling and vibration of beams.Duan et al. [7] used microstructured beam to calibrate length scalecoefficient in nonlocal beams. Wang et al. [8] and Zhang et al. [9]proposed the calibration of Eringen’s small length scale coefficientfor initially stressed vibrating nonlocal Euler beams based onmicrostructured beam model. The finite element analysis of forcedvibration for pipe conveying harmonically pulsating fluid has beenstudied by Seo et al. [10]. Hong et al. [11] studied the vibration of asingle-walled carbon nanotube embedded in an elastic mediumunder a moving internal nanoparticle. Mirramezani and Mir-damadi [12] analyzed the effects of nonlocal elasticity and Knud-sen number on fluid–structure interaction in carbon nanotubeconveying fluid. Baohui et al. [13] used the wave method to inves-tigate the free vibration analysis of micropipe conveying fluid.Ghavanloo et al. [14] studied the vibration and instability analysisof carbon nanotubes conveying fluid and resting on a linear vis-coelastic foundation. Wang et al. [15] reported the flexural vibra-tions of microscale pipes conveying fluid by considering the sizeeffects of micro-flow and micro-structure. Wang and Ni [16] pro-posed a reappraisal of the computational modeling of carbon nan-otubes conveying viscous fluid. The thermal mechanical vibrationand instability of a fluid conveying single walled carbon nanotubeembedded in an elastic medium based on nonlocal elasticity theoryhave been analyzed by Chang [17]. Ghannadpour et al. [18] studied
128 A. Azrar et al. / Composite Structures 125 (2015) 127–143
bending, buckling, and vibration problems of nonlocal Euler beamsusing Ritz method. The transverse waves propagating in carbonnanotubes via a higher-order nonlocal beam model is proposedby Huang et al. [19]. Ansari et al. [20] studied the dynamic stabilityof embedded single walled carbon nanotube including thermalenvironment effects. Vibration and dynamic instability analysesof CNT are elaborated in [21–23] and the higher order free vibra-tion of SWCNTs was extensively analyzed in [23]. The dynamic sta-bility of parametrically excited linear resonant beams underperiodic axial force has been recently studied by Jing et al. [24].To the authors’ knowledge, there are no studies in literature onthe dynamic instability of CNT based on complex modes nor onthe dynamic and parametric instabilities interactions of thesestructures. Therefore, numerical and analytical methodologicalapproaches are here proposed, proving simplified models as wellas theoretical and numerical insights into the dynamic and para-metric instabilities of CNT fluid system and on their interactions.
In the present paper, the governing equation of motion is firstderived based on the fluid–tube interaction, the nonlocal theoryand the Euler Bernoulli beams’ model. A numerical procedure basedon the differential quadrature method and multimodal formula-tions have been elaborated for the dynamic and parametric insta-bilities of viscoelastic CNT conveying pulsating and viscoelasticfluid. Frequency and time domains are considered and instabilityanalyses have been performed with respect to the considered influ-encing parameters based on analytical and numerical procedureswith an emphasis on complex modes. The influences of the nonlocalparameter, fluid viscosity, viscoelastic coefficient, foundation of theelastic medium, thermal environment and static and dynamic velo-city effects on the dynamic behaviors of the CNTs-fluid system areanalyzed. Various types of instabilities such as divergence, flutterand principal parametric instabilities as well as their interactionsare analyzed.
2. Mathematical modeling
The flexural vibration of a slender carbon nanotube conveyingfluid and subjected to axial force, thermal loading, fluid flow, andfluid pressure can be modeled based on the Euler–Bernoulli beammodel by the following partial differential equation:
@Q@x¼ mt
@2W@t2 þ Fe þ FT þ Ff þ Fp ð1Þ
where W;Q and mt are the transverse displacement, shear force andmass of the tube per unit length respectively. Fe; FT and Ff are theaxial force, the thermal force and the force per unit of lengthinduced by fluid flow. Fp represents a force due to the axial fluidpressure. The momentum-balance equation for the fluid motionmay be described by the Navier–Stokes equation: [16,25]
qdUdt¼ �rP þ #r2U ð2Þ
in which UðtÞ ¼ Ur;Uh;Uxð Þ is the time dependent fluid velocity inthe cylindrical coordinate system with components in r, h and xdirections. q, P and # are the mass density of the internal fluid,the pressure and the viscosity of the flowing fluid.
The exerted force due to the fluid flow in the nanotube can beobtained from (2). At the point of the contact between the insidetube and internal fluid, their respective velocity and accelerationin the direction of flexural displacement become equal. Theserelationships thus can be written as: [16]
Ur ¼dWdt
ð3Þ
where
dWdt¼ @
@tþ UxðrÞ
@
@x
� �W ð4Þ
Substituting Eq. (3) into Eq. (2) and using Eq. (4) one obtains:
Ff ¼ Af@pr
@r
¼ �mf@2W@t2 þ 2Ux
@2W@x@t
þ U2x@2W@x2
!
þ #Af Ux@3W@x3 þ
@3W@x2@t
!ð5Þ
in which mf is the mass of the fluid per unit axial length and Af isthe cross sectional area of the internal fluid respectively.
The axial resultant force due to the thermal loading, FT , and theforce due to the axial pressure exerted by the fluid, Fp, are given by[25]:
FT ¼EAaxTs
ð1� 2tÞ@2W@x2 ; Fp ¼ mf
dUx
dtðL� xÞ @
2W@x2 ð6Þ
in which E, A, L; t;ax and Ts represent the Young modulus, tubecross sectional area, length, Poisson ratio of the CNT, thermalexpansion and the temperature change respectively.
Based on the Euler–Bernoulli beam theory, the transverse shearforce Q and bending moment �M of the viscoelastic tube are givenby [25]:
Q ¼ @�M@x¼ E� c
@
@t
� �I@3W@x3 ð7Þ
and �M ¼R
zrxxdA
where c is the viscoelastic coefficient of the tube and rxx is its axialstress.
Based on the nonlocal elasticity theory, the following differen-tial constitutive equation for one dimensional case is adopted[4,5,17,21–23].
rxx � ðe0aÞ2 @2rxx
@x2 ¼ Eexx; ð8Þ
where e0a; exx are the nonlocal parameter and the axial strain of thenanotube respectively. Using Eq. (8), the moment �M is obtained bythe following differential equation.
�M � ðe0aÞ2 @2 �M@x2 ¼ EI
@2W@x2 ; ð9Þ
Combining Eqs. (1), (5)–(9), the partial differential equation ofmotion for a CNT conveying fluid and subjected to the consideredforces can be written as:
1þ c@
@t
� �EI@4W@x4 þ mf U2
x � NT þmf@Ux
@tL� xð Þ
� �@2W@x2
þ mf þmt� � @2W
@t2 þ 2mf Ux@2W@x@t
þ �KW � #A Ux@3W@x3 þ
@3W@x2@t
!
� e0að Þ2 mf þmt� � @4W
@x2@t2 þ mf U2x � NT
� @4W@x4
"
þ 2mf Ux@4W@x3@t
þ �K@2W@x2
#¼ 0 ð10Þ
where NT and �K are respectively the axial resultant force due to thethermal loading and the constant of the elastic medium. Note thatthe time and space (t,x) dependences are omitted for readabilitypurpose.
A. Azrar et al. / Composite Structures 125 (2015) 127–143 129
In this paper, a pulsating internal axial flow is considered andthe flow velocity is assumed to be harmonically fluctuating andto have the following form:
UxðtÞ ¼ U0 1þ g cosðXtÞ½ � ð11Þ
where U0 is the static mean flow velocity, g is the amplitude of theharmonic fluctuation and X its frequency.
The following dimensionless variables and parameters are used:
w ¼WL; y ¼ x
L; s ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiEI
mf þmt
� �st
L2 ; u ¼ffiffiffiffiffiffimf
EI
rLUx;
a ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
EImf þmt
� �sc
L2 ; l ¼ e0aL; T ¼ NT L2
EI; k ¼
�KL4
EI;
NT ¼ �EAaxTs
1� 2t; b ¼ #Affiffiffiffiffiffiffiffiffiffi
EImf
p ; Mr ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
mf
mf þmt
� �s;
V0 ¼ffiffiffiffiffiffimf
EI
rLU0; �x ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðmf þmtÞL4
EI
sx ð12Þ
Eq. (10) is thus transformed into the following dimensionlesspartial differential equation:
1þ a@
@s
� �@4w@y4 þ u2 � T þMr
duds
1� yð Þ� �
@2w@y2
þ 2Mru@2w@y@s
þ kwþ @2w@s2 � b u
@3w@y3 þMr
@2w@y2@s
!
� l2 kþ @2
@s2
!@2w@y2 þ u2 � T
� � @4w@y4 þ 2Mru
@4w@y3@s
!¼ 0 ð13Þ
The objective of this paper is to investigate the dynamic and para-metric instability behaviors as well as their interaction of theCNT-fluid system based on the partial differential Eq. (13). The timedependent fluid velocity is considered leading to a PDE with
Table 1Resonant frequencies of a simply supported CNT for V0 = 0, 2, 4 and (l = 0, b = 0, a = 0, T
V0 x DQM F
N = 7 N = 10 N = 15 N
V0 = 0 x1 3.1490 3.1415 3.1416 3x2 5.9207 6.2783 6.2832 6x3 7.8871 9.6355 9.4247 9
V0 = 2 x1 2.7513 2.7523 2.7520 2x2 6.1219 6.1207 6.1219 6x3 9.5479 9.3227 9.3205 9
V0 = 4 x1 1.9439 � 1.9439i 1.9473 + 1.9473i 1.9464 � 1.9464i 1x2 5.5630 5.5421 5.5468 5x3 9.2757 8.9954 8.9869 8
Table 2Resonant frequencies of a clamped–clamped CNT for V0 = 0, 4, 7 and (l = 0, b = 0, a = 0, T
V0 x DQM F
N = 7 N = 10 N = 15 N
V0 = 0 x1 4.7498 4.7299 4.7300 4x2 7.6435 7.8483 7.8532 7x3 9.3753 11.1148 10.9956 1
V0 = 4 x1 4.1809 4.1349 4.1354 4x2 7.1085 7.4487 7.4543 7x3 8.9864 10.8456 10.6968 1
V0 = 7 x1 1.9899 + 1.9899i 2.2783 + 2.2783i 2.2769 + 2.2769i 2x2 5.4779 6.3596 6.3697 6x3 7.8852 10.230 9.9951 9
periodically varying coefficients. Mathematical formulations basedon numerical and analytical procedures are elaborated.
It should be noted that for accurate investigation of the insta-bility of CNT conveying fluid,the used modal basis has to be care-fully selected. For this aim, the parametric free vibration analysisis first extensively studied using numerical and analytical methodswith respect to the fluid velocity and to the other considered para-meters. Complex eigenmodes and eigenfrequencies are obtained.Based on the obtained eigenmodes and the Galerkin procedure,various modal bases are used for dynamic and parametric insta-bility analyses.
3. Numerical procedure
3.1. Differential quadrature method
For numerical investigations of vibration, dynamic and para-metric instability analyses of the considered CNT-fluid system,the differential quadrature method (DQM) is adopted here.Thismethod, akin to approximate the derivative of a function at anylocation by a linear summation of all the function values along amesh line [26,27]. The procedure DQ application lies in the deter-mination of the weighting coefficients. The continuous solution isapproximated by functional values at discrete points. In the pre-sent paper, the following Chebyshev-Gauss–Lobatto quadraturepoints are used.
yi ¼12
1� cosi� 1N � 1
p� �� �
for i ¼ 1;2;3; . . . . . . ;n ð14Þ
where yi ¼ xiL and n is the number of grid points in the domain [0,1].
For a function f ðyÞ, DQ approximation of the mth order deriva-tive at the ith point is given by:
f ðyÞ ¼Xn
j¼1
ljðyÞf ðyjÞ ð15-aÞ
s = 0).
inite difference method Analytical
= 15 N = 50 N = 100
.1359 3.1411 3.1415 3.1416
.2374 6.2791 6.2822 6.2832
.2705 9.4108 9.4213 9.4248
.7453 2.7514 2.7519 2.7520
.0747 6.1175 6.1206 6.1218
.1647 9.3065 9.3171 9.3204
.9474 + 1.9474i 1.9465 � 1.9465i 1.9464 + 1.9464i 1.9463 � 1.9463i
.4930 5.5418 5.5454 5.5467
.8243 8.9722 8.9832 8.9868
s = 0).
inite difference method Analytical
= 15 N = 50 N = 100
.6875 4.7261 4.7291 4.7300
.6999 7.8390 7.8496 7.85320.6334 10.9615 10.9871 10.9956.0896 4.1313 4.1344 4.1354.2959 7.4397 7.4506 7.45430.3267 10.6621 10.6880 10.6968.2975 + 2.2975i 2.2789 + 2.2789i 2.2774 + 2.2774i 2.2769 + 2.2769i.1767 6.3523 6.3654 6.3697.5971 9.9581 9.9857 9.9949
130 A. Azrar et al. / Composite Structures 125 (2015) 127–143
dm
dym
f ðy1Þ
f ðy2Þ
..
.
f ðynÞ
8>>>>>>>><>>>>>>>>:
9>>>>>>>>=>>>>>>>>;¼ Hm
ij
f ðy1Þ
f ðy2Þ
..
.
f ðynÞ
8>>>>>>>><>>>>>>>>:
9>>>>>>>>=>>>>>>>>;; i; j ¼ 1;2; . . . ; n ð15-bÞ
in which ljðyÞ are the Lagrange interpolation polynomials and Hmij
represent the weighting coefficients given by [26].
f ðyiÞ ¼GðyÞ
ðy� yiÞG1ðyiÞ; for i ¼ 1;2; . . . ;n ð15 - cÞ
GðyÞ ¼Yn
j¼1
ðy� yjÞ ð15 - dÞ
G1ðyiÞ ¼Yn
j¼1; j–i
ðyi � yjÞ; for i; j ¼ 1;2; . . . ;n ð15 - eÞ
H1ij ¼
G1ðyiÞðyi � yjÞG1ðyjÞ
; for i; j ¼ 1;2; . . . ;n; i – j ð15 - fÞ
H1ii ¼ �
Xn
j¼1j–i
H1ij ð15-gÞ
The higher derivative, mth, can be calculated as:
Fig. 1. Real and imaginary parts of the first complex mode shape, mass normalized, o(b = 0.01, a = 0, l = 0.1, Ts = 0.1, k = 0.5).
Fig. 2. Real and imaginary parts of the first complex mode shape, mass normalized, ofa = 0, l = 0.1, Ts = 0.1, k = 0.5).
Hmij ¼ m H1
ijHm�1ii �
Hm�1ij
xi � xj
!for i ¼ 1;2; . . . ;n; j – i ð15 - hÞ
Hmii ¼ �
Xn
j¼1j–i
Hmij ð15-iÞ
The discrete classical boundary conditions at y = 0 and y = 1, usingthe DQ method, can be written as:
w1 ¼ 0 ð16 - aÞXn
k¼1
Hn01kwk ¼ 0 ð16 - bÞ
wn ¼ 0 ð16 - cÞXn
k¼1
Hn1nk
wk ¼ 0 ð16-dÞ
where n0 and nl may be taken as either 1, 2 or 3 and wk ¼ wðykÞ isthe transverse displacement of the tube at yk. Choosing the values ofn0 and nl can give the following classical boundary conditions:[26,27]
simply supported: n0 = 2; nl = 2clamped–clamped: n0 = 1; nl = 1clamped-simply supported: n0 = 1; nl = 2clamped-free: n0 = 1; nl = 3free-free: n0 = 2; nl = 3
f a simply-supported SWCNT at different dimensionless small flow velocities V0,
a simply-supported SWCNT at different dimensionless flow velocities V0, (b = 0.01,
Fig. 3. Real and imaginary parts of the first complex mode shape, mass normalized, of a clamped SWCNT at different dimensionless small flow velocities V0, (b = 0, a = 0,Ts = 0, k = 0).
Fig. 4. Real and imaginary parts of the first complex mode shapes, mass normalized, of a clamped SWCNT at different dimensionless flow velocities V0, (b = 0, a = 0, Ts = 0,k = 0).
Fig. 5. Real and imaginary parts of the second complex mode shapes, mass normalized, of a clamped SWCNT at different dimensionless flow velocities V0, (b = 0, a = 0, l = 0,Ts = 0, k = 0).
A. Azrar et al. / Composite Structures 125 (2015) 127–143 131
132 A. Azrar et al. / Composite Structures 125 (2015) 127–143
Applying Eqs. (15) and (16) to Eq. (13), one obtains the follow-ing ordinary differential system for i = 1,2,. . .,n.
Xn
k¼1
H4ikwkþa
Xn
k¼1
H4ik _wk
!þ u2�TþMr
@u@s 1�yð Þ
� �Xn
k¼1
H2ikwk
þ €wiþ2MruXn
k¼1
H1ik
_wkþkwi�b uXn
k¼1
H3ikwkþMr
Xn
k¼1
H2ik
_wk
!
�l2Xn
k¼1
H2ik
€wkþ u2�T� �Xn
k¼1
H4ikwkþ2Mru
Xn
k¼1
H3ik
_wkþkXn
k¼1
H2ikwk
" #¼0
ð17-aÞ
This system can be rewritten in the following matrix from.
½M� €wf g þ CðsÞ½ � _wf g þ KðsÞ½ �fwg ¼ 0 ð17-bÞ
where
½M� ¼ I1 �l2H2�
½K� ¼ 1�l2 uðsÞ2 � T� �
H4
þ uðsÞ2 � T þMr@uðsÞ@s
1� yð Þ� �
H2 þ k I1 �l2H2�
� buðsÞH3
½C� ¼ 2MruðsÞH1 � bMrH2 � 2l2MruðsÞH3 þ aH4
fwg ¼ w1;w2; . . . ;wnf gT
Note that the matrices C and K depend on time as well as on someinfluencing physical parameters. Numerical time response can beobtained by numerical methods such as Newmark, h-Wilson,Runge–Kutta, etc. As the matrices K and C depend on the fluid flowvelocity, considered time dependent, various types of instabilitiessuch as divergence, flutter and parametric can occur and thenmay complicate the dynamic response analysis. For the sake ofclarity, these types of instabilities are separately formulated.
3.2. Dynamic instability formulation
For static fluid velocity uðtÞ ¼ V0ðg ¼ 0Þ, the differential system(17-b) is then reduced to the following eigenvalue problem byassuming that wðsÞ ¼Wexs
x2½M� þx C½ � þ K½ �� �
fWg ¼ 0 ð18Þ
Fig. 6. Real and imaginary parts of the third complex mode shapes, mass normalized, ofTs = 0, k = 0).
where fWg denotes the unknown dynamic displacement vectordefined by:
fWg ¼ w1w2 � � �wnf gT ð19Þ
and [K], [M] and [C] are the resulting stiffness, mass and dampingmatrices respectively.
The assumed boundary conditions can also be expressed in amatrix form using (16)
KB½ � WBf g þ KC½ � WSf g ¼ 0 ð20Þ
where WBf g ¼ w1w2wn�1wnf gT and WSf g ¼ w3w4 � � �wn�2f gT . KB½ �and KC½ � are 4� 4 and 4� ðn� 4Þ matrices respectively.
Using this vector decomposition, Eq. (18) can be rewritten as:
KD½ � WBf g þ KS½ � WSf g þ x CS½ � þx2 MS½ �� �
WSf g ¼ 0 ð21Þ
Coupling Eqs. (20) and (21), one gets:
KS½ � � KD½ � KB½ ��1 KC½ � þx CS½ � þx2 MS½ �n o
WSf g ¼ 0 ð22Þ
where KS½ �; MS½ � and CS½ � are ðn� 4Þ � ðn� 4Þ matrices.This frequency dependent relation is rewritten in the following
eigenvalue problem form:
CY ¼ xY;
C ¼Ms 00 Ms
� ��1 0 Kh
Kh CS
� �; Y ¼
Ws
xWs
� �;
8><>: ð23Þ
Kh ¼ KS½ � � KD½ � KB½ ��1 KC½ �:
By solving this eigenvalue problem, eigenmodes and associatedeigenfrequencies can be numerically obtained for various types ofboundary conditions. Based on this numerical procedure, real andcomplex eigenmodes and the associated eigenfrequencies can beobtained with respect to V0, T and to the other relevant physicalparameters. As the Galerkin procedure will be based on theobtained eigenmodes, two distinct cases are considered here.
3.2.1. Case 1: Classical real modal basisIn this case, the free vibration analysis is done by considering
V0 = T = k = a ¼ b = Mr = 0. This will lead to natural frequenciesand real eigenmodes independent from V0, T, k, b and a. These realmodes are classically used in modal vibration analysis of tubesconveying fluid. For example, for simply supported case, the eigen-modes wjðxÞ ¼ sinðjpx=LÞ are classically used by many authors. It
a clamped SWCNT at different dimensionless flow velocities V0, (b = 0, a = 0, l = 0,
Fig. 7. Real and imaginary parts of the first and second complex mode shapes, mass normalized, of a clamped SWCNT at a flow velocities V0 = 9, (b = 0, a = 0, l = 0, Ts = 0,k = 0).
Fig. 8. Real and imaginary parts of dimensionless frequency x as a function of flow static velocity V0 for clamped and simply-supported CNT based on one-complex-mode oneand two-real mode approaches.
Fig. 9. Variation of dimensionless frequency of a CC-CNT with flow velocity fordifferent temperature changes (e0a/L = 0.05, K = 0 MPa), based on the complexmode.
A. Azrar et al. / Composite Structures 125 (2015) 127–143 133
should be noted that the simplified modes can not be used foraccurate prediction of the dynamic instability analysis of CNTand particularly at large fluid velocity. In this simplified real modalbasis, the imaginary parts are disregarded as well as the fluid velo-city effect.
3.2.2. Case 2: General modal basisBy considering all parameters, complex eigenfrequencies and
eigenmodes are resulted. These complex characteristics have beencarefully computed for the considered boundary conditions. Usingthe numerically obtained eigenmodes, Galerkin’s method will beapplied for dynamic and parametric instability analyses.
4. Multi-modal formulation
For multimodal analysis, the CNT deflection can be approximat-ed by N modes as:
WðsÞ ¼XN
i¼1
ZiqiðsÞ ð24Þ
Fig. 10. Real and imaginary dimensionless first, second and third frequency of a cantilever SWCNT with flow velocity for different physical parameters.
134 A. Azrar et al. / Composite Structures 125 (2015) 127–143
where Zi and qiðsÞ are the vector eigenfunctions that depend on theconsidered physical parameters and the generalized coordinatesrespectively. Eqs. (17-b) and (24) lead to the following time depen-dent differential system for j = 1–N.
XN
i¼1
MZi;�Zj �
€qiðsÞþXN
i¼1
C0þC1 cosðXsÞ½ �Zi;�Zj �
_qiðsÞ
þXN
i¼1
K0þK1 cosðXsÞþK2 cos2ðXsÞþK3 sinðXsÞ� �
Zi;�Zj �
qiðsÞ¼0 ð25Þ
where
M ¼ I1 � l2H2�
; K0 ¼ H4 þ V20 � T
� H2 þ k I1 � l2H2
� � l2 V2
0 � T�
H4 � bV0H3
K1 ¼ 2gl2V20H4 þ 2gl2V2
0H2 � bV0gH3;
K2 ¼ g2V20 þ H2 � l2H4
� ; K3 ¼ MrV0gX y� 1ð ÞH2
C0 ¼ aH4 � bMrH2 þ 2MrV0 H1 � l2H3
� ;
C1 ¼ 2MrV0g H1 � l2H3�
and �Zj stands complex conjugate of Zj.For the dynamic response, numerical solution of the time
dependent system (25) can be conducted based on a numericalprocedure for the considered mode number. The effects of the con-sidered parameters can be investigated.
For a deep analysis of instability behaviors of the consideredCNT-fluid system, models of dynamic and parametric instabilitybehaviors are formulated hereafter.
Static fluid case (g=0).For the dynamic instability analysis of CNT conveying static flu-
id velocity, the critical fluid velocities and velocity–frequencydependence can be investigated based on Eq. (23). The timeresponse at any static velocity level can be obtained by numericallysolving the following reduced differential system.
I1�l2H2h i
€qrf gT þ aH4þ2MrV0H1�bMrH2
h�2l2MrV0H3
i_qrf gT
þ V20�T�l2k
� H2þkI1�bV0H3þ 1�l2V2
0
� H4
h iqrf g
T ¼ 0 ð26Þ
fqg ¼ q1; q2; . . . :; qNf g
where
Fig. 10 (continued)
A. Azrar et al. / Composite Structures 125 (2015) 127–143 135
I1 ¼ Zi; �Zj �
; H1 ¼ C1ijZi; �Zj
D E; H2 ¼ C2
ijZi; �Zj
D E;
H3 ¼ C3ijZi; �Zj
D E; H4 ¼ C4
ijZi; �Zj
D E;
5. Parametric instability formulation
The parametric instability behaviors of carbon nanotubes con-veying pulsating fluid can be investigated at various fixed values
of the influencing parameters V0, T, k, l and b. Here, the effectsof g and X on the system stability are the main focuses at variousvelocity V0 levels.
The generalized coordinate qjðsÞ is assumed periodic and isexpressed in the form:
qjðsÞ ¼X�N
k¼0
ak sinkXs
2
� �þ bk cos
kXs2
� �� �ð27Þ
Fig. 11. One-mode based instability regions of the CNT with different values of the viscous parameter b and viscoelastic coefficient a for a simply supported CNT.
Fig. 12. One-mode based instability regions of the CNT with different values of the nonlocal parameter l and the thermal coefficient Ts for simply supported CNT.
Fig. 13. One and two-real-modes based instability regions of a simply-supportedCNT with different values of the static velocity V0, (l = 0, b = 0, a = 0, T = 0, k = 0).
136 A. Azrar et al. / Composite Structures 125 (2015) 127–143
Based on the multi-modal formulation and substituting Eq. (27)into Eq. (25) the following algebraic system is obtained
½G�fXg ¼ f0g ð28Þ
where [G] is a ð2�N þ 1Þ � ð2�N þ 1Þ matrix and X is a ð2�N þ 1Þ vector
fXg ¼ b0; a1; b1; . . . ; a�N; b�Nf gT
For the sake of clarity, the matrix G is given in the Appendix Afor �N ¼ 5. The characteristic equation of this problem, detðGÞ ¼ 0,leads to a highly nonlinear algebraic relationship on X given theinstability boundaries. For the sake of simplicity, the characteristicequation is given for some particular simple cases.
5.1. Particular cases
The stability regions can be obtained based on the algebraic Eq.(28) by considering the required modes. Various modes and var-ious harmonic decompositions can be considered for the requiredaccuracy.
5.2. One-mode approach
Based on the one-mode and one-harmonic approachðN ¼ �N ¼ 1Þ, the instability boundaries can be obtained by solvingthe following algebraic equation.
X2
� �4
1þ l2ðH211Þ
2� 2
� X2
� �2
a1 þ a4ð Þ 1þ l2ðH211Þ
2�
þ a2a3
h iþ a1a4 ¼ 0 ð29aÞ
This leads to
n0g4 þ ðn2V20X
2 þ f0Þg2 þ kX4 þ c2X2 þ c0 ¼ 0 ð29bÞ
g ¼ �
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�ðn2V2
0X2 þ f0Þ �
ffiffiffiffiDp
2n0
sð29cÞ
where
D ¼ ðn2V20X
2 þ f0Þ2� 4n0ðkX4 þ c2X
2 þ c0Þ
a1 ¼Y2
2� Y4
2þ Y6; a2 ¼
Y4
2þ Y6; a3 ¼
Y1
2þ Y3
4� Y5
2;
a4 ¼Y1
2þ Y3
4þ Y5
2;
A. Azrar et al. / Composite Structures 125 (2015) 127–143 137
k ¼ � l2H211 � 1
� 2=16
n0 ¼ ð�ð1=4ÞðH411Þ
2l4 � ð1=4ÞðH2
11Þ2 þ ð1=2Þl2H4
11H211ÞV
40;
n2 ¼ ð1=4ÞMr2F211 � ð1=4ÞðH2
11Þ2l2 þ ð1=4ÞH2
11
� ð1=2ÞMr2F11H211 þ ð1=4ÞðH2
11Þ2Mr2
� ð1=4Þl2H411 þ ð1=4Þl4H2
11H411
f0 ¼ ð�H211H4
11Tl2 þ ðH211Þ
2T�H2
11H411 þ ðH
411Þ
2l2ÞV2
0
c0 ¼ ð�ðH211Þ
2 � ðH411Þ
2l4 þ 2l2H2
11H411ÞV
40 þ 2H2
11H411T
� ðH211Þ
2T2 þ ð�2H2
11H411Tl2 þ 2ðH2
11Þ2T� 2H2
11H411
þ 2ðH411Þ
2l2ÞV2
0 � c211
c2 ¼ ðð1=2ÞH211 � ð1=2ÞðH2
11Þ2l2 � ð1=2Þl2H4
11
þ ð1=2Þl4H211H4
11ÞV20 � ð1=4Þa2ðH4
11Þ2 þ ð1=2ÞbMraH2
11H411
þ ð1=2ÞH411 þ ð1=2ÞðH2
11Þ2Tl2 � ð1=2ÞH2
11T
� ð1=4Þb2Mr2ðH211Þ
2 � ð1=2Þl2H211H4
11
Yj are given in the Appendix A.
5.3. Two-modes approach
Based on the two-modes and one-harmonic approachesðN ¼ 2; �N ¼ 1Þ, the instability boundaries associated to the firsttwo modes can be obtained by solving the following algebraicequation.
X2
� �8
b8 þX2
� �6
b6 þX2
� �4
b4 þX2
� �2
b2 þ b0 ¼ 0 ð30Þ
where the coefficients b0;b2;b4; b6;b8 are given by:
b0 ¼ ðS11S14 � S12S13ÞðS41S44 � S43S42Þ
b2 ¼14ðS12S13 � S11S14ÞðS41m4 þm1S44Þ
� 14ðS11m4 þm1S14ÞðS41S44 � S43S42Þ
þ ðS11S23 � S21S13ÞðS42S34 � S32S44Þþ ðS11S24 � S22S13ÞðS32S43 � S34S41Þþ ðS21S14 � S12S23ÞðS42S33 � S31S44Þþ ðS22S14 � S12S24ÞðS31S43 � S33S41Þ
b4 ¼1
16m2
1S14S44 þ S11m24S41
� �þm1
4S24ðS11S34 � S12S33Þð þ S23ðS32S44 � S42S34Þ
þ S24ðS34S41 � S32S43ÞþS22ðS14S33 � S13S34ÞÞ
þm4m1
16ðS11 þ S41ÞðS14 þ S44Þ � S12S13 � S43S42ð Þ
þm4
4S21ðS14S31 � S13S32Þð S23ðS11S32 � S12S31Þ
þ S21ðS31S44 � S42S33Þ þ S22ðS33S41 � S31S43ÞÞþ S22S23ðS32S33 � S31S34Þ þ S21S24ðS31S34 � S32S33Þ
b6 ¼�164ðS11 þ S41Þm1m2
4 þ ðS14 þ S44Þm4m21
� �� 1
16ðS22S33 þ S23S32Þm1m4 þ S24S34m2
1 þ S21S31m24
� �b8 ¼
1256
m21m2
4
in which:
S11 ¼12
V20g
2 � 2V20g
� H2
11 þ 2l2V20g� l2V2
0g2
� H4
11
� þ k I1
11 þ ðV20 � T� l2KÞH2
11 þ ð1� l2V20ÞH
411
S12 ¼12
bV0g� bV0
� �H3
21; S13 ¼12
bV0g� bV0
� �H3
12
S14 ¼12
V20g
2 � 2V20g
� H2
22 þ 2l2V20g� l2V2
0g2
� H4
22
� þ k I1
22 þ ðV20 � T� l2kÞH2
22 þ ð1� l2V20ÞH
422
S21 ¼12� MrV0gþ bMrð ÞH2
11 þMrV0gF11 þ aH411
�
S22 ¼12
MrV0gF21 þ 2MrV0D21 � 2l2MrV0H321
�
þ 14
2MrV0gD21 � 2l2MrV0gH321
�
S23 ¼12
MrV0gF12 þ 2MrV0D12 � 2l2MrV0H312
�
þ 14
2MrV0gD12 � 2l2MrV0gH312
�
S24 ¼12� MrV0g� bMrð ÞH2
22 þMrV0gF22 þ aH422
� ;
S31 ¼12
MrV0gF11 � aH411 þ bMr�MrV0gð ÞH2
11
�
S32 ¼12
MrV0gF21 � 2MrV0D21 þ 2l2MrV0H321
�
þ 14
2MrV0gD21 � 2l2MrV0gH321
�
S33 ¼12
MrV0gF12 � 2MrV0D12 þ 2l2MrV0H312
�
þ 14
2MrV0gD12 � 2l2MrV0gH312
�
S34 ¼12
MrV0gF22 � aH422 þ bMr�MrV0gð ÞH2
22
�
S41 ¼12
2V20gH2
11 � 2l2V20gH4
11
� þ k I1
11 þ ðV20 � T� l2kÞH2
11
þ ð1� l2V20ÞH
411
S42 ¼ �12
bV0gþ 2V20g
� �H3
21; S43 ¼ �12
bV0g� bV0
� �H3
12
S44 ¼12
2V20gH2
22 � 2l2V20gH4
22
� þ k I1
22 þ ðV20 � T� l2kÞH2
22
þ ð1� l2V20ÞH
422
F ¼ yiC3ijZi; �Zj
D Emi ¼ I1
ii � l2H2ii
For more general cases using more eigenmodes and harmonics, ahighly nonlinear algebraic equation f(XÞ = det(G) = 0 will be result-ed and thus has to be numerically solved.
Based on these relationships, the dynamic and parametric insta-bility analyses can be numerically investigated. The effects of theconsidered physical parameters on the divergence, flutter andparametric instability zones as well as the associated time respon-se can be numerically analyzed.
138 A. Azrar et al. / Composite Structures 125 (2015) 127–143
6. Analytical procedure
For the sake of accuracy and comparison, analytical proceduresare elaborated for some simple cases.As various parameters suchas fluid velocity, thermal effect, tube viscoelasticity and fluid velo-city and viscosity are considered, the free dynamic behaviors of theCNT can be affected by all these parameters. To easily handle theeffect of these parameters, it is important to obtain some associat-ed analytical relationships. To this aim, the transverse displace-ment is assumed of the form:
Wðy; sÞ ¼ wðyÞeixs ð31Þ
The complex characteristic equation associated to the main govern-ing Eq. (13), for a static fluid velocity ðg ¼ 0Þ, is given for, wðyÞ ¼ eky,by:
ixaþ1�l2ðV20�TÞ
� k4þ ibV0�2xMrV0l2� �
k3
� V20�T� ibMr�l2ðk�x2Þ
� k2þ2xMrV0kþk�x2¼0 ð32Þ
For distinct solutions of (32), the general solution associated to (13)can be written in the form:
wðyÞ ¼ A1eik1y þ A2eik2y þ A3eik3y þ A4eik4y ð33Þ
where the arbitrary constants Ai are determined by the consideredboundary conditions.
It should to be noted that the resulting frequencies and eigen-modes depend on the considered parameters such as V0, k, a, etcand on the temperature T. This will lead to complex eigenfrequen-cies and eigenmodes that may be extensively changed with respectto these parameters.
For simply supported and clamped CNT, the associated eigen-modes are given by:
ujðyÞ ¼ A1eik1jy þ A2eik2jy þ A3eik3jy þ A4eik4jy ð34Þ
where
A2 ¼ �ðc4j � c1jÞeik3j þ ðc1j � c3jÞeik4j þ ðc3j � c4jÞeik1j
ðc3j � c4jÞeik2j þ ðc4j � c2jÞeik3j þ ðc2j � c3jÞeik4jA1;
A3 ¼ �ðc4j � c1jÞeik2j þ ðc1j � c2jÞeik4j þ ðc2j � c4jÞeik1j
ðc3j � c4jÞeik2j þ ðc4j � c2jÞeik3j þ ðc2j � c3jÞeik4jA1;
A4 ¼ �ðA1 þ A2 þ A3Þ
in which cij ¼ kij and cij ¼ k2ij for clamped and simply supported
boundary conditions respectively. The resulting transcendentalequation is thus given by:
Fig. 14. One and two-real-modes based instability regions of a clamped CNT withdifferent values of the static velocity V0, (l = 0, b = 0, a = 0, T = 0, k = 0).
eðk1jþk2jÞ þ eðk3jþk4jÞ� �
ðc2j � c1jÞðc4j � c3jÞþ eðk1jþk3jÞ þ eðk2jþk4jÞ� �
ðc2j � c4jÞðc3j � c1jÞþ eðk2jþk3jÞ þ eðk1jþk4jÞ� �
ðc1j � c4jÞðc2j � c3jÞ ¼ 0 ð35Þ
It has to be noted that Eqs. (32) and (35) have to be simultaneouslynumerically solved in order to get the frequency and eigenmodes atrequired flow velocity levels. A numerical iterative-incrementalprocedure based on the Newton–Raphson algorithm has beenelaborated for that goal. The real as well as the complex eigenmodescan be obtained based on this semi-analytical methodologicalapproach.
6.1. Simplified case
The analytical formulation is more simple when a ¼ b = Mr = 0.In this case, Eq. (32) is reduced to:
1� l2ðV20 � TÞ
� k4 � V2
0 � T � l2ðk�x2Þ�
k2 þ k�x2 ¼ 0
ð36Þ
and its solutions are:
k1 ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiV2
0 � T � l2ðk�x2Þ�
þffiffiffiffiDp
2 1� l2ðV20 � TÞ
� vuuut ; k2 ¼ �k1;
k3 ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiV2
0 � T � l2ðk�x2Þ�
�ffiffiffiffiDp
2 1� l2ðV20 � TÞ
� vuuut ; k4 ¼ �k3
where: D ¼ V20 � T � l2ðk�x2Þ
� 2� 4 1� l2ðV2
0 � T�
k�x2� �
:
In this case, Eq. (35) is reduced to:
FðxÞ¼ ðk1�k3Þ2ðeiðk1þk3Þ þe�iðk1þk3ÞÞ�ðk1þk3Þ2ðeiðk1�k3Þ þe�iðk1�k3ÞÞþ8k1k3¼0 ð37Þ
The mode shapes ujðyÞ can be classically obtained for the consid-ered boundary conditions by:
ujðyÞ ¼ A1eik1jy þ A2e�ik1jy þ A3eik3jy þ A4e�ik3jy ð38Þ
6.2. Buckling problem
For the sake of completeness, the static buckling problem is alsoconsidered when the foundation is disregarded, (k = 0). In this case,Eq. (13) is reduced to:
1� l2ðV2c � TÞ
� d4w
dy4 þ V2c � T
� d2w
dy2 ¼ 0 ð39Þ
Fig. 15. One and two-real-modes based instability regions of a clamped CNT withdifferent values of the static velocity V0, b = 0.1, (l = 0, a = 0, T = 0, k = 0).
Fig. 16. Two-real-modes based instability regions of the CNT with different values of the fluid viscosity b for simply supported (left) and clamped (right) boundary conditionsfor l = 0.1, a = 0.001.
A. Azrar et al. / Composite Structures 125 (2015) 127–143 139
if 1� l2ðV2c � TÞ
� – 0
d4w
dy4 þ c2 d2w
dy2 ¼ 0 c2 ¼V2
c � T�
1� l2ðV2c � TÞ
� ð40Þ
The general solution associated to (40) can be written in the form:
wðyÞ ¼ A1 þ A2yþ A3 sinðcyÞ þ A4 cosðcyÞ ð41Þ
For the clamped boundary condition, the transcendental equation isgiven by:
c sinc2
� tan
c2
� � c
2
� ¼ 0 ð42Þ
The solutions of Eq. (42) are given by:
c ¼ 2np n 2 N or c
¼ 8:986; 15:451; 21:808; 28:132; 34:442; :::½ � ð43Þ
in which N is the set of integers. The associated buckling modeshapes are given by:
wðyÞ ¼ A1 1� cðcosðcÞ � 1Þ�cþ sinðcÞ yþ ðcosðcÞ � 1Þ
�cþ sinðcÞ sinðcyÞ � cosðcÞ� �
ð44Þ
For the simply supported boundary condition, the transcendentalequation is given by:
c4 sin cð Þ ¼ 0 ð45Þ
The solutions are given by:
c ¼ np n 2 N
and the associated buckling mode shapes are given by:
wðyÞ ¼ A sinðcyÞ ð46Þ
The critical buckling flow velocity is given by:
Vc ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffic2 1þ l2Tð Þ þ T
1þ c2l2ð Þ
sð47Þ
Note that, when the nonlocal effect is neglected, this equation isreduced to the simply supported critical Euler buckling value:
V2c � T ¼ ðnpÞ2 [25].
A multimodal dynamic analysis can be elaborated based on theanalytical modes ujðyÞ by assuming:
wðy; sÞ ¼XN
j¼1
ujðyÞqjðsÞ ð48Þ
Substituting Eqs. (11) and (38) into the main governing Eq. (13) andintegrating over [0,1], the following second-order ordinary differen-tial system is analytically obtained.
��A� l2��Bh i
€qrf g þ a��C þ 2MrV0 1þ g cosðXsÞð Þ��D� bMr��B
h
� 2l2MrV0 1þ g cosðXsÞð Þ��Ei
_qrf g
þ V20 1þ 2g cosðXsÞ þ g2 cos ðXsÞ2� �h
� T � l2kþMrV0gX sinðXsÞ���B�MrV0gX sinðXsÞ��F
þ k��A� bV0 1þ g cosðXsÞð Þ��Eþ 1� l2V20 1þ 2g cosðXsÞð
�
þ g2 cos ðXsÞ2
��Ci
qrf g ¼ 0 ð49Þ
Where ��A; ��B; ��C; ��D; ��E and ��F are matrices with elements given by: (s,r = 1,2,. . .,N).
��Asr ¼Z 1
0ur �usdy; ��Dsr ¼
Z 1
0
dur
dy�usdy;
��Bsr ¼Z 1
0
d2ur
dy2�usdy; ��Esr ¼
Z 1
0
d3ur
dy3�usdy;
��Csr ¼Z 1
0
d4ur
dy4�usdy; ��Fsr ¼
Z 1
0y
d2ur
dy2�usdy
It should be stated that when the modal basis is availableanalytically, the dynamic and parametric instability analyses couldbe investigated based on the same procedures developed in the pre-vious sections.
7. Numerical results and discussion
Let us note that due to the fluid velocity the dynamic instabilitywill occur at increased static velocity V0. This will lead to diver-gence and flutter instabilities. Then, even if the viscosities of thefluid and tube are disregarded, the complex eigenmodes and fre-quencies may occur. This paper will focus on complex eigenmodes
Fig. 17. Instability regions of a clamped CNT with different values of the staticvelocity V0 based on various modal approaches, b = 0.1 and (l = 0, a = 0, T = 0, k = 0).
Fig. 19. Parametric instability bifurcation points for different values of b.
140 A. Azrar et al. / Composite Structures 125 (2015) 127–143
and on the coupling dynamic and parametric instabilities withrespect to static velocity and dynamic fluid pulsation. For the sakeof clarity, the one-mode and two-modes approaches are used forthe analysis. Based on these approaches, the first and second nat-ural frequencies, critical flow velocity and stability regions areobtained. The time responses are computed for various instabilityregions.
For numerical analysis, the following material and geometricalparameters of CNT-fluid system are used. The Young’s modulusof carbon nanotube is assumed to be E = 1TPa with an effectivethickness about h = 0.34 nm. The diameter d, the mass density qc
and the aspect ratio L/d of SWCNT are 1 nm, 2300 kg/m3, and 10respectively. The fluid inside the nanotube is assumed to be thewater with the mass density qf and viscosity # are 1000 kg/m3
and 1.12 � 103 Pa respectively [2]. Two cases of the low and hightemperature are considered. At low and room temperature, thethermal expansion coefficients ax = �1.6e�6 K�1 and at high tem-perature ax = 1.1e�6 K�1.
For comparison and validation of the presented numerical andanalytical methodological approaches numerical results are pre-sented in Tables 1 and 2. The finite difference method (FDM) hasbeen also used and programmed here for numerical comparison.The details of this classical numerical approach have been omittedhere. Numerical results of the three first natural frequencies ofsimply supported and clamped boundary conditions CNT are givenfor various values of V0. Numerical results are obtained based onthe DQM for point numbers N = 7, 10, 15 and on the FDM forN = 15, 50, 100 and compared in Tables 1 and 2. The analyticalresults, obtained based on Eqs. (32) and (35), are also given for
Fig. 18. One-complex mode based parametric instability regions of a clamped CNT with
convergence test. It is demonstrated that the DQM converges for15 points while the finite difference method needs largely morediscretization points (N = 100) to converge.
Complex eigenmodes will be classically induced by the fluidviscosity, the CNT viscoelasticity and particularly by the static fluidvelocity V0. To clarify the fluid viscosity and velocity effects on theCNT vibration behaviors, numerical results of the obtained eigen-modes are presented.
Figs. 1 and 2 present real and imaginary parts of the first modeshape, mass normalized, of a simply supported CNT forV0 ¼ 0;1;2;3;4;5;6;7 as well as the classical simplified mode‘‘sinðpx=LÞ’’. Note that, for these small values of V0, sinðpx=LÞ isgood enough for the real part of the first mode. The imaginary part,neglected by this classical mode, is not zero even for V0 ¼ 0 as thefluid viscosity is considered. At large V0, the shape of the real partchanges and the imaginary part is highly increased as shown inFig. 2. The first mode of a clamped CNT at various values of V0ispresented in Fig. 3. It is demonstrated that for small values of V0;ðV0 ¼ 0;1;2;3;4;5Þ the real part is almost unchanged but there isan imaginary part that is considerably increasing with V0. To showthe effect of V0 on the first modes a large range of values of V0isconsidered and real and imaginary parts of the first, second andthird eigenmodes are presented in Figs. 4–6 for a clamped CNT(b = a = Ts = K = 0). These figures show that there is a transition ofthe real part of the first mode to the second one and that the ima-ginary parts of all modes become very significant when the staticvelocity V0 exceeds a critical value. To show the coalescence behav-ior at a critical fluid velocity V0, the first and second modes are pre-sented in Fig. 7 for clamped case.
respect to the static velocity V0 for different values of b, (l = 0, a = 0, T = 0, k = 0).
A. Azrar et al. / Composite Structures 125 (2015) 127–143 141
The transition from mode to mode and the growing of imagin-ary parts with respect to V0can be explained by the frequency-ve-locity dependence curves giving the dynamic instability behaviorof the CNT-fluid system. This dynamic behavior is investigatedhere based on the one and two-real-mode as well as on the com-plex modes approaches. Fig. 8 demonstrates the divergence andflutter instability types for the considered simply supported andclamped CNT. The real frequency parts decreases with increasingof the flow velocity V0up to the divergence instability (x = 0).The critical values of V0 for the divergence instability associatedto complex-mode are V0= 3.14 for the considered simply supportedCNT and V0 = 6.29 for the clamped CNT (T = 0, a = 0). The criticalvalues of the flutter instability are V0 = 6.38 and V0 = 9.01 for S-Sand C-C CNT respectively.
It should to be stated that a one-real mode approach leads toerroneous results for flutter analysis but good enough for diver-gence detection. The predictions obtained for the dynamic insta-bility analysis based on the two-real-modes and the one-complexmode approaches are very close for simply supported case butleads to different values for the clamped case at large values of V0.
The temperature effect on the dynamic instability of a clampedCNT predicted by a one complex-mode approach is presented inFig. 9 and only small effects are observed.
Fig. 20. One real-mode time responses for l = 0.1, b = 0.01, V0 = 2, a = 0.001, Ts = 20
Fig. 21. Two-real-mode time responses for l = 0.1, b = 0.01, V0 = 4, a =
The effects of the static fluid velocity V0, the viscosity b, the vis-coelastic parameter a, the nonlocal parameter l and the thermalcoefficient effect Ts on real and imaginary parts of the first, secondand third frequencies of cantilever CNTs conveying fluid are pre-sented in Fig. 10. This effect is more significant at higher velocities.
A parametric study with respect to all considered physical andmaterial parameters can be easily done by the presented method-ological approaches.
Figs. 11 and 12 demonstrate the effects of the static fluid velo-city V0, the viscosity b, the viscoelastic parameter a, the nonlocalparameter l and the thermal coefficient effect Ts on the instabilityboundaries in the principal parametric resonance based on theone- real-mode approach. In Fig. 11 the instability areas, originsof parametric instability are reduced with increase in fluid vis-cosity b and the viscoelastic coefficient a. It should be noted thata larger flow velocity V0 leads to a large instability region. InFig. 12, the effects of the nonlocal parameter l and the thermalcoefficient Ts are presented. The parametric resonance regionsmove significantly backwards by increasing the nonlocal para-meter l and slightly upwards by increasing Ts.
The one and two-modes based parametric instability regions ofthe CNT with different values of the static flow velocity V0 of a sim-ply supported and clamped CNT are presented in Figs. 13 and 14
, g = 0.1 and different values of XðX/2 = 7, 7.5, 8) for a simply supported CNT.
0.001, Ts = 20, g = 0.14 and different values of X for a C-C- CNT.
G=
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142 A. Azrar et al. / Composite Structures 125 (2015) 127–143
respectively. The influence of the nonlocal parameter on the para-metric frequency is presented in Fig. 13. The difference betweenthe one and two modes predictions instability regions are present-ed in Fig. 15. It is observed that the critical parametric frequencyand dimensionless pulsation amplitude g, associated to bifurca-tions point are increased when two-modes is used. The fluid vis-cosity b effect is demonstrated in Fig. 16. Moreover, comparisonbetween the real and the complex mode approaches are given inFig. 17. It is demonstrated that for fixed V0 the parametric insta-bility regions are shifted to the higher parametric frequency whenthe complex mode are used. For a general representation, Fig. 18shows the parametric instability regions of the clamped CNT inthree dimensions (X, V0, gÞ based the on one-complex-modeapproach the interaction between dynamic and parametric insta-bilities is shown. The evolutions of the parametric instability fre-quencies and the associated bifurcation points for differentviscous parameter b are presented in Fig. 19. These analyses allowdetermining the stability boundaries and zones with respect to thestatic fluid velocity V0 and to the other physical parameters.
To clearly demonstrate the dynamic behaviors in these regions,time responses are presented in Fig. 20 for SS-CNT and in Fig. 21 forCC- CNT based on the one-real mode and two-real modes approach-es. These figures show the time responses associated to various val-ues of X for fixed values of V0. It is observed that the dynamicresponse is very sensitive to the static velocity V0 as well to theparameters g and X. The stability and instability behaviors areclearly demonstrated with respect the parametric instability zones.
8. Conclusion
Vibration, dynamic and parametric instabilities of CNT convey-ing pulsating fluid are analyzed based on the nonlocal elasticity,fluid interaction and Euler–Bernoulli beam theory. A numericalmethodological approach based on the differential quadraturemethod has been formulated and used for numerical solution.For comparisons in some simple cases, a semi analytical procedurehas been also developed. It has been demonstrated that theobtained eigenmodes are complex and the amplitude of their ima-ginary parts, mass normalized, is increasing with V0. For small V0,this amplitude is too small related to the amplitude of the corre-sponding real parts. But, for V0 greater than the critical divergenceand flutter values, the amplitudes of imaginary and real parts ofthe resulting eigenmodes are almost of the same magnitude. Themultimode approach has been formulated based on the numerical-ly computed eigenmodes, for dynamic and parametric instabilities.For simplified models the one-real mode, two-real mode and one-complex mode approaches have been developed for both types ofinstabilities. The inaccuracy of the one-real mode approach isdemonstrated. The influences of the internal fluid velocity, thenonlocal parameter, the viscosity, the viscoelastic coefficient aswell as the thermal effects on the dynamic behaviors and flow-in-duced structural instability of CNTs are studied. Various types ofinstabilities such as divergence, flutter and parametric instabilityand their interactions are investigated.
Acknowledgement
This project was funded by the Deanship of Scientific Research(DSR) at King Abdulaziz University, Jeddah, under Grant No. 20-135-35-RG. The Authors, therefore, acknowledge with thanksDSR technical and financial support.
Appendix A
A. Azrar et al. / Composite Structures 125 (2015) 127–143 143
where Yi are given by:
Y1 ¼MrV0g F � H2�
;
Y2 ¼ V20g2 H2 � l2H4�
;
Y3 ¼ 2MrV0g H1 � l2H3�
;
Y4 ¼ 2V20g H2 � l2H4�
� bV0gH3;
Y5 ¼ 2MrV0 H1 � l2H3�
þ aH4 � bMrH2;
Y6 ¼ kI1 þ ðV20 � T� l2kÞH2 þ ð1� l2V2
0ÞH4 � bV0H3;
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