Accepted Manuscript Nonlinear thermal and flow-induced vibration analysis of fluid-conveying carbon nanotube resting on Winkler and Pasternak foundations M.G. Sobamowo PII: DOI: Reference:
S2451-9049(17)30034-3 http://dx.doi.org/10.1016/j.tsep.2017.08.005 TSEP 54
To appear in:
Thermal Science and Engineering Progress
Received Date: Revised Date: Accepted Date:
19 February 2017 2 August 2017 30 August 2017
Please cite this article as: M.G. Sobamowo, Nonlinear thermal and flow-induced vibration analysis of fluidconveying carbon nanotube resting on Winkler and Pasternak foundations, Thermal Science and Engineering Progress (2017), doi: http://dx.doi.org/10.1016/j.tsep.2017.08.005
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Nonlinear thermal and flow-induced vibration analysis of fluid-conveying carbon nanotube resting on Winkler and Pasternak foundations
M. G. Sobamowo Department of Mechanical Engineering, University of Lagos, Akoka, Lagos, Nigeria. E-mail:
[email protected],
[email protected],
[email protected]
Abstract The thermal-mechanical modeling of single-walled carbon nanotube results in nonlinear equations that are difficult to solve exactly and analytically with any existing general method of solution. However, in many cases, recourse is made to approximate analytical methods in which their accuracies largely depend on the number of terms included in the solutions. In this work, approximate analytical solutions are presented using differential transformation method with after-treatment method for the nonlinear equations arising in thermal and flow-induced vibrations of carbon nanotubes. The developed analytical solutions are used to investigate the effects of nonlocal parameter, Knudsen number, temperature, foundation parameters and boundary conditions on the dynamic behaviour of the nanotube. Also, the developed analytical solutions are verified with the numerical solutions and validated with experimental results. Good agreements are established between the analytical solutions and numerical solutions, and also between the analytical solutions and experimental results. The analytical solutions as presented in this work can serve as benchmarks for other methods of solutions of the problem. They can also provide a starting point for a better understanding of the relationship between the physical quantities of the problems. Keyword:
Non-linear Vibration; Differential Transformation method; Aftertreatment techniques; Nanotubes; Thermal effects; Elastic
foundations.
1.0 Introduction Following the discovery of carbon nanotubes (CNT) by Iijima [1], there have been tremendous interests in the analysis and accurate predictions of the dynamic behaviours of the nanotubes. Indisputably, the dynamic analysis of flow and thermal induced vibrations in nanotubes has become a subject of vast interests as it has attracted a large number of studies in literatures [226]. This is because, modeling the dynamic behaviours of the structures under the influence of some thermo-fluidic or thermo-mechanical parameters often results in nonlinear equations and such are difficult to solve exactly and analytically with any existing or general analytical method of solution. In some cases where decomposition procedures into spatial and temporal parts are carried out, the resulting nonlinear equations for the temporal part come in form of Duffing equations which have been solved by several analytical and numerical methods in literature. However, application of analytical methods such as Exp-function method, He’s Exp-function method, improved F-expansion method, Lindstedt-Poincare techniques, quotient trigonometric function expansion method to the nonlinear equations presents analytical solutions either in implicit or explicit form which often involved complex mathematical analysis leading to analytic expression involving a large number terms. Furthermore, most of the methods are time-
consuming and they required possession of high skills in mathematics. Also, they do not provide general analytical solutions since the solutions often come with conditional statements which make them limited in used as many of the conditions with the exact solutions do not meet up with the practical applications. In practice, analytical solutions with large number of terms and conditional statements for the solutions are not convenient for use by designers and engineers [27]. Consequently, recourse has always been made to numerical methods or approximate analytical methods in solving the problems. However, the classical way for finding exact analytical solution is obviously still very important since it serves as an accurate benchmark for numerical solutions. Additionally, exact analytical solutions for specified problems are also essential for the development of efficient applied numerical simulation tools. Inevitably, exact analytical expressions are required to show the direct relationship between the models parameters. When such exact analytical solutions are available, they provide good insights into the significance of various system parameters affecting the phenomena as it gives continuous physical insights than pure numerical or computation methods. Furthermore, most of the purely numerical methods that are applied in literature to solve nonlinear problems are computationally intensive. Exact analytical expression is more convenient for engineering calculations compare with experimental or numerical studies and it is obvious starting point for a better understanding of the relationship between physical quantities or properties. It is convenient for parametric studies, accounting for the physics of the problem and appears more appealing than the numerical solutions. It helps to reduce the computation costs, simulations and task in the analysis of real life problems which are nonlinear problems virtually in all cases. Therefore, an exact analytical solution is required for the problem. The relative new approximate analytical method, differential transformation method has proven proved to be more effective than most of the other approximate analytical solutions as it does not require many computations as carried out in Adomian decomposition method (ADM), homotopy analysis method (HAM), homotopy perturbation method (HPM), and variational iteration method ((VIM) and variational of parameter method (VPM). The differential transformation method as introduced by Ζhou [28] has fast gained ground as it appeared in many engineering and scientific research papers [29-47]. This is because of its comparative advantages over the other known approximate analytical methods. It is a method that could solve differential equations, difference equation, differential-difference equations, fractional differential equation, pantograph equation and integro-differential equation. It solves nonlinear integral and differential equations without linearization, discretization, restrictive assumptions, perturbation and discretization or round-off error. It reduces complexity of expansion of derivatives and the computational difficulties of the other traditional methods. Using DTM, a closed form series solution or approximate solution can be obtained as it provides excellent approximations to the solution of non-linear equation with high accuracy. It is capable of greatly reducing the size of computational work while still accurately providing the series solution with fast convergence rate. It is a more convenient method for engineering calculations compared with other approximate analytical or numerical methods. It appears more appealing than the numerical solution as it helps to reduce the computation costs, simulations and task in the analysis of nonlinear problems. Moreover, the need for small perturbation parameter as required in traditional perturbation methods (regular and singular perturbation methods), the rigour of the determination of Adomian polynomials as experienced in ADM, the restrictions of HPM to weakly nonlinear problems as established in literatures, the lack of rigorous theories or proper guidance for choosing initial approximation, auxiliary linear operators, auxiliary functions,
auxiliary parameters, and the requirements of conformity of the solution to the rule of coefficient ergodicity as done in HAM, the difficulty in the search for Langrange multiplier as carried in VIM and Wronkian multiplier in VPM , and the challenges associated with proper construction of the approximating functions for arbitrary domains or geometry of interest as in Galerkin weighted residual method (GWRM), least square method (LSM) and collocation method (CM) are some of the difficulties that DTM overcomes. Also, Galerkin’s decomposition procedure provides a very powerful and novel procedure of decomposing linear and non-linear time-spatial dependent partial differential equations into spatial and temporal parts, thereby, reducing the partial differential equation to ordinary differential equation. Combining the Galerkin’s method with differential transform method in the analysis of non-linear initial-boundary value problems, provides complementary advantages of higher accuracy, reduced computation cost and task as compared to the direct applications of multi-dimensional differential transformation method and other methods as found in literatures. Although, the hybrid method of Galerkin-differential transform method (GDTM) solves the differential equations without linearization, discretization or approximation, linearization restrictive assumptions or perturbation, complexity of expansion of derivatives and computation of derivatives symbolically, a well-known fact is that, the differential transformation method (DTM) gives the solution in the form of a truncated series. Unfortunately, in the case of oscillatory systems, the truncated series obtained by the method is periodic only in a very small region. This drawback is not only peculiar to DTM, other approximate analytical methods such as ADM, HAM, VIM, VPM have the same short-coming for oscillatory systems [48, 49]. To overcome this difficulty and obtain approximate periodic solutions for a large or infinite region, an aftertreatment technique (AT) was established [48-51]. In modifying ADM to provide periodic solutions in a large region, Venkatarangan and Rajakshmi [48] and Jiao et al. [49] developed AT techniques which are based on using Pade approximates, Laplace transform and its inverse. Although, their AT techniques were found to be effective in many cases, it has some disadvantages, not only it required a huge amount of computational work to provide accurate approximations for the periodic solutions but also come with difficulties of obtaining the inverse Laplace transform which greatly restricts the applications of the after-treatment technique [49]. Consequently, Ebaid [50] and Ebaid and Ali [51] developed very effective AT techniques (Sine-AT technique and Cosine-AT technique) that avoid the use of Pade approximates, Laplace transform and its inversion problems. The technique has shown to be better than previously developed aftertreatment techniques which are based on using Pade approximates and Laplace transform, basis function and domain decomposition techniques. Therefore, in this work, the Galerkin-differential transformation method with Ebaid’s cosine aftertreatment is applied to provide analytical solutions to the nonlinear partial differential equations arising in thermal and flow-induced vibration of singlewalled carbon nanotubes resting on Winkler and Pastnetak foundations. The nonlinear partial differential equations are converted to nonlinear ordinary differential equations and then differential transformation method with aftertreatment technique is utilized to provide the exact analytical solutions to the nonlinear ordinary differential equations of vibration of the structures. The developed analytical solutions are verified with the numerical solutions and validated with experimental results. Good agreements are established between the analytical solutions and numerical solutions, and also between the analytical solutions and experimental results. The analytical solutions can serve as benchmarks for other methods of solutions of the problem. They can also provide a starting point for a better understanding of the relationship between the physical quantities of the problems.
2.0 The Governing equations and boundary conditions Consider a single-walled carbon nanotube conveying hot fluid, subjected to stretching effects and resting on linear and nonlinear elastic foundations (Winkler and Pasternak foundations) under external applied tension, global pressure and slip boundary conditions. Using the Eringen’s nonlocal elasticity theory [5255], Euler-Bernoulli beam theory and Hamilton’s principle, the governing equation of motion for the single-walled carbon nanotube (SWCBT) is developed as EI
2 ∂4 w ∂2 w ∂w ∂2 w EAα∆θ ∂2 w EA L ∂w ∂2w 2 + ( m + m ) + µ + 2 um + m u + PA − T − k − − p f f p f dx ∂x4 ∂t 2 ∂t ∂x∂t 1− 2ν ∂x2 2L ∫0 ∂x ∂x2
2 ∂4 w ∂3w ∂2 w ∂4 w ∂w 2 ∂w + 2umf 3 (mp + mf ) 2 2 + µ 2 + k1 2 + 6k3w + 3w ∂x ∂t ∂x ∂t ∂x ∂x ∂x ∂t ∂x 2 +k1w + k3w3 − ( eoa) 2 =0 4 4 L ∂w EA α ∆ θ ∂ w EA ∂ w + mf u2 + PA − T − k p − 4 − ∫0 dx 4 1 − 2 ∂ x 2 L ∂ x ∂ x ν
(1)
Fig. 1 Single-walled carbon nanotube conveying hot fluid resting on Winkler and Pasternak elastic foundations
If the nanotube is slightly curved, then the governing equation for the nanotube becomes EI
∂4w ∂2 w ∂w ∂2 w EAα∆θ ∂ 2 w 2 + ( m + m ) + µ + 2 um + m u + PA − T − k − p f f f p ∂x 4 ∂t 2 ∂t ∂x∂t 1 − 2ν ∂x 2
2 L EA ∂Zo ∂w ∂w ∂ 2 w ∂2 Zo 3 − No + + dx 2 + 2 + k1w + k3 w ∫ 2 L 0 ∂x ∂x ∂x ∂x ∂x 2 ∂4w ∂3w ∂2w ∂4w ∂w 2 ∂w + 2um f 3 (m p + m f ) 2 2 + µ 2 + k1 2 + 6k3 w + 3w ∂x ∂t ∂x ∂t ∂x ∂x ∂x ∂t ∂x 2 − ( eo a ) =0 2 L 4 4 4 ∂ Z ∂ Z EA α ∆ θ ∂ w EA ∂ w ∂ w ∂ w o o + m u 2 + PA − T − k − − No + + dx + p f 1 − 2ν ∂x 4 2L ∫0 ∂x ∂x ∂x ∂x 4 ∂x4
(2) For nanotube conveying fluid, the radius of the tube is assumed to be the characteristics length scale, Knudsen number is larger than 10-2. Therefore, the assumption of no-slip boundary conditions does not hold and modified model should be used. Therefore, we have VCF =
u U avg , no − slip
=
U avg , slip U avg , no − slip
2 −σv = (1 + ak Kn ) 4 σv
Kn + 1 1 + Kn
Where Kn is the Knudsen number (slip parameter), σvis tangential moment accommodation coefficient which is considered to be 0.7 for most practical purposes [56].
(3)
ak = ao
ao =
2
tan −1 ( a1 Kn B ) π
(4)
64 4 3π 1 − b
(5)
Therefore, 2 −σv U avg , slip = u = (1 + ak Kn ) 4 σv
Kn + 1 U avg , no − slip = VCF (U avg , no − slip ) 1 + Kn
(6)
Substituting Eq. (6) into Eq. (2), we have EI
2 ∂4w ∂2w ∂w ∂2w EAα∆θ ∂2w + ( m + m ) + µ + 2 m VCF U + m VCF U + PA − T − k − ( avg,no−slip ) ∂x∂t f ( avg,no−slip ) p f f p ∂x4 ∂t 2 ∂t 1− 2ν ∂x2
2 L EA ∂Z ∂w ∂w ∂2w ∂2Z − No + ∫ o + dx 2 + 2o + k1w+ k3w3 2L 0 ∂x ∂x ∂x ∂x ∂x 2 ∂4w ∂3w ∂2w ∂4w ∂w 2 ∂w + 2mf VCF (Uavg,no−slip ) 3 (mp + mf ) 2 2 + µ 2 + k1 2 + 6k3w + 3w ∂x ∂t ∂x ∂t ∂x ∂x ∂x ∂t 2 ∂x −( eoa) =0 2 EAα∆θ ∂4w + mf VCF (Uavg,no−slip ) + PA −T − kp − 1− 2ν ∂x4
(7) Using the Galerkin’s decomposition procedure to separate the spatial and temporal parts of the lateral displacement functions as
w( x, t ) = φ ( x)u(t )
(8)
Where u(t ) the generalized coordinate of the system and φ ( x) is a trial/comparison function that will satisfy both the geometric and natural boundary conditions. Applying one-parameter Galerkin’s solution given in Eq. (9) to Eq. (7) L
∫ R ( x, t ) φ ( x ) dx 0
Where
(9)
2 ∂4w ∂2w ∂w ∂2w EAα∆θ ∂2w R(x, t) = EI 4 + (mp + mf ) 2 + µ + 2mf VCF (Uavg,no−slip ) + mf VCF (Uavg,no−slip ) + PA−T − kp − ∂x ∂t ∂t ∂x∂t 1− 2ν ∂x2 2 L EA ∂Z ∂w ∂w ∂2w ∂2Z − No + ∫ o + dx 2 + 2o + k1w+ k3w3 2L 0 ∂x ∂x ∂x ∂x ∂x 2 ∂4w ∂3w ∂2w ∂4w ∂w 2 ∂w ( m + m ) + µ + k + 6 k w + 3 w + 2 m VCF U p f 2 2 ( 1 3 f avg,no−slip ) 2 2 3 ∂x ∂t ∂x ∂t ∂x ∂x ∂x ∂t ∂x 2 −( eoa) 2 L 4 ∂Zo ∂w ∂w ∂4w ∂4Zo 2 α θ EA ∆ ∂ w EA + m VCF (U − No + ∫ + dx + avg,no−slip ) + PA−T − kp − f 1− 2ν ∂x4 2L 0 ∂x ∂x ∂x ∂x4 ∂x4
We have Mus (t ) + Gu s (t ) + ( K + C )us (t ) − Vus3 (t ) = 0
where L d 2φ M = ∫ (m p + m f )φ ( x ) φ ( x ) − (eo a )2 2 dx 0 dx
L dφ d 3φ d 2φ G = ∫ φ ( x ) 2m f us − (eo a )2 3 + µ φ ( x) − (eo a) 2 2 dx 0 dx dx dx
2 L d 4φ d 2φ d 2φ 2 d φ 2 K = ∫ φ ( x) EI 4 + k1φ ( x) − k p 2 − k1 ( eo a ) e a k + ( o ) p 2 dx 2 0 dx dx dx dx
L EAα∆θ C = ∫ m f us2 + PA − T − 0 1 − 2ν
4 d 2φ 2 d φ x e a dx φ ( ) − ( ) 2 o 4 dx dx
2 2 EA L ∂φ d 2φ d 3φ ∂φ 2 − k3 3 ( x) + 6k3 (eo a ) φ ( x) No + dx 2 L ∫0 ∂x dx2 dx ∂x L V = ∫ φ ( x) dx 2 0 4 L ∂φ EA ∂φ dφ 2 2 2 −3k3 (eo a ) φ ( x) ∂x − (eo a) N o + 2 L ∫0 ∂x dx dx 4
Here, we have
us = VCF (U avg ,no− slip )
(10)
For the slightly curved nanotube, M, G, K and C are the same but 2 2 L EA ∂Z o ∂w ∂w ∂ 2 w ∂ 2 Z o d 3φ ∂φ 2 No + + + dx − k3 3 ( x ) + 6k3 (eo a ) φ ( x) 2 L ∫0 ∂x ∂x ∂x ∂x 2 ∂x 2 dx ∂x L V = ∫ φ ( x) dx 0 2 L 4 4 φ ∂ EA ∂ Z ∂ w ∂ w ∂ w ∂ Z 2 2 o o − (eo a )2 N o + + −3k3 (eo a) φ ( x) dx 4 + 4 ∫ ∂x 2 L 0 ∂x ∂x ∂x ∂x ∂x
and the circular fundamental natural frequency gives
ωn =
K + C∗ M
(11)
where L EAα∆θ C ∗ = ∫ PA − T − 0 1 − 2ν
4 d 2φ 2 d φ x e a φ ( ) − ( ) 2 dx o dx 4 dx
3. The initial and boundary conditions The nanotube may be subjected to any of the following boundary conditions i. Clamped-Clamped (doubly clamped) Where the trial/comparison function are given as
sinhβ n L + sinβ n L ( sinhβ n x − sinβ n x ) coshβ n L − cosβ n L
(12a)
coshβ n L − cosβ n L ( sinhβ n x − sinβ n x ) sinhβ n L − sinβ n L
(12b)
φ ( x) = coshβn x − cosβn x − or
φ ( x) = coshβn x − cosβn x −
where β n are the roots of the equation cosβ n Lcoshβ n L = 1 The initial and the boundary conditions are w (0, x ) = A w (0, x ) = 0
(13) w (0, t ) = w '(0, t ) = 0 w ( L , t ) = w '( L , t ) = 0
The applications of space function as given above for clamped-clamped will involve long calculations and expressions in Finding M, G, K, C, and V, alternatively, a polynomial function of the form Eq. (14) can be applied for this type of support system.
φ ( x) = ao + a1 X + a2 X 2 + a3 X 3 + a4 X 4 Where X =
(14)
x L
Applying the boundary conditions
φ ( x) = ( X 2 − 2 X 3 + X 4 ) a4
(15)
Orthogonal function should satisfy the equation
∫
a
0
φ ( X )φ ( X )dx = 1
(16)
Substitute Eq. (15) into Eq. (16), we have 1 a4 = 3 70 5 a ( 70a 4 − 315a 3 + 540a 2 − 420a + 126 )
(17)
For a =1, we arrived at a4 = 25.20 for the first mode ii. Clamped-Simple supported
The trial/comparison function are given as coshβ n L − cosβ n L ( sinhβ n x − sinβ n x ) sinhβ n L − sinβ n L
φ ( x) = coshβn x − cosβn x −
(18)
β n are the roots of the equation tanβ n L = tanhβ n L
The initial and the boundary conditions are w (0, x ) = A w (0, x ) = 0
(19) w (0, t ) = w '(0, t ) = 0 w ( L , t ) = w ''( L , t ) = 0
Alternatively, a polynomial function of the form Eq. (20) can be applied for this type of support system. 3 2
5 2
φ ( x ) = X 2 − X 3 + X 4 a4
(20)
On using orthogonal functions, a4 = 11.625 for the first mode iii. Simple-Simple supported
φ ( x) = sinβn x sinβ L = 0 ⇒ β n =
(21) nπ L
The initial and the boundary conditions are w (0, x ) = A w (0, x ) = 0
(22) w (0, t ) = w ''(0, t ) = 0 w ( L , t ) = w ''( L , t ) = 0
Alternatively, a polynomial function of the form Eq. (22) can be applied for this type of support system.
φ ( x) = ( X − 2 X 3 + X 4 ) a4
(23)
On using orthogonal functions, a4 = 3.20 for the first mode
iv.
Clamp-Free (cantilever)
coshβ n L + cosβ n L ( sinhβ n x − sinβn x ) sinhβn L + sinβ n L
(24a)
sinhβ n L − sinβ n L ( sinhβ n x − sinβ n x ) coshβ n L − cosβ n L
(24b)
φ ( x) = coshβ n x − cosβ n x − or
φ ( x) = coshβn x − cosβn x −
β n are the roots of the equation cos β n Lcoshβ n L = −1
The initial and the boundary conditions are
w (0, x ) = A w (0, x ) = 0
(25) w (0, t ) = w '(0, t ) = 0 w ''( L , t ) = w '''( L , t ) = 0
Alternatively, a polynomial function of the form Eq. (26) can be applied for this type of support system.
φ ( x) = ( 6 X 2 − 4 X 3 + X 4 ) a4
(26)
Also, with the aid of orthogonal functions, a4 = 0.6625 for the first mode
4. Determination of natural frequencies The natural frequency analysis is the sine qua non for the analysis of stability, it must therefore be carried out in the dynamic response analysis of the structures.
Under the transformation, τ = ωt , Eq. (10) turns out to be M ω 2 u(τ ) + Gωu (τ ) + ( K + C )u (τ ) − Vu 3 (τ ) = 0
(27)
For the undamped clamped-clamped, clamped-simple and simple-simple supported structures, G=0 M ω 2 u(τ ) + ( K + C )u (τ ) − Vu 3 (τ ) = 0
(28)
In order to find the periodic solution of Eq. (28), we assume an initial approximation for zeroorder deformation as uo (τ ) = Acos τ
(29)
Substituting Eq. (29) into Eq. (28), we have − M ωo2 Acos τ + ( K + C ) Acos τ − VA3cos 3τ = 0
(30)
which gives 3cos τ + cos 3τ − M ω 2 Acos τ + ( K + C ) Acos τ − VA3 4
Collecting like terms in Eq. (31), gives
=0
3VA3 1 ( + ) − − M ω 2 A cos τ − VA3cos 3τ = 0 K C A 4 4 In order to eliminate the secular term, we have 3VA3 ( + ) − − M ωo2 A = 0 K C A 4
(31)
(32)
Thus, zero-order nonlinear natural frequency becomes
K + C 3VA2 ωo ≈ − M 4M
(33)
Therefore, the ratio of the zero-order nonlinear natural frequency, ωo to the linear frequency, ωb
ωo 3VA2 ≈ 1− 4( K + C ) ωb
(34)
Following the same procedural approach, the first-order nonlinear natural frequency was found to be 2 K + C 3VA2 3V 2 A4 1 K + C 3VA2 ω1 ≈ − + − 4M − 32 M 2 2 M 4 M M
(35)
The ratio of the first-order nonlinear frequency, ω1 to the linear frequency, ωbgives
3VA2 3V 2 A4 ω1 1 3VA2 ≈ 1 − + 1 − − 2 4( K + C ) ωb 4( K + C ) 32( K + C ) 2
(36)
On a general note, following Lai et al. [57], it can easily be shown that the exact natural frequency of the fluid-conveying structures is given as
ω1,exact =
πξ1 1 − 2 ∫ (1 + ξ2 sin 2t ) 2 dt 0
π 2
For the general case in this work,
(37)
VA2 M K + C VA2 ξ1 = − ξ = 2 2 VA K + C M 2 M − 2 M M 1 2
When the nonlinear term, V is set to zero, the linear natural frequency was recovered. It is very difficult to generate exact solution of Eq. (37). However, on using series integration method, an approximation analytical solution was developed for the nonlinear natural frequency as
ω1 =
ξ1
(1 − 0.25000ξ
+ 0.11680ξ2 − 0.59601ξ 23 − 1.90478ξ2 4 − 2.04574ξ 25 ) 2
2
(38)
The ratio of the nonlinear frequency, ω1 to the linear frequency, ωb
ω1 1 = 2 ωb (1 − 0.25000ξ 2 + 0.11680ξ 2 − 0.59601ξ23 − 1.90478ξ 2 4 − 2.04574ξ 25 )
(39)
Also, it is difficult to generate any closed form solution for the above nonlinear Eq. (8). In finding simple, direct and practical solutions to the problem, differential transformation with after treatment technique is applied to the nonlinear equation. 5. Differential transformation method
As pointed previously, the differential transformation method is an approximate analytical method for solving differential equations. However, a closed form series solution or approximate solution can be obtained for non-linear differential equations with the use of DTM. The basic definitions of the method is as follows If u (t ) is analytic in the domain T, then it will be differentiated continuously with respect to time t. d K u (t ) = φ (t , k ) dt K
for
all t ∈ T
(40)
for t = ti , then φ (t , k ) = φ (t i , K ) , where K belongs to the set of non-negative integers, denoted as the K-domain. Therefore Eq. (40) can be rewritten as d k u (t ) U ( k ) = φ (t i , k ) = K dt t =t i
(41)
Where U k is called the spectrum of u (t ) at t = t i If u (t ) can be expressed by Taylor’s series, the u (t ) can be represented as ∞ (t − t i ) K u (t ) = ∑ k k!
U (k )
(42)
Where Equ. (42) is called the inverse of U (k ) using the symbol ‘D’ denoting the differential transformation process and combining (41) and (42), it is obtained that
(t − t i )K u (t ) = ∑ K =0 K! ∞
−1 U (k ) = D U (k )
(43)
5.1 Operational properties of differential transformation method
If u (t ) and v (t ) are two independent functions with time (t) where U (k ) and V (k ) are the transformed function corresponding to u (t ) and v (t ) , then it can be proved from the fundamental mathematics operations performed by differential transformation that. i.
If z (t ) = u (t ) ± v (t ),
then Ζ ( k ) = U ( k ) ± V ( k )
ii.
If z (t ) = α u (t ), then Z ( k ) = α U ( k )
iii.
If z (t ) =
iv.
If z (t ) = u (t )v (t ), then Ζ (t ) =
du(t ) , then Ζ ( k ) = ( k − 1) U ( k + 1) dt K
∑ V (l )U (k − l ) i =0 K
v.
If z (t ) = u m (t ) , then Ζ(t ) = ∑ U m −1 (l )U (k − l ) I =0
vi.
k l Z ( t ) = If z (t ) = u (t )v (t ), then ∑ [V ( j )U (l − j )] ∑ l − 0 j =0 n
n
k −l
j =0
∑ [V ( j )U (k − l − j)]
k
vii.
If z (t ) = u (t )v (t ), then Ζ (k ) = ∑ (l + 1)V (l + 1)U (k − l ) l =0
5.2 Solution of the temporal nonlinear equation using differential transformation method
The above DTM described is applied to solve the temporal nonlinear differential equation. Applying DTM to Eq. (10), we have
M [ ( p + 2)( p + 1)U ( p + 2) ] + G [ ( p + 1)U ( p + 1) ] p m + ( K + C )U ( p ) − V ∑∑ U (n)U (m − n)U ( p − n) = 0 m n= 0
(44)
Then p m V ∑∑U (n)U (m − n)U ( p − n) 1 U ( p + 2) = m n =0 ( p + 2)( p + 1) M −G [ ( p + 1)U ( p + 1)] − ( K + C )U ( p )
(45)
For p = 0, 1, 2, 3, 4, 5, 6, 7, 8. We have the following U (0) = A U (1) = 0 U (2) =
U (4) =
−GU (2) A VA2 − ( K + C )} U (3) = { 2M 3M
1 1 3VA2U (2) − 3GU (3) − (K + C)U (2) U(5) = 3VA2U(3) − 4GU(4) − (K + C)U(3) 12M 20M
U (6) =
1 3VA2U (4) + 3 AU (2) − 5GU (5) − ( K + C )U (4) 30M
(46) U (7) =
1 3VA2U (5) + 6VAU (2)U (3) − 6GU (6) − ( K + C )U (5) 42M
U (8) =
1 56M
U (9) =
2 1 3VA U (7) + 6VAU (2)U (5) + 6VAU (3)U (4) 72 M +3VAU 2 (2)U (3) − 8GU (8) − ( K + C )U (7)
3VA2U (6) + 6VAU (2)U (4) + 3VAU 3 (3) 3 +VAU (2) − 7GU (7) − ( K + C )U (6)
1 U (10) = 90 M
3VA2U (8) + 3VAU (2)U 2 (3) + 3VAU 2 (4) + 3VAU 2 (2)U (4) +6VAU (2)U (6) − 9GU (9) − ( K + C )U (8)
Applying the comparison function in Eqs. (10)-(26), where G=0 for the clamped-clamped, clamped-simple and simple-simplesupported structures, the above equations reduced to
U (0) = A U (1) = 0 U (2) =
A VA2 − ( K + C )} U (3) = 0 { 2M
U (4) =
1 3VA2U (2) − ( K + C )U (2) U (5) = 0 12 M
U (6) =
1 3VA2U (4) + 3 AU (2) − 5GU (5) − ( K + C )U (4) U (7) = 0 30M
(47) U (8) =
2 3 1 3VA U (6) + 6VAU (2)U (4) + 3VAU (3) U (9) = 0 56M +VAU 3 (2) − 7GU (7) − ( K + C )U (6)
1 U (10) = 90 M
3VA2U (8) + 3VAU (2)U 2 (3) + 3VAU 2 (4) + 3VAU 2 (2)U (4) +6VAU (2)U (6) − 9GU (9) − ( K + C )U (8)
Therefore, ∞
u (t ) = ∑ U (k )t k = U (0) + U (1)t + U (2)t 2 + U (3)t 3 + U (4)t 4 + U (5)t 5 k =0
+ U (6)t 6 + U (7)t 7 + U (8)t 8 + U (9)t 9 + U (10)t10 + ...
Substituting Eq. (47) into Eq.(48), we have
(48)
A {VA 2 − ( K + C )} A 3VA 2 − ( K + C ) t 4 VA 2 − ( K + C )} t 2 + { 2M 24 M 2 2 VA 3 VA 2 − ( K + C )}{3VA 2 − ( K + C )} + 3 A {VA 2 − ( K + C )} 2 { 2M 1 8M t6 + 30 M A ( K + C ) {VA 2 − ( K + C )}{3VA 2 − ( K + C )} − 2 24 M
u (t ) = A +
VA 3 3 A2 VA 2 − ( K + C )}{3VA 2 − ( K + C )} + {VA 2 − ( K + C )} 2 { 2 8 M 2 M VA 2 2 10 M A ( K + C ) {VA − ( K + C )}{3VA − ( K + C )} − 2 24 M 2 2 4 A {VA − ( K + C )} 3 1 3VA VA 2 2 2 8 + + {VA − ( K + C )} 3VA − ( K + C ) + 8 M 3 {VA − ( K + C )} t 56 M M 24 M 2 VA 3 3 A2 2 2 2 VA − ( K + C ) 3 VA − ( K + C ) + VA − ( K + C ) { }{ } { } ( K + C ) 8M 2 2M − 2 2 30 M A ( K + C ) {VA − ( K + C )}{3VA − ( K + C )} − 2 24 M VA 3 3 A2 VA 2 − ( K + C )}{3VA 2 − ( K + C )} + {VA 2 − ( K + C )} 2 { 2 8 M 2 M VA 10 M A ( K + C ) {VA 2 − ( K + C )}{3VA 2 − ( K + C )} − 2 24 M 2 2 2 4 A {VA − ( K + C )} 3 VA 3VA 3VA 2 2 2 3VA − ( K + C ) + {VA − ( K + C )} 56 M + M {VA − ( K + C )} 24 M 2 8M 3 3 2 VA 3A 2 2 2 VA − ( K + C ) 3 VA − ( K + C ) + VA − ( K + C ) { }{ } { } ( K + C ) 8M 2 2M − 2 2 30 M A ( K + C ) {VA − ( K + C )}{3VA − ( K + C )} − 2 24 M 2 2 VA 2 {VA 2 − ( K + C )} 3 A VA − ( K + C ) 2 { } 3VA 2 − ( K + C ) 3VA + 3VA 2 − ( K + C ) + VA 2 − ( K + C )} 2 2 { 2 8M 4M 24 M 1 t 10 + ... VA 3 3 A2 + 2 2 2 8 M 2 {VA − ( K + C )}{3VA − ( K + C )} + 2 M {VA − ( K + C )} 90 M 2 + VA 2 VA − ( K + C ) } A ( K + C ) VA 2 − ( K + C ) 3VA 2 − ( K + C ) 10 M 2 { { }{ } − 24 M 2 3 2 VA 3A 2 2 2 VA − ( K + C ) 3 VA − ( K + C ) + VA − ( K + C ) { }{ } { } 2 2 2M 8 M VA 2 2 10 M A ( K + C ) {VA − ( K + C )}{3VA − ( K + C )} − 2 24 M ( K + C ) 3VA 2 4 A {VA 2 − ( K + C )} 3 VA 2 2 2 − 3VA − ( K + C ) + + VA − ( K + C )} VA − ( K + C )} { 2 3 { 56 M M 24 M 8M 3 2 VA 3 A 2 {VA 2 − ( K + C )}{3VA 2 − ( K + C )} + VA − ( K + C ) { } 2 (K + C ) 8M 2M − 2 2 30 M A ( K + C ) {VA − ( K + C )}{3VA − ( K + C )} − 2 24 M
(49) The DTM solution has 11-terms including the zero-term
Apart from the fact that the solution of the DTM given above is too long for practical applications, the solution is given in form of truncated series. This truncated series is periodic only in a very small region. In order to make the solution periodic over a large region, we applied Cosine-after treatment (CAT-technique) as established by Ebaid and Ebaid and Ali [50, 51]. If the truncated series in Eq. (49) is expressed only in even-power of the independent variable t, i.e. N N (50) Φ N (t ) = ∑ U (2 k )t 2 k U (2 k + 1) = 0, ∀k = 0,1,..., − 1, where N is even 2 k =0 then CAT- technique is based on the assumption that this truncated series can be expressed as another finite series in terms of the cosine-trigonometric functions with different amplitude and arguments [49, 50] N
Φ N (t ) = ∑ λ j cos ( Ω j t ) , where n is finite
(51)
j =1
On expanding both sides of Eq. (51) as power series of t and equating the coefficient of like powers, we have t0; t2; t4; t6; t8; t10 ;
∑ λ = U (0), ∑ λ Ω = −2!U (2), ∑ λ Ω = 4!U (4), ∑ λ Ω = −6!U (6), ∑ λ Ω = 8!U (8), ∑ λ Ω = −10!U (10), n
j
j
n j
j
2 j
j
4 j
j
6 j
j
8 j
n j
n j
n j
n
j
j
(52)
10 j
...
For practical application, it is sufficient to express the truncated series Φ N (t ) in terms of two cosines with different amplitudes and arguments as 2
Φ 6 (t ) = ∑ λ j cos ( Ω j t ) .
(53)
j =1
Therefore, we have
λ1 + λ2 = U (0)
(54a)
λ1Ω12 + λ2 Ω 22 = −2U (2)
(54b)
λ1Ω14 + λ2 Ω 42 = 24U (4)
(54c)
λ1Ω16 + λ2 Ω62 = −720U (6)
(54d)
Substituting the DTM results in Eq. (47) into Eq. (54a-d), we have
λ1 + λ2 = A
(55a)
λ1Ω12 + λ2 Ω 22 = −
A VA2 − ( K + C )} { M
A {VA2 − ( K + C )}
(55b)
3VA2 − ( K + C )
(55c)
VA3 3 A2 2 2 VA − ( K + C )}{3VA − ( K + C)} + VA2 − ( K + C )} { 2 { 2M 1 8M λ1Ω16 + λ2Ω26 = − 2 2 24M A( K + C ) {VA − ( K + C )}{3VA − ( K + C)} − 24M 2
(55d)
4 1
4 2
λ1Ω + λ2 Ω =
M2
Solving the above Eq. (54a-54d), gives 4( K + C ) − 5VA2 + 2 ( K + C ) − 3VA2 8 ( K + C ) − 9VA2 λ1 = A 2 2 ( K + C ) − 3VA2 8 ( K + C ) − 9VA2
(56a)
4( K + C ) − 5VA2 − 2 ( K + C ) − 3VA2 8 ( K + C ) − 9VA2 λ2 = − A 2 2 ( K + C ) − 3VA2 8 ( K + C ) − 9VA2
(56b)
5( K + C ) − 6VA2 − 2 ( K + C ) − 3VA2 8 ( K + C ) − 9VA2 Ω1 = ± M
(56c)
5( K + C ) − 6VA2 + 2 ( K + C ) − 3VA2 8 ( K + C ) − 9VA2 Ω2 = ± M
(56d)
Therefore, the approximated periodic solution for u(t) is given as
4(K + C) − 5VA2 + 2 ( K + C ) − 3VA2 8 ( K + C ) − 9VA2 5( K + C ) − 6VA2 − 2 ( K + C ) − 3VA2 8 ( K + C ) − 9VA2 u(t ) = A cos t 2 2 M 2 2 ( K + C ) − 3VA 8 ( K + C ) − 9VA 4( K + C ) − 5VA2 − 2 ( K + C ) − 3VA2 8 ( K + C ) − 9VA2 5( K + C ) − 6VA2 + 2 ( K + C ) − 3VA2 8 ( K + C ) − 9VA2 − A cos t M 2 2 ( K + C ) − 3VA2 8 ( K + C ) − 9VA2
(57) Substituting Eq. (12a) and Eq. (57) into Eq. (8), we have (i) Clamped-clamped (doubly clamped) supported nanotube 2 2 2 4( K + C ) − 5VA + 2 ( K + C ) − 3VA 8 ( K + C ) − 9VA 2 2 2 2 K + C − 3 VA 8 K + C − 9 VA ( ) ( ) 5( K + C ) − 6VA2 − 2 ( K + C ) − 3VA2 8 ( K + C ) − 9VA2 ×cos t M w( x, t ) = A 4( K + C ) − 5VA2 − 2 ( K + C ) − 3VA2 8 ( K + C ) − 9VA2 2 ( K + C ) − 3VA2 8 ( K + C ) − 9VA2 2 − 5( K + C ) − 6VA2 + 2 ( K + C ) − 3VA2 8 ( K + C ) − 9VA2 ×cos t M
coshβn x − cosβ n x sinhβn L + sinβ n L − coshβ L − cosβ L ( sinhβn x − sinβ n x ) n n
(58) where β n are the roots of the equation cosβ n Lcoshβ n L = 1 Also, substituting Eq. (18) and Eq. (57) into Eq. (8), we have Clamped-Simple supported 2 2 2 4( K + C ) − 5VA + 2 ( K + C ) − 3VA 8 ( K + C ) − 9VA 2 2 2 2 K + C − 3 VA 8 K + C − 9 VA ( ) ( ) 5( K + C ) − 6VA2 − 2 ( K + C ) − 3VA2 8 ( K + C ) − 9VA2 ×cos t M w( x, t ) = A 4( K + C ) − 5VA2 − 2 ( K + C ) − 3VA2 8 ( K + C ) − 9VA2 2 ( K + C ) − 3VA2 8 ( K + C ) − 9VA2 2 − 2 2 2 5( K + C ) − 6VA + 2 ( K + C ) − 3VA 8 ( K + C ) − 9VA ×cos t M
coshβn x − cosβ n x coshβ n L − cosβ n L sinh x sin x − β − β ) ( n n sinhβ L − sinβ L n n
(59)
where β n are the roots of the equation tanβ n L = tanhβ n L Furthermore, substituting Eq. (21) and Eq. (57) into Eq. (8), we have (ii) Pinned-Pinned (simply supported) nanotube 4( K + C ) − 5VA2 + 2 K + C − 3VA2 8 K + C − 9VA 2 ) ) ( ( 2 2 2 2 K + C − 3 VA 8 K + C − 9 VA ( ) ( ) 2 2 2 5( K + C ) − 6 VA − 2 K + C − 3 VA 8 K + C − 9 VA ( ) ( ) ×cos t M w( x, t ) = A 4( K + C ) − 5VA2 − 2 ( K + C ) − 3VA2 8 ( K + C ) − 9VA2 2 2 2 2 K + C − 3 VA 8 K + C − 9 VA ) ) ( ( − 5( K + C ) − 6VA2 + 2 ( K + C ) − 3VA2 8 ( K + C ) − 9VA 2 ×cos t M
Where β n =
nπ x s in L
(60)
nπ L
Note: M, K, C and V are respective integral values of the respective space functions of the respective boundary conditions and structure considered. Going back to the natural frequencies analysis, it can easily be seen that as the nonlinear term, V tends to zero, the frequency ratio of the nonlinear frequency to the linear frequency,
ω =1 V →0 ω b lim
ω tends to 1. ωb (61)
Where (K + C) M
ωb =
It should be pointed out that when the nonlinear term, V is set to zero, we recovered the linear natural frequency Also, as the amplitude, A tends to zero,the frequency ratio of the nonlinear frequency to the ω linear frequency, tends to 1. ωb
ω =1 A→0 ω b
(62)
lim
Furthermore, it can easily be shown that lim u (t ) = A
(63)
t →0
And for very large values of the amplitude A, we have
ω =∞ A →∞ ω b
(64)
lim
where M, K, C and V have been defined previously. Their respective values from the given integrals depend on the boundary conditions considered. 6. Results and Discussion In this section, the results of the simulation and the effects of various parameters of the developed models on the dynamic response of the nanotube are presented and discussed. 6.1 Effects of modal number and boundary conditions on the mode shapes of nanotube Figs. 2-5 show the first five normalized mode shapes of the beams for clamped-clamped, simplesimple, clamped-simple and clamped-free supports. Also, the figures show the deflections of the beams along the beams’ span at five different buckled and mode shapes. Fig. 2 The first five normalized mode shaped of the beams under clamped-clamped supports
Fig. 3 The first five normalized mode shaped of the beams under simple-simple supports
Fig. 4 The first five normalized mode shapes of the beams under clamped-simple supports
Fig. 5 First five normalized mode shaped of the beams under clamped-free (cantilever) supports
6.2 Comparisons of the developed approximate polynomial functions with the established hyperbolic-Trigonometric functions for different boundary conditions of the nanotube
Figs. 6-9 show the comparison of hyperbolic-trigonometric and the polynomial functions for the normalized mode shapes of the beams for clamped-clamped, simple-simple, clamped-simple and clamped-free supports i.e. different boundary conditions of the structure. The figures depict the validity of the developed polynomial functions in this work as there are very good agreements between the established hyperbolic-trigonometric in literature and the developed approximate polynomial functions in this work. Fig. 6 Comparison of the normalized mode shaped functions of the structure for clamped-clamped supports
Fig. 7 Comparison of the normalized mode shaped functions of the structure for simple-simple supports Fig. 8 Comparison of the normalized mode shaped functions of the structure for clamped-free supports Fig. 9 Comparison of the normalized mode shaped functions of the structure for clamped-simple supports
6.3 Effects of boundary conditions, fluid flow velocity and axial tension on the frequency ratio of the nanotube
Figs. 10-12 illustrate the effects of boundary conditions, fluid flow velocity and axial tension on the nonlinear amplitude-frequency response curves of the nanotube. Fig. 10 shows the variation of frequency ratio of the nanotube with the dimensionless maximum amplitude of the structure under different boundary conditions. From, the result, it shown that frequency ratio is highest in the clamped-free (cantilever) supported beam and lowest with clamped-clamped beam. The lowest frequency ratio of the clamped-clamped beam is due to higher stiffness of the beam than the other types of boundary conditions. Also, it should be pointed out that the fundamental linear vibration frequency is the lowest root of the resulting characteristics equation. It can also be inferred from the figure, in contrast to linear systems, the nonlinear frequency is a function of amplitude so that the larger the amplitude, the more pronounced the discrepancy between the linear and the nonlinear frequencies becomes. Fig. 10 Effects of boundary conditions on the nonlinear amplitude-frequency response curves of the nanotube
Fig. 11 Effects of fluid-flow velocity on the nonlinear amplitude-frequency response curves of the nanotube
Fig. 12 Effects of axial tension on the nonlinear amplitude-frequency response curves of nanotube
Fig. 11 shows the effects of fluid-flow velocity on the nonlinear amplitude-frequency response curves of nanotube. It is observed that as the fluid-flow velocity increases, the nonlinear vibration frequency ratio increases. Also, Fig. 12 shows that as the axial tension increases, the nonlinear vibration frequency ratio increases and the difference between the nonlinear and linear frequency becomes pronounced. The same trend was recorded for the effects of pressure on the nonlinear frequency ratio. The results in Figs. 10-12 reveal that the boundary conditions, fluid flow velocity and axial tension have significant effects on the nonlinear behaviour of the nanotube and therefore, these parameters can be used to control the nonlinearity of the structure. 6.4 Effects of Knudsen number and flow velocity on the deflection of the nanotube Effects of slip parameter called Knudsen number and the fluid flow velocity on the deflection of the the nanotube are presented in Fig. 13-17. Fig. 13 illustrates the midpoint deflection time history for the nonlinear analysis of SWCBT when Kn = 0.03 and U= 100 m/s while Fig. 14 presents the midpoint deflection time history for the nonlinear analysis of SWCBT when Kn = 0.03 and U= 500 m/s. Also, Fig. 15 depicts the midpoint deflection time history for the linear analysis of SWCBT when Kn = 0.03 and U= 500 m/s while Figs.16 and 17 shows the midpoint
deflection time history when Kn = 0.05, U= 500 m/s and when Kn = 0.1 and U= 500 m/s, respectively. Fig. 18 shows the comparison of the linear vibration with nonlinear vibration of the SWCNT. It could be seen in the figure that the discrepancy between the linear and nonlinear amplitudes increases as the value of the maximum vibration increases. Fig.13 Midpoint deflection time history for the nonlinear analysis of SWCBT when Kn=0.03 and U= 100 m/s Fig. 14 Midpoint deflection time history for the nonlinear analysis of SWCBT when Kn=0.03 and U= 500 m/s
Fig. 15 Midpoint deflection time history for the linear analysis of SWCBT when Kn=0.03 and U= 500 m/s
Fig. 16 Midpoint deflection time history for the linear analysis of SWCBT when Kn=0.05 and U= 500 m/s
Fig. 17 Midpoint deflection time history for the linear analysis of SWCBT when Kn=0.1 and U= 500 m/s
Fig. 18 Comparison of midpoint deflection time history for the linear and nonlinear analysis of CBT when Kn=0.03 and U= 500 m/s
Fig. 19 Effects of Knudsen number on the dimensionless frequency of simply supported single-walled nanotube
Effects of slip parameter, Knudsen number on the dimensionless frequency ratio of the nanotube are shown in Figs. 19. It is depicted that increase in the slip parameter leads to decrease in the dimensionless frequency ratio of vibration of the SWCNT. It should be pointed out that the Knudsen number predicts various flow regimes in the fluid-conveying nanotube. The Knudsen number with zero value has the highest frequency as shown in the figure. As the Knudsen number increases, the bending stiffness of the nanotube decreases and in consequent, the critical continuum flow velocity decreases as the curves shift to the lowest frequency zone. 6.5 Effects of fluid velocity and change in temperature on the dimensionless frequency of the nanotube Fig. 18a-b shows that as the continuum flow velocity in the nanotube is increased from zero, the nanotube becomes more flexible and the fundamental frequencies decreases. The natural frequency continues to decrease as the fluid flow velocity increase and the natural frequency becomes zero when the conveyed fluid flow velocity exceeds a certain value which corresponds to the critical flow velocity,a point of bounded neutral stability. Also, the figures show the influence of temperature change on the frequencies of the CNT. Jiang et al. [58] pointed out that the coefficient of thermal expansion of carbon nanotube is negative at low/room temperature and positive at high temperature. For the low or room temperature, as the change in temperature increases, the natural frequencies and the critical flow velocity of the structure increase. This is because the Young modulus and the flexural rigidity of the nanotube, which are functions of temperature, decrease at low temperature and consequently, the nanotube becomes increasingly flexible, thereby, increases the frequency of the vibration. However, at high temperature, increase in temperature change, decreases the natural frequencies and the critical flow velocity of
the structure. This is because the Young modulus and the flexural rigidity of the nanotube, which are functions of temperature as stated before, increase at high temperature and consequently, the nanotube becomes increasingly rigid as the change in temperature increase, thereby, decreases the frequency of the vibration of the structure. Also, it should be stated that the critical flow velocity in the low temperatures are higher than that of high temperatures as shown in the figures. These results depict similar trends and tendencies with Yoon et al. [3], Wang et al. [12] and Chang [16]. Fig. 20 Effects of change in temperature on the natural frequency of the nonlinear vibration at low temperature Fig. 21 Effects of change in temperature on the natural frequency of the nonlinear vibration at high temperature
6.6 Effects of foundations and foundation parameters on the dimensionless frequency of the Nanotube The effects of foundations and foundation parameters on the dimensionless frequency of the flow-induced vibration of the fluid-conveying nanotube are shown in Figs. 22-26. Fig. 22 shows that as the Winkler foundation parameter i.e. the shear stiffness of the elastic medium increases, the frequency of the vibration of the fluid-conveying structure and the continuum critical velocity of the fluid increase. This is because the increased value of the elastic medium constant i.e. the shear stiffness makes the fluid-conveying nanotube stronger and vibrates at higher frequency. This same effects are recorded when the fluid flow at high temperature under the same Winkler foundation and also, under Pasternak foundation as shown in Fig. 23-25. However, the natural frequencies of vibration of the nanotube and the critical velocities of the fluid have lower values at high temperature than at low temperature. Also, it was established that the natural frequencies of vibration of the nanotube and the critical velocities of the fluid have higher values when the structure is on Pasternak foundation than when it is laid on Winkler foundation. This is clearly shown in Fig. 26 as it presents the comparative and the combined effects of the Pasternak and Winkler foundations. Fig. 22 Effects of Winkler foundation parameter on the natural frequency of the nonlinear vibration at low temperature
Fig. 23 Effects of Winkler foundation parameter on the natural frequency of the nonlinear vibration at high temperature
Fig. 24 Effects of Pasternak foundation parameter on the natural frequency of the nonlinear vibration at low temperature
Fig. 25 Effects of Pasternak foundation parameter on the natural frequency of the nonlinear vibration at high temperature
Fig. 26 Effects of Winkler and Pasternak foundations parameters on the natural frequency of the nonlinear vibration at low temperature
6.7 Effects of nonlocal parameter on the dimensionless frequency of the nanotube The studies and the investigations of the dynamic and stability behaviours of the structure are largely dependent on the effects of fluid flow velocity and amplitude on the natural frequencies
of the vibration. Effects of nonlocal parameter on the vibration of the nanotube are shown in Fig. 27-30. It is depicted that increase in the slip parameter leads to decrease in the frequency of vibration of the structure and the critical velocity of the conveyed fluid. It should be pointed out as shown in the figures that the zero value for the nonlocal parameter, i.e. eo a = 0 , represents the results of the classical Euler-Bernoulli model which has the highest frequency and critical fluid velocity (a point where the structure starts to experience instability). When the flow velocity of the fluid
attains the critical velocity, both the real and imaginary parts of the frequency are equal to zero. Also, the figures present the critical speeds corresponding to the divergence conditions for different values of the nonlocal parameters. It is shown in figures 33 and 34, the real and imaginary parts of the eigenvalues related to the two lowest modes with different nanotube parameters. Fig. 27 Effects of nonlocal parameter on the natural frequency of the nonlinear vibration in low or room temperature
Fig. 28 Effects of nonlocal parameter on the natural frequency of the nonlinear vibration for high temperature
Fig. 29 Effects of nonlocal parameter on the natural frequency of the nonlinear vibration for the first two modes of vibration for high temperature Fig. 30 Effects of nonlocal parameter on the natural frequency of the nonlinear vibration for the first two modes of vibration for high temperature
6.8 Model Verification and Validation
Fig. 18 and 19 show the comparison of the results of exact analytical solution and the results of the present study for the linear models while Fig. 13 presents the comparison of numerical results models and the results of present work for the nonlinear models. The results show that good agreements are established between the solutions. Fig. 31 Comparison between the obtained results and the exact solution for the linear vibration
Fig. 32 Comparison between the obtained results and the numerical solution for the nonlinear vibration Fig. 33 Comparison of experimental results and the obtained results by the solution developed in this work
As stated previously that the zero value for the nonlocal parameter, i.e. eo a = 0 , represents the results of the classical Euler-Bernoulli model. Under such scenario, Dodds and Runyans [59] presented experimental studies on the effects of high velocity fluid flow in the bending vibrations and static divergence of a simply supported pipe. The validity of the developed analytical solution was tested and established using the Dodds and Runyans’ experiment [57]. Fig. 23 shows the comparison of the
experimental results and the results of the present study for the developed analytical solutions where the effects of nonlocal parameter, elastic foundations, thermal loads, pressure and axial tension are not considered. From the figure, it shows that the analytical solutions as presented in this study are in good agreements with the experiment results.
7. Conclusion In this work, analytical solutions have been developed for the nonlinear equations arising in flow-induced vibration of in nanotubes using Galerkin-differential transformation method with aftertreatment technique. The results show that the alteration of nonlinear flow-induced frequency from linear frequency is significant as the amplitude, flow velocity, and aspect ratio increase. The developed analytical solutions are verified with the exact and numerical solutions as well as the experimental results and good agreements are established. The analytical solutions can serve as benchmarks for other methods of solutions of the problem. They can also provide a starting point for a better understanding of the relationship between the physical quantities of the problems. Nomenclature A Area of the nanotube E Young Modulus of Elasticity G Shear Modulus I moment of area kp Pasternak foundation coefficient k1 linear Winkler foundation coefficient k3 nonlinear Winkler foundation coefficient Kn Knudsen number, L length mp mass of the nanotube mf mass of fluid N axial/Longitudinal force P Pressure r radius of the structure t time T tension u (t ) generalized coordinate of the system w transverse displacement/deflection x axial coordinate Zo(x) arbitrary initial rise function. Symbols σv tangential moment accommodation coefficient φ ( x ) trial/comparison function υ Poisson’ ratio µ damping coefficient ∆θ change in temperature α coefficient of expansion (eoa) nonlocal parameter/nano parameter
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Fig. 1 Single-walled carbon nanotube conveying hot fluid resting on Winkler and Pasternak elastic foundations
4 1st mode shape 2nd mode shape 3rd mode shape
3
4th mode shape 5th mode shape
Mode shape function
2
1
0
-1
-2
0
0.1
0.2
0.3 0.4 0.5 0.6 0.7 Dimensionless beam lenght
0.8
0.9
1
Fig. 2 The first five normalized mode shaped of the beams under clamped-clamped supports
3
1st mode shape 2nd mode shape
2.5
3rd mode shape
2
4th mode shape 5th mode shape
Mode shape function
1.5 1 0.5 0 -0.5 -1 -1.5 -2
0
0.1
0.2
0.3 0.4 0.5 0.6 0.7 Dimensionless beam lenght
0.8
0.9
1
Fig. 3 The first five normalized mode shaped of the beams under simple-simple supports
4
1st mode shape 2nd mode shape 3rd mode shape
3
4th mode shape 5th mode shape
Mode shape function
2
1
0
-1
-2
0
0.1
0.2
0.3 0.4 0.5 0.6 0.7 Dimensionless beam lenght
0.8
0.9
1
Fig. 4 The first five normalized mode shapes of the beams under clamped-simple supports
1st mode shape
4
2nd mode shape 3rd mode shape 4th mode shape
3 Mode shape function
5th mode shape 2
1
0
-1
-2
0
0.1
0.2
0.3 0.4 0.5 0.6 0.7 Dimensionless beam lenght
0.8
0.9
1
Fig. 5 First five normalized mode shaped of the beams under clamped-free (cantilever) supports 2
Hyperbolic-Trigonometric function Polynomial function
1.8 1.6
Mode shape function
1.4 1.2 1 0.8 0.6 0.4 0.2 0
0
0.1
0.2
0.3 0.4 0.5 0.6 0.7 Dimensionless beam lenght
0.8
0.9
1
Fig. 6 Comparison of the normalized mode shaped functions of the structure for clamped-clamped supports
Hyperbolic-Trigonometric function Polynomial function
Mode shape function
1
0.8
0.6
0.4
0.2
0
0
0.1
0.2
0.3 0.4 0.5 0.6 0.7 Dimensionless beam lenght
0.8
0.9
1
Fig. 7 Comparison of the normalized mode shaped functions of the structure for simple-simple supports 2
Hyperbolic-Trigonometric function Polynomial function
1.8 1.6
Mode shape function
1.4 1.2 1 0.8 0.6 0.4 0.2 0
0
0.1
0.2
0.3 0.4 0.5 0.6 0.7 Dimensionless beam lenght
0.8
0.9
1
Fig. 8 Comparison of the normalized mode shaped functions of the structure for clamped-free supports 2 Hyperbolic-Trigonometric function Polynomial function
1.8 1.6
Mode shape function
1.4 1.2 1 0.8 0.6 0.4 0.2 0
0
0.1
0.2
0.3 0.4 0.5 0.6 0.7 Dimensionless beam lenght
0.8
0.9
1
Fig. 9 Comparison of the normalized mode shaped functions of the structure for clamped-simple supports
1.5
Clamped-Simple supported Clamped-Clamped supported Simple-Simple supported Clamped-Free supported
1.45 1.4
Frequency ratio
1.35 1.3 1.25 1.2 1.15 1.1 1.05 1
0
0.5
1 1.5 2 Dimensionless maximum amplitude
2.5
3
Fig. 10 Effects of boundary conditions on the nonlinear amplitude-frequency response curves of the nanotube
1.35 u = 10 m/s u = 30 m/s u = 50 m/s
1.3
Frequency ratio
1.25
1.2
1.15
1.1
1.05
1
0
0.5
1 1.5 2 Dimensionless maximum amplitude
2.5
3
Fig. 11 Effects of fluid-flow velocity on the nonlinear amplitude-frequency response curves of the nanotube
1.35 T = 0 N/m T = 2 x108 N/m 1.3
T = 6 x108 N/m
Frequency ratio
1.25
1.2
1.15
1.1
1.05
1
0
0.5
1 1.5 2 Dimensionless maximum amplitude
2.5
3
Fig. 12 Effects of axial tension on the nonlinear amplitude-frequency response curves of nanotube
8
Deflection, w(x,t) (x10-3 nm)
6 4 2 0 -2 -4 -6 -8
0
2
4
6
8
10 12 Time (sec)
14
16
18
20
Fig.13 Midpoint deflection time history for the nonlinear analysis of SWCBT when Kn=0.03 and U= 100 m/s 8
Deflection, w(x,t) (x10-3 nm)
6 4 2 0 -2 -4 -6 -8
0
2
4
6
8
10 12 Time (sec)
14
16
18
20
Fig. 14 Midpoint deflection time history for the nonlinear analysis of SWCBT when Kn=0.03 and U= 500 m/s
8
Deflection, w(x,t) (x10-3 nm)
6 4 2 0 -2 -4 -6 -8
0
2
4
6
8
10 12 Time (sec)
14
16
18
20
Fig. 15 Midpoint deflection time history for the linear analysis of SWCBT when Kn=0.03 and U= 500 m/s
8
Deflection, w(x,t) (x10-3 nm)
6 4 2 0 -2 -4 -6 -8
0
2
4
6
8
10 12 Time (sec)
14
16
18
20
Fig. 16 Midpoint deflection time history for the linear analysis of SWCBT when Kn=0.05 and U= 500 m/s 8
Deflection, w(x,t) (x10-3 nm)
6 4 2 0 -2 -4 -6 -8
0
2
4
6
8
10 12 Time (sec)
14
16
18
20
Fig. 17 Midpoint deflection time history for the linear analysis of SWCBT when Kn=0.1 and U= 500 m/s
8 Linear vibration Non-linear vibration
Deflection, w(x,t) (x10-3 nm)
6 4 2 0 -2 -4 -6 -8
0
2
4
6
8
10 12 Time (sec)
14
16
18
20
Fig. 18 Comparison of midpoint deflection time history for the linear and nonlinear analysis of CBT when Kn=0.03 and U= 500 m/s
2 Kn = 0.00 Kn = 0.01 Kn = 0.10
1.8
Dimensionless Frequency
1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0
0
0.2
0.4 0.6 0.8 Dimensionless flow velocity
1
1.2
Fig. 19 Effects of Knudsen number on the dimensionless frequency of simply supported single-walled nanotube
6 Change in Temp. = 0K Change in Temp. = 15K Change in Temp. = 30K
Dimensionless Frequency
5
4
3
2
1
0
0
0.2
0.4
0.6 0.8 1 1.2 1.4 Dimensionless flow velocity
1.6
1.8
2
Fig. 20 Effects of change in temperature on the natural frequency of the nonlinear vibration at low temperature 5
Change in Temp. = 0K Change in Temp. = 15K Change in Temp. = 30K
4.5
Dimensionless Frequency
4 3.5 3 2.5 2 1.5 1 0.5 0
0
0.5 1 Dimensionless flow velocity
1.5
Fig. 21 Effects of change in temperature on the natural frequency of the nonlinear vibration at high temperature
5.5 5
Winkler constant, k 1 = 0.00 MPa Winkler constant, k 1 = 0.05 MPa
4.5
Winkler constant, k 1 = 0.10 MPa
Dimensionless Frequency
4 3.5 3 2.5 2 1.5 1 0.5 0
0
0.2
0.4
0.6 0.8 1 1.2 1.4 Dimensionless flow velocity
1.6
1.8
2
Fig. 22 Effects of Winkler foundation parameter on the natural frequency of the nonlinear vibration at low temperature 5
Winkler constant, k 1 = 0.00 MPa
4.5
Winkler constant, k 1 = 0.05 MPa Winkler constant, k 1 = 0.10 MPa
Dimensionless Frequency
4 3.5 3 2.5 2 1.5 1 0.5 0
0
0.2
0.4
0.6 0.8 1 1.2 Dimensionless flow velocity
1.4
1.6
Fig. 23 Effects of Winkler foundation parameter on the natural frequency of the nonlinear vibration at high temperature 8 Pasternak constant, k p = 0.00 MPa Pasternak constant, k p = 0.05 MPa
7
Pasternak constant, k p = 0.10 MPa Dimensionless Frequency
6 5 4 3 2 1 0
0
0.2
0.4
0.6 0.8 1 1.2 1.4 Dimensionless flow velocity
1.6
1.8
2
Fig. 24 Effects of Pasternak foundation parameter on the natural frequency of the nonlinear vibration at low temperature
8
Pasternak constant, k = 0.00 MPa p
Pasternak constant, k = 0.05 MPa
7
p
Pasternak constant, k p = 0.10 MPa Dimensionless Frequency
6 5 4 3 2 1 0
0
0.2
0.4
0.6 0.8 1 1.2 Dimensionless flow velocity
1.4
1.6
1.8
Fig. 25 Effects of Pasternak foundation parameter on the natural frequency of the nonlinear vibration at high temperature
7 Winkler Foundation ony Pasternak foundation only Combined Winkler and Pasternak foundations
Dimensionless Frequency
6
5
4
3
2
1
0
0
0.2
0.4
0.6 0.8 1 1.2 1.4 Dimensionless flow velocity
1.6
1.8
2
Fig. 26 Effects of Winkler and Pasternak foundations parameters on the natural frequency of the nonlinear vibration at low temperature
3.5 eoa/L = 0.000 eoa/L = 0.075
Dimensionless Frequency
3
eoa/L = 0.100
2.5
2
1.5
1
0.5
0
0
0.5
1
1.5 2 2.5 Dimensionless flow velocity
3
3.5
4
Fig. 27 Effects of nonlocal parameter on the natural frequency of the nonlinear vibration in low or room temperature
7 Dimensionless Frequency (Imaginary, mode 2)
eoa/L = 0.000 eoa/L = 0.050
6
eoa/L = 0.100
5
4
3
2
1
0
0
0.5
1
1.5 2 2.5 Dimensionless flow velocity
3
3.5
4
Fig. 28 Effects of nonlocal parameter on the natural frequency of the nonlinear vibration for high temperature
6 (eoa/L)w1 = 0.000 (eoa/L)w1 = 0.075 5
(eoa/L)w1 = 0.100
Dimensionless Frequency
(eoa/L)w2 = 0.000 (eoa/L)w2 = 0.075
4
(eoa/L)w2 = 0.100
3
2
1
0
0
0.5
1
1.5 2 2.5 Dimensionless flow velocity
3
3.5
4
Dimensionless Frequency(Imaginary, mode 1 and mode 2)
Fig. 29 Effects of nonlocal parameter on the natural frequency of the nonlinear vibration for the first two modes of vibration for high temperature 10
(eoa/L)mode1 = 0.000 (eoa/L)mode1 = 0.050
8
(eoa/L)mode1 = 0.100 (eoa/L)mode2 = 0.000
6
(eoa/L)mode2 = 0.050 (eoa/L)mode2 = 0.100
4
2
0
-2
-4
0
0.5
1
1.5 2 2.5 Dimensionless flow velocity
3
3.5
4
Fig. 30 Effects of nonlocal parameter on the natural frequency of the nonlinear vibration for the first two modes of vibration for high temperature
8
Exact solution GDTM-AT solution
Deflection, w(x,t) (x10-3 nm)
6 4 2 0 -2 -4 -6 -8
0
2
4
6
8
10 12 Time (sec)
14
16
18
20
Fig. 31 Comparison between the obtained results and the exact solution for the linear vibration
8 Numerical solution GDTM-AT
Deflection, w(x,t) (x10-3 nm)
6 4 2 0 -2 -4 -6 -8
0
2
4
6
8
10 12 Time (sec)
14
16
18
20
Fig. 32 Comparison between the obtained results and the numerical solution for the nonlinear vibration
Experiment DTM-ATT
Dimensionless frequency
1
0.8
0.6
0.4
0.2
0
0
0.2
0.4 0.6 0.8 Dimensionless velocity
1
1.2
Fig. 33 Comparison of experimental results and the obtained results by the solution developed in this work
Nonlinear vibration equations of in fluid-conveying carbon nanotube are developed. Thermal-Fluidic parameter effects are considered The Galerkin- differential transformation method with CAT-aftertreatment technique is used. The developed solutions are used to carry out parametric studies on the nonlinear vibration