Free vibration of functionally graded SWNT reinforced aluminum alloy beam
Abdellatif Selmi^{1} , Awni Bisharat^{2}
^{1}Prince Sattam Bin Abdulaziz University, College of Engineering, Civil Engineering Department, B.P. 655, AlKharj 11942, Saudi Arabia
^{1}Ecole Nationale d’Ingénieurs de Tunis (ENIT), Civil Engineering Laboratory, B.P. 37, Le belvédère 1002, Tunis, Tunisia
^{2}Prince Sattam Bin Abdulaziz University, College of Engineering, Mechanical Engineering Department, B.P. 655, AlKharj 11942, Saudi Arabia
^{1}Corresponding author
Journal of Vibroengineering, Vol. 20, Issue 5, 2018, p. 21512164.
https://doi.org/10.21595/jve.2018.19445
Received 21 November 2017; received in revised form 16 March 2018; accepted 20 April 2018; published 15 August 2018
Aluminum alloy (Alalloy) reinforced with Single walled carbon nanotubes (SWNT), which represents an important industrial application, is studied. Different beam theories (BT) are applied to investigate functionally graded (FG) beams made of Alalloy reinforced with randomly oriented, straight and long SWNT. The RayleighRitz method is used to estimate the beam frequencies. First, the MoriTanak (MT) homogenization technique is used to predict the effective material properties of the beams. Second, results from BT are verified against finite element (FE) simulations. Next, a parametric study is carried out in order to investigate the influence of SWNT volume fractions, SWNT distributions and beam edgetothickness ratios on the vibration behavior of the FG beam. Results demonstrate the important effect of the studied parameters on the dynamic behavior of the FG SWNT reinforced Alalloy composite beams.
Keywords: aluminum alloy, SWNT, functionally graded beam, free vibration.
1. Introduction
SWNT are an important variety of carbon nanotubes. SWNT possesses exceptional mechanical, electrical and thermal properties. The application of the concept of FGM to SWNT composites has led to design different components satisfying particular properties [1]. FG SWNT reinforced composites are a new composite material having different applications in aerospace, defense, energy, automobile, medicine, structural and chemical industry [2]. The efficiency of SWNT as reinforcement can be attributed to the load transfer mechanism from matrix to SWNT at nanoscale. Interfacial bonding in the interphase region between embedded SWNT and its surrounding polymer is a key factor for the load transfer and reinforcement phenomena [3].
Alalloys are broadly used in diverse applications like aerospace, automotive as well as chemical industries due to its distinct properties as compared to other metals [4]. Because of the extraordinary physical and chemical properties of SWNT, reinforcement of Alalloys with this type of material leads to huge changes in its properties like greater strength, improved stiffness, reduced density, improved high temperature properties, improved abrasion and wear resistance [5, 6].
Most of the studies in SWNT reinforced composites are concentrated on the effect of the reinforcement volume fraction on the mechanical properties [7, 8] but research on vibration analysis of FG SWNT reinforced composite structures is limited to a few published articles. In the following, some of the recent published works on the dynamic characteristics of FG SWNT reinforced composite beams.
Heshmati and Yas [9] investigated the improvement of the fundamental natural frequency of FG SWNT reinforced polymer composite beam. The governing equations are found using the EulerBernoulli beam theory. The effect of SWNT agglomeration, distribution and boundary conditions on the dynamic behavior of the beam is found to be very important. The work carried out recently by Shenas et al. [10] presented the free vibration behavior of the pretwisted FG SWNT reinforced composite beams in thermal environment. The thirdorder shear deformation beam theory is used to obtain the governing equations. The ChebyshevRitz method is used to determine the free vibration eigenvalue equations. The FG beam is found to be sensitive to the pretwist angle and to the temperature. The free vibration of nanocomposite Timoshenko FG beams reinforced with SWNT resting on an elastic foundation is studied by Yas and Samadi [11]. The SWNT are assumed to be aligned and straight with a uniform layout. The rule of mixture is used to describe the effective material properties of the nanocomposite beams. The governing equations are derived through Hamilton's principle and then solved by using the generalized differential quadrature method. Effects of SWNT volume fraction, foundation stiffness parameters, slenderness ratios, SWNT distribution and boundary conditions on natural frequency are estimated. Ke et al. [12] investigated the nonlinear vibration of aligned straight SWNT FG beams based on Timoshenko beam theory and von Karman geometric nonlinearity. The effect of the vibration amplitude, volume fraction of SWNT, ratio of length to thickness, boundary conditions and SWNT distribution were taken into account to characterize the nonlinear vibration of the beams. In light of von Karman geometric nonlinearity assumptions and the firstorder shear deformation beam theory, Wu et al. [13] studied the nonlinear vibration of FG SWNT reinforced composite beams with initial imperfection. The Ritz method is applied to derive the nonlinear eigen frequency. Ansari et al. [14] dealt with the nonlinear vibration behavior of nanocomposite beams reinforced with SWNT based on the Timoshenko beam theory along with von Karman geometric nonlinearity. Poly methyl methacrylate (PMMA) is considered as the matrix. The vibration of cantilever FG SWNT beam subjected to compressive axial force is studied by Nejati et al. [15]. The twodimensional elasticity theory and Hamilton’s principle are used to determine the stability and motion equations. These equations are discretized using the generalized differential quadrature method. The influence of graded agglomerated SWNT, and the effect of axial forces exerted on the natural frequencies of FG beam are investigated. Yas and Heshmati [16] worked on dynamics of FG nanocomposite beams reinforced with randomly oriented SWNT. The dynamic characteristics of the FG beam are predicted using Timoshenko and EulerBernoulli beam theories. It is found that under the action of moving load, FG SWNT beam with symmetrical distribution gives superior properties than that of unsymmetrical distribution.
To the author’s knowledge, the free vibration of FG SWNT reinforced Alalloy composite beams, which represent an important industrial application, has not been investigated.
In a variety of dynamic problems, exact solutions may not be obtained, and one has to employ approximate methods. The RayleighRitz method is an approximate numerical method used extensively in several research sectors, but especially in the analysis of structural members [17]. The method can be used for both continuous and discrete systems. It is based on a linear expansion of the solution in terms of admissible functions [18]. In vibration problems, the frequencies are deduced from a quotient with potential energy being the numerator and kinetic energy function being the denominator. This quotient is called the Rayleigh quotient. The expansion coefficients are obtained using the principle of minimum potential energy [18].
Based on the classical beam theory (CBT), the Timoshenko beam theory (TBT) and the parabolic shear deformation beam theory (PSDBT), the present research focuses on the free vibration of FG SWNT reinforced Alalloy composite beams. The RayleighRitz method is used to determine the frequencies.
The objective is to study the effects of SWNT volume fraction, SWNT distribution pattern, beam slenderness ratio and the BT on the natural frequencies of FG SWNT reinforced Alalloy composite beams. To validate the present analysis, comparative studies are carried out with available results from the existing literature and with performed FE simulations.
The paper has the following outline. In Section 2, the homogenized properties of SWNT reinforced Alalloy composite are determined using the two level (MT, MT) scheme. Section 3 details the mathematical modeling on vibration of randomly oriented FG SWNT reinforced Alalloy composite beams based on the BT and the RayleighRitz method. Section 4 presents the numerical results of free vibration of FG SWNT reinforced Alalloy composite beams using the mentioned BT and FE simulations. The validation of obtained results and parametric study are discussed in the same Section 4. Lastly, Section 5 summarizes the results and conclusions.
2. Material properties of FG SWNT reinforced Alalloy composite beams
2.1. Mechanical properties of Alalloy, SWNT and SWNT/Alalloy composites
The Young’s modulus, the Poisson’s ratio and the density of Alalloy taken in this paper are respectively; $E=$ 70 GPa, $\nu =$ 0.33 and $\rho =$ 2700 Kg/m^{3} [19] while the Young’s modulus and the Poisson's ratio of the homogenized graphene sheet that constitute the SWNT are determined by the author using a homogenization method based on the energy equivalence and have been found to be 2520 GPa and 0.25, respectively [7]. The density of the armchair SWNT with chiral index of (10, 10) is 1330 Kg/m^{3}^{}[20].
A mean field homogenization scheme named twolevel (MT/MT) is selected to predict the mechanical properties of SWNT/Alalloy composites.
The twolevel procedure was proposed by Friebel et al. [21] for coated inclusionreinforced materials. The methodology is illustrated on Fig. 1. Each SWNT is seen (deep level) as a twophase composite (graphene sheet with cavities) which, once homogenized, plays the role of a homogeneous reinforcement for the matrix material (high level). In the first level of the twolevel procedure, the graphene matrix containing many small ellipsoidal cavities (having the same shapes and aspect ratios as the actual ones) is homogenized. The homogenization of the matrix material reinforced with homogeneous reinforcement is performed in the second level. In this paper, choosing MoriTanaka (MT) for both levels, the scheme is labeled “twolevel (MT/MT)”.
Fig. 1. Schematic view of the twolevel homogenization procedure for the effective properties of SWNT composites. For each level a twophase homogenization model is required
The rule of mixture is adopted to determine the density of the mentioned composite. The variations of Young’s modulus, Poisson’s ratio and density, as a function of reinforcement volume fraction, $V$, of 3D randomly oriented and long SWNT/Alalloy composites are reported in Table 1.
Interpretation: From Table 1, it is deduced that the reinforcing effect of SWNT is very important and as the SWNT volume fraction in the Alalloy increases, the Young’s modulus increases rapidly. For an Alalloy comprising 10 % of SWNT, the Young’s modulus, $E$, is increased by a factor of 1.4 but the Poisson’s ratio and the density decreases by a factor of 1.06 and 1.05, respectively.
Table 1. SWNT/Alalloy composite with 3D randomly oriented and long reinforcements
$V$ (%)

$E$ (GPa)

${\rm N}$

$\rho $ (Kg/m^{3})

0

70

0.33

2700

1

72.88

0.327

2686.3

3

78.60

0.323

2658.9

5

84.28

0.319

2631.5

10

98.33

0.311

2563

2.2. Properties of FG materials
A straight simply supported FG beam of length $L$, width $b$, and thickness $h$ is shown in Fig. 2.
In the present study, the volume fraction of SWNT is assumed to be graded in the thickness direction so that the material properties of the beam vary continuously according to powerlaw form as shown in Figs. 3, 4.
Fig. 2. Beam element with Cartesian coordinates
Fig. 3. Variation of Young’s modulus through the thickness direction of the beam
Fig. 4. Variation of mass density through the thickness direction of the beam
The considered powerlaw variation is:
${P}_{a}$ and ${P}_{c}$ stand for the values of the material properties of Alalloy and SWNT/Alalloy composite, respectively. Therefore, the bottom surface of the beam is Alalloy, whereas its top surface is SWNT reinforced Alalloy composite. Hereafter, for the top surface, 5 % and 10 % SWNT volume fractions are considered.
3. Numerical modeling and formulation
Let’s assume the deformation of FG beam in the $x$$z$ plane and designate the displacement components in the $x$ and $z$ directions by ${u}_{x}$ and ${u}_{z}$, respectively. Based on the BT, the axial and transverse displacement of any point of the beam are respectively:
where $u$ and $w$ denote the axial and the transverse displacement of any point on the neutral axis respectively, while $v$ represents the effect of transverse shear strain on the neutral axis. $\varphi \left(z\right)$ stands for the shape function and ($x$) indicates the partial derivative in terms of $x$. The present study is concerned with classical beam theory, CBT, Timoshenko beam theory, TBT, and parabolic shear deformation beam theory, PSDBT [22]. $\varphi \left(z\right)$ for these BT are given as below [23]:
$\mathrm{T}\mathrm{B}\mathrm{T}:\varphi \left(z\right)=z,$
$\mathrm{P}\mathrm{S}\mathrm{D}\mathrm{B}\mathrm{T}:\varphi \left(z\right)=z\left(1\frac{4{z}^{2}}{3{h}^{2}}\right).$
The strain energy, $S$, and the kinetic energy, $T$, of the beam at any moment are given as:
where $A$ and $\rho $ are the area of crosssection and the mass density of the beam, respectively. ${\sigma}_{xx}$, ${\tau}_{xz}$, ${\epsilon}_{xx}$ and ${\gamma}_{xz}$ denote the normal stress, the shear stress, the normal strain and the shear strain, respectively
Assuming the harmonic displacement components:
where $U\left(x\right)$, $V\left(x\right)$ and $W\left(x\right)$ are the respective amplitudes of the displacement components and $\mathrm{\omega}$ is the natural frequency.
The substitution of the expressions of displacement components into Eqs. (3), (4) yields the maximum strain energy, ${S}_{max}$, and the maximum kinetic energy, ${T}_{max}$ as:
where:
and:
The transformed stiffness constants are:
Using the RayleighRitz method, the amplitudes of vibration can be expanded in terms of polynomial functions by the following series as:
where ${c}_{i}$, ${d}_{j}$ and ${e}_{k}$ are constants to be determined and ${\phi}_{i}$, ${\psi}_{j}$ and ${\chi}_{k}$ are known functions that must satisfy the boundary conditions of the problem. These admissible functions can be written as:
For simply supported beams: $f={x}^{2}(L/2{)}^{2}$ [24].
Consequently, by equating ${S}_{max}$ and ${T}_{max}$, the Rayleigh Quotient ${\omega}^{2}$ can be deduced. The principle of minimum potential energy implies that the partial derivatives of the Rayleigh Quotient with respect to each of the constants ${c}_{i}$, ${d}_{j}$, and ${e}_{k}$ are nul. Accordingly, one can write:
Eq. (10) represents a set of 3$n$ algebric equations in 3$n$ unknowns ${c}_{1}$, ${c}_{n}$, ${d}_{1}$, ${d}_{n}$, and ${e}_{1}$, ${e}_{n}$. Its resolution requires considerable computation duration. Eq. (10) is then written in matricial form as:
where [$K$] and [$M$] are the dynamic stiffness and inertia matrices, respectively and [$\mathrm{\Delta}$] is the vector of unknown coefficients. The frequency parameters for free vibration problem are given by $\lambda $. Frequency parameters determined from this eigenvalue problem are investigated in the next section with the mentioned BT. Validation with the existing literature and with FE results are also reported.
4. Numerical results
In this section, the first three frequency parameters for the free vibration of simply supported FG beam are investigated using the abovementioned beam theories. The effect of the distribution fashion of the SWNT volume fraction throughout the beam thickness, the effect of SWNT content in the beam top surface and the beam slenderness ratio are analyzed. The frequency parameter is expressed as:
4.1. Convergence and validation of the analysis
4.1.1. Convergence analysis
In Tables 24, the convergence behavior of first three frequency parameters of simply supported FG beam with $L/h=$ 5 and 5 % SWNT volume fraction on the beam top surface are reported. The increase in the number n of polynomial items in the admissible functions is checked using CBT, TBT and PSDBT.
Table 2. Convergence of first three frequency parameters of simply supported FG SWNT/Alalloy beams using CBT for ($L/h=$ 5) and 5 % SWNT content on the beam top surface
CBT

Alalloy

$k=$ 5


$N$

${\lambda}_{1}$

${\lambda}_{2}$

${\lambda}_{3}$

${\lambda}_{1}$

${\lambda}_{2}$

${\lambda}_{3}$

2

3.2956

14.3779

16.7497

3.4988

14.7756

17.1114

3

2.9706

14.3779

16.6402

3.0621

14.774

17.0014

4

2.9706

11.3964

16.6401

3.0621

11.7342

16.9624

5

2.9697

11.3964

16.6401

3.0612

11.7342

16.9623

6

2.9697

11.3491

16.6401

3.0612

11.6859

16.9618

7

2.9697

11.3491

16.6401

3.0612

11.6859

16.9618

8

2.9697

11.3491

16.6401

3.0612

11.6857

16.9618

9

2.9697

11.3491

16.6401

3.0612

11.6857

16.9618

10

2.9697

11.3491

16.6401

3.0612

11.6857

16.9618

Table 3. Convergence of first three frequency parameters of simply supported FG SWNT/Alalloy beams using TBT for ($L/h=$ 5) and 5 % SWNT content on the beam top surface
TBT

Alalloy

$k=$ 5


$n$

${\lambda}_{1}$

${\lambda}_{2}$

${\lambda}_{3}$

${\lambda}_{1}$

${\lambda}_{2}$

${\lambda}_{3}$

2

3.2955

13.5286

16.7497

3.3975

14.0428

17.0927

3

2.8384

13.4888

16.6402

2.9449

14.0038

16.9816

4

2.8300

9.5017

16.6402

2.9364

10.0211

16.9529

5

2.8225

9.4253

16.5059

2.9304

9.9397

16.9525

6

2.8216

9.4035

16.3217

2.9295

9.9230

16.9524

7

2.8165

9.3492

16.2094

2.9019

9.6689

16.7125

8

2.8163

9.3433

16.2061

2.9016

9.6467

16.6947

9

2.8163

9.3431

16.2061

2.8999

9.6458

16.6947

10

2.8163

9.3428

16.2060

2.8998

9.6455

16.6948

It is observed that increasing the number of polynomial items, n, improves the accuracy of results and leads to convergent solutions at $n=$ 10. Hence $n=$ 10 is used in the following numerical calculations.
4.1.2. Validation of the analysis
The validation analysis is done through direct comparison with previously published results and with finite element results obtained by using commercial finite element software package ANSYS [25].
4.1.2.1. Comparison with available results
In order to check the above written formulation, the first three frequency parameters of simply supported FG beams are compared with [26] for slenderness ratios ($L/h=$ 20, 50 and 100) and powerlaw index ($k=$ 0, 0.1 and 0.2). The material properties of steel are $E=$ 210 GPa; $\rho =$ 2700 Kg/m^{3} and those of Alumina (Al_{2}O_{3}) are $E=$ 390 GPa; $\rho =$ 3960 Kg/m^{3}. Table 5 shows that the results given by the BT are very close to those available in [26].
Table 4. Convergence of first three frequency parameters of simply supported FG SWNT/Alalloy beams using PSDBT for ($L/h=$ 5) and 5 % SWNT content on the beam top surface
PSDBT

Alalloy

$k=$ 5


$n$

${\lambda}_{1}$

${\lambda}_{2}$

${\lambda}_{3}$

${\lambda}_{1}$

${\lambda}_{2}$

${\lambda}_{3}$

2

3.2955

13.5290

16.7497

3.3975

13.8882

17.09

3

2.8392

13.4896

16.5612

2.9222

13.8474

16.9789

4

2.8308

9.5187

16.4352

2.9136

9.7509

16.9519

5

2.8238

9.4449

16.3954

2.9059

9.6756

16.9328

6

2.8230

9.4275

16.3915

2.9051

9.6569

16.7576

7

2.8187

9.4247

16.3895

2.9011

9.6552

16.7579

8

2.8172

9.3961

16.3821

2.9008

9.6546

16.7575

9

2.8169

9.3957

16.3805

2.9005

9.6541

16.7574

10

2.8168

9.3955

16.3802

2.9005

9.6539

16.7574

Table 5. First three frequency parameters. A comparison with data after Amal et al. [26]
$L/hk$

$L/hk$

$L/hk$

$L/hk$


200

500

1000.1

1000.2


CBT

Ref

CBT

Ref

PSDBT

Ref

TBT

Ref

4.4019

4.3425

4.4038

4.3444

4.2526

4.2838

4.1372

4.2336

8.7903

8.6716

8.8054

8.6866

8.5009

8.5671

8.265

8.4666

13.152

12.975

13.2027

13.025

12.748

12.849

12.3952

12.699

4.1.2.2. Comparison with finite element resulats
As already mentioned, the material properties of the FG beam vary continuously throughout its thickness. The beam bottom surface is made of Alalloy, whereas the top one is made of SWNT/Alalloy composite. In order to model the FG beam, the numerical model has been divided into several layers so that the changes in properties can be made. Each layer has the finite portion of the thickness and treated like isotropic material. Material properties of each layer have been calculated at its midplane by using the chosen powerlaw distribution. To study the convergence analysis, various number of layers has been taken; 2, 4, 8 and 10. The FE modeling has been performed using ANSYS (2013). Higher order 3D, 10node elements (SOLID187) has been used for modeling of FG beams. SOLID187 has a quadratic displacement behavior. This element has three degrees of freedom at each node: translations in the nodal $x$, $y$, and $z$ directions. To simulate the pin support, the $x$, $y$ and $z$ bottom edge displacements are constrained while the roller support is free to move in the axial direction. 208797 elements with 292166 nodes are needed. Fig. 5 shows the fundamental mode shape for 10 layers of simply supported FG SWNT reinforced Alalloy composite beam. The materials used for modeling and analysis of beams are Alalloy and Alalloy comprising 10 % of SWNT. Their properties are given in the second and sixth lines of Table 1.
Fig. 5. Fundamental mode shape for 10 layers of FG beam
Table 6 reports the fundamental frequency parameters delivered by the FE analysis. Table 7 exposes the comparison between FE results and those obtained using the formulation explained in Section 3.
Interpretation: It can be interpreted that for different powerlaw distributions, the number of layers has a great influence on the fundamental frequencies of the modeled FG beam. From Table 6, one can consider that the convergence is reached for 10 layers. The FE results gotten for ten layers beams are confronted against those obtained using CBT, TBT and PSDBT. It is found that the present results are very close (see Table 7). The satisfactory results concerning the frequency parameters give confidence in the predictions reported in next sections.
Table 6. FE predictions of fundamental frequency parameters of simply supported FG SWNT/Alalloy beams with ($L/h=$ 30) and different powerlaw exponents ($k$)
$N$

$k=$ 0

$k=$ 0.4

$k=$ 1

$k=$ 2

$k=$ 5

$k=$ 10

Alalloy

2

3.4668

3.3045

3.1576

3.0405

2.9247

2.8680

2.8500

4

3.4668

3.2825

3.1509

3.0686

2.9839

2.9184

2.8500

8

3.4668

3.2727

3.1496

3.0781

3.0057

2.9499

2.8500

10

3.4668

3.2718

3.1496

3.0795

3.0084

2.9546

2.8500

Table 7. Comparison of fundamental frequency parameters of simply supported FG SWNT/Alalloy beams with FE results for different powerlaw exponents and slenderness ratio $L/h=$30
Theory

$k=$ 0

$k=$ 0.4

$k=$ 1

$k=$ 2

$k=$ 5

$k=$ 10

Alalloy

CBT

3.6698

3.4645

3.3477

3.2764

3.1982

3.1406

3.0168

TBT

3.6642

3.4593

3.3426

3.2713

3.1931

3.1355

2.9726

PSDBT

3.6642

3.4593

3.3426

3.2713

3.1930

3.1355

2.9726

FE

3.4668

3.2718

3.1496

3.0795

3.0084

2.9546

2.8500

4.2. Parametric study
A parametric study is carried out with CBT, TBT and PSDBT theories in order to predict the frequency parameters of simply supported FG SWNT reinforced Alalloy composite beams. In this parametric investigation, various values of SWNT volume fractions, powerlaw index and slenderness ratios of the beams are taken into consideration.
Tables 8 and 9 show the fundamental frequency parameters of FG SWNT/Alalloy composite beams for slenderness ratios $L/h=$ 5 and 30, respectively and for different powerlaw exponents ($k$). The SWNT volume fraction incorporated in Alalloy varies from 0 % on the bottom to 5 % on the beam top surface.
Tables 10 and 11 show the fundamental frequencies of FG SWNT/Alalloy composite beams for slenderness ratios $L/h=$ 5 and 30, respectively and for different powerlaw exponents ($k$). The SWNT volume fraction incorporated in Alalloy varies from 0 % on the bottom to 10 % on the beam top surface.
Table 8. Fundamental frequency parameters of simply supported FG SWNT/Alalloy beams with ($L/h=$ 5) for 5 % SWNT content on the beam top surface
Theory

$k=$ 0

$k=$ 0.4

$k=$ 1

$k=$ 2

$k=$ 5

$k=$ 10

Alalloy

CBT

3.3007

3.1954

3.1366

3.1009

3.0612

3.0319

2.9697

TBT

3.1305

3.0317

2.9748

2.9390

2.8998

2.8725

2.8163

PSDBT

3.1327

3.0344

2.9771

2.9403

2.9005

2.8736

2.8168

Table 9. Fundamental frequency parameters of simply supported FG SWNT/Alalloy beams with ($L/h=$ 30) for 5 % SWNT content on the beam top surface
Theory

$k=$ 0

$k=$ 0.4

$k=$ 1

$k=$ 2

$k=$ 5

$k=$ 10

Alalloy

CBT

3.353

3.2461

3.1864

3.1499

3.1096

3.0798

3.0168

TBT

3.3478

3.2412

3.1814

3.1450

3.1046

3.0121

2.9726

PSDBT

3.3478

3.2412

3.1814

3.1449

3.1046

3.0121

2.9726

From Tables 811, it may be noted that for different powerlaw exponents and various SWNT contents, the three BT predict almost the same fundamental frequency parameters for $L/h=$ 30 but for $L/h=$ 5, predictions given by CBT are comparatively greater than that given by both TBT and PSDBT. As can be expected, for thick beams, effect of shear deformation becomes more significant and affects the results greatly. Thus, TBT and PSDBT are more efficient for thicker beams.
Table 10. Fundamental frequency parameters of simply supported FG SWNT/Alalloy beams with ($L/h=$ 5) for 10 % SWNT content on the beam top surface
Theory

$k=$ 0

$k=$ 0.4

$k=$ 1

$k=$ 2

$k=$ 5

$k=$ 10

Alalloy

CBT

3.6125

3.4103

3.2955

3.2254

3.1486

3.0918

2.9697

TBT

3.4262

3.2366

3.1255

3.0555

2.9795

2.9263

2.8136

PSDBT

3.4289

3.2398

3.1279

3.0560

2.9784

2.9263

2.8168

Table 11. Fundamental frequency parameters of simply supported FG SWNT/Alalloy beams with ($L/h=$ 30) for 10 % SWNT content on the beam top surface
Theory

$k=$ 0

$k=$ 0.4

$k=$ 1

$k=$ 2

$k=$ 5

$k=$ 10

Alalloy

CBT

3.6698

3.4645

3.3477

3.2764

3.1982

3.1406

3.0168

TBT

3.6642

3.4593

3.3426

3.2713

3.1931

3.1355

2.9726

PSDBT

3.6642

3.4593

3.3426

3.2713

3.1930

3.1355

2.9726

It is seen that for different SWNT volume fractions, the fundamental frequency parameter is decreasing with increasing $k$. It is illustrated that the variation in the low values of power exponent is more effective on the fundamental frequency parameter than the variation in high values of power exponent. The change in frequency parameter with respect to slenderness ratio is detrimental. The variation in fundamental frequency parameter is relatively high for higher slenderness ratio.
It can be observed that for a constant power exponent and a constant slenderness ratio, an increase in SWNT content on the beam top surface causes the increase in fundamental frequencies. This augmentation becomes more important for high slenderness ratio.
It is clearly shown that reinforcing Alalloy with randomly oriented SWNT does not only improve the material properties of Alalloy but also increase the fundamental frequency parameters. For 0.4 powerlaw exponent, compared to simply supported Alalloy beam, the continuous variation of SWNT volume fraction to reach 5 % on the beam top surface increases the fundamental frequency parameter to 7.2. This improvement increases with both SWNT volume fraction and slenderness ratio.
Tables 12 and 13 present the first three frequency parameters of FG SWNT/Alalloy composite beams for slenderness ratios $L/h=$ 5 and 30, respectively and for different powerlaw exponents. The SWNT volume fraction incorporated in Alalloy varies from 0 % on the bottom to 5 % on the beam top surface.
Tables 14 and 15 present the first three frequency parameters of FG SWNT/Alalloy composite beams for slenderness ratios $L/h=$ 5 and 30, respectively and for different powerlaw exponents. The SWNT volume fraction incorporated in Alalloy varies from 0 % on the bottom to 10 % on the beam top surface.
Tables 16 and 17 summarize the first three frequency parameters of FG SWNT/Alalloy composite beams considering 0.4 powerlaw exponent and two slenderness ratios $L/h=$5 and 30. For the top surface, 5 % and 10 % SWNT volume fractions are taken into account.
Table 12. First three frequency parameters of simply supported FG SWNT/Alalloy beams with ($L/h=$ 5) for 5 % SWNT content on the beam top surface
Theory

$\lambda $

$k=$ 0

$k=$ 0.4

$k=$ 1

$k=$ 2

$k=$ 5

$k=$ 10

Alalloy

CBT

${\lambda}_{1}$

3.3007

3.1954

3.1366

3.1009

3.0612

3.0319

2.9697

${\lambda}_{2}$

12.614

12.1989

11.9612

11.824

11.6857

11.582

11.3491


${\lambda}_{3}$

18.4948

17.9791

17.5889

17.2794

16.9618

16.8153

16.6401


TBT

${\lambda}_{1}$

3.1305

3.0317

2.9748

2.939

2.8998

2.8725

2.8163

${\lambda}_{2}$

10.4479

10.1178

9.9131

9.7804

9.6455

9.5613

9.3428


${\lambda}_{3}$

18.2282

17.5708

17.202

16.9484

16.6948

16.5544

16.2060


PSDBT

${\lambda}_{1}$

3.1327

3.0344

2.9771

2.9403

2.9005

2.8736

2.8168

${\lambda}_{2}$

10.444

10.1489

9.3009

9.7963

9.6539

9.5747

9.3955


${\lambda}_{3}$

18.2597

17.7054

17.3196

17.0344

16.7574

16.6318

16.3802

Table 13. First three frequency parameters of simply supported FG SWNT/Alalloy beams with ($L/h=$ 30) for 5 % SWNT content on the beam top surface
Theory

$\lambda $

$k=$ 0

$k=$ 0.4

$k=$ 1

$k=$ 2

$k=$ 5

$k=$ 10

Alalloy

CBT

${\lambda}_{1}$

3.353

3.2461

3.1864

3.1499

3.1096

3.0798

3.0168

${\lambda}_{2}$

13.3939

12.9596

12.713

12.567

12.4132

12.2991

12.0507


${\lambda}_{3}$

30.0753

29.1018

28.5499

28.2223

27.8752

27.6179

27.0593


TBT

${\lambda}_{1}$

3.3478

3.2412

3.1814

3.145

3.1046

3.0121

2.9726

${\lambda}_{2}$

13.312

12.8811

12.6355

12.4894

12.3357

12.2224

11.8202


${\lambda}_{3}$

29.6658

28.7089

28.1621

27.8343

27.4875

27.2344

26.3414


PSDBT

${\lambda}_{1}$

3.3478

3.2412

3.1814

3.1449

3.1046

3.0121

2.9726

${\lambda}_{2}$

13.312

12.8814

12.6356

12.489

12.3348

11.9771

11.8202


${\lambda}_{3}$

29.666

28.7103

28.1622

27.8321

27.4835

26.6912

26.3414

Table 14. First three frequency parameters of simply supported FG SWNT/Alalloy beams with ($L/h=$ 5) for 10 % SWNT content on the beam top surface
Theory

$\lambda $

$k=$ 0

$k=$ 0.4

$k=$ 1

$k=$ 2

$k=$ 5

$k=$ 10

Alalloy

CBT

${\lambda}_{1}$

3.6125

3.4103

3.2955

3.2254

3.1486

3.0918

2.9697

${\lambda}_{2}$

13.8058

12.9904

12.505

12.231

11.9811

11.7951

11.3491


${\lambda}_{3}$

20.2422

19.2589

18.5128

17.9138

17.2861

16.9914

16.6401


TBT

${\lambda}_{1}$

3.4262

3.2366

3.1255

3.0555

2.9795

2.9263

2.8136

${\lambda}_{2}$

11.435

10.7922

10.3847

10.1237

9.8692

9.7117

9.3428


${\lambda}_{3}$

19.8507

18.7512

18.0276

17.5329

17.0461

16.7779

16.206


PSDBT

${\lambda}_{1}$

3.4289

3.2398

3.1279

3.056

2.9784

2.9263

2.8168

${\lambda}_{2}$

11.4648

10.8288

10.4117

10.1311

9.8605

9.713

9.3955


${\lambda}_{3}$

19.9862

18.9062

18.1507

17.5943

17.0578

19.8185

16.3802

Table 15. First three frequency parameters of simply supported FG SWNT/Alalloy beams with ($L/h=$ 30) for 10 % SWNT content on the beam top surface
Theory

$\lambda $

$k=$ 0

$k=$ 0.4

$k=$ 1

$k=$ 2

$k=$ 5

$k=$ 10

Alalloy

CBT

${\lambda}_{1}$

3.6698

3.4645

3.3477

3.2764

3.1982

3.1406

3.0168

${\lambda}_{2}$

14.6593

13.8149

13.3206

13.0321

12.7448

12.5322

12.0507


${\lambda}_{3}$

32.9168

31.0264

29.9232

29.2765

28.6256

28.1438

27.0593


TBT

${\lambda}_{1}$

3.6642

3.4593

3.3426

3.2713

3.1931

3.1355

2.9726

${\lambda}_{2}$

14.5697

13.6704

13.24

12.9515

12.664

12.4527

11.8202


${\lambda}_{3}$

32.4687

30.6109

29.5193

28.8724

28.2207

27.7461

26.3414


PSDBT

${\lambda}_{1}$

3.6642

3.4593

3.3426

3.2713

3.193

3.1355

2.9726

${\lambda}_{2}$

14.5698

13.7324

13.2401

12.9505

12.6621

12.4515

11.8202


${\lambda}_{3}$

32.469

30.6134

29.5196

28.8679

28.2121

27.7401

26.3414

Table 16. First three frequency parameters of simply supported FG SWNT/Alalloy beams with $k=$ 0.4 for 5 % SWNT content on the beam top surface
Theory

$L/h$

${\lambda}_{1}$

${\lambda}_{2}$

${\lambda}_{3}$

CBT

5

3.4103

12.9904

19.2589

30

3.4645

13.8149

31.0264


TBT

5

3.2366

10.7922

18.7512

30

3.4593

13.6704

30.6109


PSDBT

5

3.2398

10.8288

18.9062

30

3.4593

13.7324

30.6134

Table 17. First three frequency parameters of simply supported FG SWNT/Alalloy beams with $k=$ 0.4 for 10 % SWNT content on the beam top surface
Theory

$L/h$

${\lambda}_{1}$

${\lambda}_{2}$

${\lambda}_{3}$

CBT

5

3.1096

12.4132

27.8752

30

3.1096

12.4132

27.8752


TBT

5

2.8998

9.6455

16.6948

30

3.1046

12.3357

27.4875


PSDBT

5

2.9005

9.6539

16.7574

30

3.1046

12.3348

27.4835

Same remarks and conclusions, deduced from Tables 811, concerning the effect of different variables (SWNT distribution, SWNT volume fraction and slenderness ratio) on the fundamental frequency parameters can be noted from Tables 1217 for the three first frequency parameters. The frequency parameters are increasing with increase in slenderness ratios ($L/h$) and are decreasing with increase in powerlaw exponents ($k$). It is also seen that for $L/h=$ 5, the results for FG beam using CBT are comparatively greater than those found using other BT, where as for $L/h=$ 30, one may experience mere coincidence of frequency parameters. This due to the deficiency in Euler beam theory for consideration the shear effect, which affects significant on the frequencies especially for the short beam. Moreover, it is very important to see that the variable parameters have more influence on the second and third frequency parameters than that on the fundamental frequency parameter. For 0.4 powerlaw exponent, compared to simply supported Alalloy beam, the continuous variation of the SWNT volume fraction to reach 10 % on the beam top surface increases the frequency parameters to at least 13.5. This improvement increases with both the SWNT volume fraction and the slenderness ratio.
5. Conclusions
The free vibration of simply supported FG SWNT/Alalloy beam was investigated based on various BT using the RayleighRitz method. This helps fill in a gap in the literature where the available mechanical results about SWNT reinforced Alalloy composites did not concern the free vibration. The RayleighRitz method is found to be efficient when compared to FE simulations. The objective of improving the dynamic characteristics of Alalloy in order to prevent problems such as failures associated with resonance and fatigue is achieved by reinforcing with functionally graded SWNT. Reinforcement with SWNT allows a significant increase of the Alalloy dynamic properties without compromising other factors such as mass of the structure. The improvement of the natural frequencies of beams made of ALalloy was attained by increasing the amount of the reinforcement, increasing the slenderness ratio and decreasing the powerlaw exponent.
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