Abstract
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.
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. 1Schematic 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 1SWNT/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. 2Beam element with Cartesian coordinates
Fig. 3Variation of Young’s modulus through the thickness direction of the beam
Fig. 4Variation 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 2Convergence 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 3Convergence 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 4Convergence 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 5First 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. 5Fundamental 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 6FE 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 7Comparison 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 8Fundamental 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 9Fundamental 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 10Fundamental 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 11Fundamental 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 12First 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 13First 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 14First 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 15First 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 16First 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 17First 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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