Torabi, K., Rahi, M., Afshari, H. (2015). Transverse Vibration for Nonuniform Timoshenko Nanobeams. Mechanics of Advanced Composite Structures, 2(1), 116. doi: 10.22075/macs.2015.327
Torabi, K., Rahi, M., Afshari, H. (2015). 'Transverse Vibration for Nonuniform Timoshenko Nanobeams', Mechanics of Advanced Composite Structures, 2(1), pp. 116. doi: 10.22075/macs.2015.327
Torabi, K., Rahi, M., Afshari, H. Transverse Vibration for Nonuniform Timoshenko Nanobeams. Mechanics of Advanced Composite Structures, 2015; 2(1): 116. doi: 10.22075/macs.2015.327
Transverse Vibration for Nonuniform Timoshenko Nanobeams
^{}Department of Solid Mechanics, Faculty of Mechanical Engineering, University of Kashan, Kashan, Iran
Abstract
In this paper, Eringen’s nonlocal elasticity and Timoshenko beam theories are implemented to analyze the bending vibration for nonuniform nanobeams. The governing equations and the boundary conditions are derived using Hamilton’s principle. A Generalized Differential Quadrature Method (GDQM) is utilized for solving the governing equations of nonuniform Timoshenko nanobeam for pinnedpinned, clamped–clamped, clamped–pinned, clamped–free, clamped–slide, and pinnedslide boundary conditions. The nondimensional natural frequencies and the normalized mode shapes are obtained for short and stubby nanobeams where influences varying crosssection area, small scale, shear deformation, rotational moment of inertia, acceleration gravity and the selfweight of the nonuniform Timoshenko nanobeam are discussed. The present study illustrates that the small scale effects are more significant for smaller size of nanobeam, larger nonlocal parameter and higher vibration modes. Further, the compression forces due to gravity and the selfweight of the nanobeam also like the small scale effect are reduced the magnitude of the frequencies of the nanobeam.
Transverse Vibration for Nonuniform Timoshenko Nanobeams
K. Torabi^{*}, M. Rahi, H. Afshari
Department of Solid Mechanics, Faculty of Mechanical Engineering, University of Kashan, Kashan, Iran
Paper INFO
ABSTRACT
Paper history:
Received 30 October 2014
Received in revised form 21 October 2015
Accepted 27 October 2015
In this paper, Eringen’s nonlocal elasticity and Timoshenko beam theories are implemented to analyze the bending vibration for nonuniform nanobeams. The governing equations and the boundary conditions are derived using Hamilton’s principle. A Generalized Differential Quadrature Method (GDQM) is utilized for solving the governing equations of nonuniform Timoshenko nanobeam for pinnedpinned, clamped–clamped, clamped–pinned, clamped–free, clamped–slide, and pinnedslide boundary conditions. The nondimensional natural frequencies and the normalized mode shapes are obtained for short and stubby nanobeams where influences varying crosssection area, small scale, shear deformation, rotational moment of inertia, acceleration gravity and the selfweight of the nonuniform Timoshenko nanobeam are discussed. The present study illustrates that the small scale effects are more significant for smaller size of nanobeam, larger nonlocal parameter and higher vibration modes. Further, the compression forces due to gravity and the selfweight of the nanobeam also like the small scale effect are reduced the magnitude of the frequencies of the nanobeam.
The progress of nanotechnology has enthused scientists in their search of producing all sorts of micro/nanostructures such as atomic force microscope cantilever tips, nanowires, nanoactuators, nanoprobes and nanobeams. Whenever the Euler–Bernoulli beam theory and the Timoshenko beam theory are applied to the analysis of the small beamlike structures, the researchers are found to be inadequate. Being scale free, these classical beam theories could not capture the small scale effect in the mechanical properties. For example, Wang and Hu [1] showed that the classical beam theories are not able to predict the decrease in phase velocities of wave propagation in carbon nanotubes when the wave number is so large that the nanostructure has a significant influence on the flexural wave dispersion.
In 1972, Eringen [2], Eringen and Edelen [3] and later Eringen [4,5] initiated the nonlocal continuum mechanics to allow for the small scale effect by specifying the stress state at a given point to be a function of the strain states at all points in the body. Then the nonlocal theory of elasticity has been used to study lattice dispersion of elastic waves, wave propagation in composites, dislocation mechanics, fracture mechanics, surface tension fluids, etc. [610] , and Wang et al. [11] used the nonlocal elasticity constitutive equations to investigate the vibration of carbon nanotubes. Wang et al. [11] neglected the nonlocal effect in writing the shear stress–strain relation of the Timoshenko beam theory (TBT), and therefore the effect of including nonlocal constitutive behavior amounted to using an equivalent shear correction factor.
Reddy [12] used various available beam theories, including the Timoshenko beam theory, are reformulated using the nonlocal differential constitutive relations of Eringen. Wang et al. [13] analyzed the vibration of nonlocal Timoshenko beams.
This paper presents a formulation of nonlocal elasticity theory for the transverse vibration analysis of nonuniform Timoshenko nanobeams (NUTNB). The differential equations of motion are employed to derive from the variational procedure (Hamilton’s principle) and the solutions numerically calculated using GDQM for pinnedpinned, clampedclamped, clampedpinned, clampedfree, clampedslide and pinnedslide boundary conditions. This study makes the first attempt to study the axially distributed forces such as an integrationform varying compression force due to gravity acceleration on NUTNB. In addition the small scale effects, Timoshenko parameters and axial loading are investigated for nonuniform (tapered) nanobeam. The main purpose of this article is to investigate the vibrational response of NUTNB with axial loading for arbitrary boundary conditions and for various values of the small scale effects. This study is organized as follows. Firstly, in Section 2, the nonlocal elasticity theory of Eringen is presented using the nonlocal constitutive differential equations. Secondly, in Section 3, the governing differential equations describing the transverse vibrations will be derived for NUTNB. Also, the present paper is concerned with the gravity effect on the vibration behavior of nanobeams. Then, in Section 4, the solution procedure and dimensionless equations will be derived from some of the various boundary conditions. In Section 5 the numerical solution of the problem will be solved approximately using GDQM and the matrix form will be transformed for equations of motion. Also in Section 6, the numerical results obtained from the present analysis are discussed in many cases of nanobeam. Finally, in Section 7 conclusions will be drawn and remarks the results of presenting work.
Nonlocal Elasticity Theory of Eringen
The classical elasticity theory presents the constitutive equation in the form of an algebraic relationship between the stress and strain tensors while that of the nonlocal elasticity theory according to Eringen [25], is that the stress field at a reference point x in an elastic continuum depends not only on the strain at that point but likewise on the strains at all other points in the body. Eringen’s nonlocal elasticity involves spatial integrals which represents weighted averages of the contributions of strain tensors of all points in the body to the stress tensor at the given point. Eringen attributed this fact to the atomic theory of lattice dynamics and experimental observations on phonon dispersion [12]. The scale effects are accounted for in the theory by considering the internal size as a material parameter [14].
2.1. The Constitutive Relations
The basic equations for linear, homogeneous, isotropic, nonlocal elastic solid with zero body force are given by [15],
(1)
where σ is the nonlocal stress tensor, t(x´) is the classical, the macroscopic stress tensor at point x and the kernel function K(x´– x, τ) are the nonlocal moduli, or attenuation function incorporating into constitutive equations the nonlocal effects at the reference point x are produced by local strain at the source x´, τ is the material constant which is defined as τ=e_{0}a/l where e_{0} is a constant appropriate to each material, a is an internal characteristics length (e.g., lattice parameter, granular distance) and l is an external characteristics length (e.g., crack length, wavelength).
The constitutive Eq. (1) defines the nonlocal constitutive behavior of a Hookean solid and represents the weighted average of the contributions of the strain field of all points in the body to the stress field at a point. Though the integral constitutive relation in Eq. (1) makes mathermatical difficulties to obtain the solution of nonlocal elasticity problems, Eringen [4] represents this integral constitutive equation to equivalent differential constitutive equations under certain conditions. For an elastic material in the one dimensional case, the nonlocal constitutive relations may be simplified as [4,13].
2.2. Stress Resultant
The nonlocal constitutive can be approximated to a onedimensional form, in terms of the strains in the Timoshenko beam theory (TBT) [16],
(2)
(3)
where E and G are the Young’s and shear moduli, respectively, and γ is the shear strain. Hence, the nonlocal parameter μ=(e_{0}a)^{2}, a in the theory will be led to smallscale effect on the response of structures of nanosize and when μ is zero, the constitutive relations will be derived of the local theories.
Nonlocal Equations of NUTNB
3.1. The Governing Equations of Motion
The Timoshenko beam theory (TBT), which is based on the displacement field at some points, can be found in [17,18].
All applied loads and geometry are such that the displacements (u_{1},u_{2},u_{3}) along the coordinates (x,y,z) are only functions of the x and z coordinates and time t. The displacement u_{2} is supposed identically zero. The terms u and w are the axial and transverse displacements, respectively, of the point (x,0) on the midplane (i.e., z = 0) of the beam and the ϕ denotes the rotation of the crosssection. The nonzero strains according to TBT are expressed as,
(4)
The nonlocal bending moment M, and the shear force Q, can be written in the following form,
(5)
(6)
where σ_{xx} is the normal stress and σ_{xz} the transverse shear stress. A is the crosssectional area of the beam. The following relations are introduced for using in the coming sections,
(7)
Therefore, the xaxis is taken along the geometric centric of the beam, where I is the second moment of area of the crosssection.
In this stage, by multiplying both sides of Eq. (2) by z and integrating over the crosssection area of the beam, then using Eqs. (4) to (7) , the nonlocal Timoshenko constitutive relations yields,
(8)
Also, by integrating Eq. (3) over the area, and using Eqs. (4) to (7), one obtains,
(9)
where K_{s} denotes the shear correction factor of TBT in order to compensate for the error in assuming a constant shear strain (stress) through the thickness of the beam. The shear correction factors depend not only on the material and geometric parameters but also on the load and boundary conditions.
The shear correction factor is 9/10 for a circular shape crosssection and 5/6 for a rectangular crosssection [11]. A value of 0.877 was used by Reddy and Pang [19] for the analysis of carbon nanotubes (CNTs) with the relation K_{s} = (5+5ν)/(6+5ν) for the rectangle and K_{s} = (6+12ν+6ν^{2})/(7+12ν+4ν^{2}) for the circle in which ν is Poisson’s ratio with value of ν = 0.3 [20]. The strain energy U and the kinetic energy T of the beam are obtained, respectively, by the following equations [21],
(10)
where ρ is the mass density of the nanobeam material.
By substituting Eqs. (4) to (7), into the above energy statements, and neglecting axial displacement of the neural web u(x,t), the kinetic and strain energies with respect to the displacement field may be expressed as,
(11)
In addition, the work done by the external axial force is denoted by,
(12)
where P is the distributed axial load along x axis. The Hamilton’s principle is the most powerful variational principle of mechanics, hence the principle of virtual displacements for the TBT is given by,
(13)
Therefore, by substituting Eqs. (11) and (12) into Eq. (13),then, integrating by parts and since δw and δϕ are arbitrary in the domain of nanobeam and then setting the coefficients of δw and δϕ to zero lead to the Euler–Lagrange equations of motion in 0 < x < L as [22],
(14)
The corresponding boundary conditions involve specifying one element of each of the following three pairs at the end of x = 0 and x = L,
(15)
where V denotes the equivalent shear force. By substituting Eq. (14) into the implicit Eqs. (8) and (9), the explicit expressions of the nonlocal bending moment M and shear force Q can be obtained as,
(16)
Then, the Euler–Lagrange equations of motion for the nonlocal NUTNB can be derived by inserting Eq. (16) into Eq. (14),
(17)
3.2. The Gravity Acceleration
One kind of the axial force is the influence of gravity. Now in the present study, the transverse vibration characteristics of NUTNB will be analyzed in two cases, with or without axial force. In case with axial force, there are also two cases, one is a NUTNB subjected to the constant axial force and the other is of a cantilever NUTNB under axially distributed gravity force.
Due to the gravity and the selfweight of the nanobeam an integrationform varying compression or tensile force is acting on the nanobeam. The force of nonuniform cantilever nanobeam (ClampedFree) due to acceleration gravity given by,
(18)
(19)
where compressive force P_{S} is used for a standing beam and tensile force P_{H} is used for a hanging one.
The Solution Procedure and Dimensionless Equations
A general solution is assumed in the form,
(20)
where W and Ф are the amplitude of the generalized displacements and rotation of beam, respectively, ω the vibration frequency of NUTNB and i^{2}=1. For convenience and simplification, the governing equations can be expressed in the nondimensional form by introducing the following dimensionless parameters [13,14,23],
(21)
where λ is the nondimensional natural frequency, ζ is the slenderness ratio, α is the scaling effect parameter of the nanobeam, Ω is the shear deformation parameter, is the nondimensional bending moment, and V are the nondimensional axial and shear forces, respectively, ε is the gravity parameter, and are the nondimensional coefficient and function of the axial force due to gravity, respectively. It should be noted that in the case with gravity, the following statements are used,
(22)
By substituting Eqs. (20) and (21), into governing equations and corresponding boundary conditions, the nonlocal governing equations for NUTNB in dimensionless form yields,
(23)
The natural boundary conditions are as,
(24)
The geometry boundary conditions are as,
(25)
Note that the nonlocal governing equations given in Eqs. (23) to (25), reduce to that of the counterpart local Timoshenko model when the nonlocal parameter or small scale effect is set to zero (α = 0). Finally, the corresponding dimensionless boundary conditions with nanobeam are listed in Table 1.
Generalized Differential Quadrature Method (GDQM)
Gauss quadrature is a numerical integration method. Its basic idea is to approximate a definite integral with a weighted sum of integrand values of a group of nodes in the form of,
(26)
where x_{j} are nodes and w_{j} are weighting coefficients.
In the differential quadrature method (DQM) the weighting coefficients are determined by solving a system of linear equations. Extending Gauss quadrature to find the derivatives of various orders of a differentiable function gives a rise to the differential quadrature [24,25].
In other words, the derivatives of a function are approximated by weighted sums of the function values in a group of nodes [26,27].
The two major disadvantages of DQM are: the first limits, the small number of grid points and requires solving sets of linear equations, the second limits the distribution of the grid points which is critical in structural dynamic analysis. In GDQM the weighting coefficients for derivative approximations are given by a simple algebraic expression and a recurrence relationship, together with arbitrary choices of grid points [2730]. Consider the discretization of mth order derivative of w(x), the following DQ approximation is assumed as,
(27)
where C_{mij} are weighting coefficients for mth derivative and w(x_{j}) are function values at grid points x_{j} (i = 1,2,…,N). Therefore, explicit formulas for these coefficients are found to be [26,29],
(28)
By choosing the Lagrange interpolated polynomial M(x) as the set of test functions, yields,
(29)
Higher order coefficient matrices at each grid point can be obtained in GDQM from the first order weighting matrix as follows, [31, 32],
(30)
To choose the distribution of the grid points, among nonuniform spacing of nodes which ensure the convergence, the Chebyshev nodes defined by the following equation are nearly optimal [26, 27].
Table 1. The dimensionless boundary conditions of NUTNB
Type
Boundary conditions (BCs)
Geometry
Natural
Pinned (P)


Clamped (C)


Free (F)


Slide (S)


(31)
By the expansion of Eqs. (23) and (24), and then rewriting the resultant equations by generalizing differential quadrature (GDQ) model, the governing equations were obtained for the nonlocal NUTNB.
(32)
Natural boundary conditions may be rewritten in the following statements in the form of GDQM,
(33)
where the displacement vector and the rotation vector are expressed as,
(34)
Eqs. (32) to (34) can be easily transformed into an eigenvalue problem to obtain the nondimensional natural frequency. For NUTNB model,
(35)
In simple form, yields
(36)
And K, M are the stiffness and mass matrices, respectively. Also the nonlocal boundary conditions are expressed as,
(37)
The present problem has been solved for some of the Boundary Conditions (BCs.) as: PinnedPinned (PP), ClampedClamped (CC), ClampedPinned (CP), ClampedFree (CF), ClampedSlide (CS) and PinnedSlide (PS). The BCs. have been implemented simply using GDQM technique in the following matrix form. To illustrate the technique, Consider the NUTNB pinned at the two ends,
(38)
Eq. (37) is converted to the following form for imposition the natural BCs.
(39)
Therefore, Eq. (36) is transformed to,
(40)
(41)
Finally, for imposition the geometry BCs., one obtains,
(42)
(43)
where the matrices in Eq. (42) have been extracted by omitting the first and Nth the row and column, respectively. Hence, using this technique Eq. (42) can easily be reduced to an eigenvalue problem as,
(44)
(45)
Eq. (44) can be solved by a standard eigenvalue solver, and then the nondimensional nonlocal natural frequencies of the NUTNB are obtained.
Numerical Results and Discussions
6.1. The Technique Verification
Firstly, the bending vibration of a uniform clamped–free Timoshenko beam with the rotary inertia parameter r^{2 }= 0.01 and 0.001; and the shear deformation parameter s^{2 }= 2.8 and 0.5 [23] is considered by following statements,
(46)
The first ten nondimensional natural frequencies are shown in Table 2. The result in this Table, yields highly agreement with the result of the exact solution of Hijmissen and Horssen [23]. The difference between the two sets of results is very small and is well within 0.0036%. Secondly, the natural frequencies of a standing Timoshenko beam with clampedfree BCs. subjected to the gravity acceleration are listed in Table 3. The first ten dimensional frequencies ω (Hz) have been compared with Hijmissen and Horssen [23]. The excellent accuracy is detected and the differences between the results are very small and are within 0.0384%, then the proposed technique is very close to the exact solutions.
Table 2. The nondimensional natural frequency λ^{2} for uniform CF local beam
Thirdly, the first five dimensionless natural frequencies of a Timoshenko beam are displayed for PP, CP, CC and CF boundary conditions are presented for various small scale effects α and L/d = 10, in Table 4. Where the other parameters of beam according to [13] are: diameter d = 0.678 nm, lengths L = 10d, and the following assumed mechanical parameters: Young’s modulus E = 5.5 TPa, Poisson’s ratio ν = 0.19, effective tube thickness t = 0.066 nm, shear correction factor K_{s} = 0.563. In Table 4, the frequencies have been compared by Wang et al. [13]. Note that the results associated with α = 0 correspond to the local TBT frequencies. The excellent accuracy and agreement are achieved between the results of the present method and the exact method [13] for local frequencies and also there is a rate of convergence for an adequate number of grid points. Although, for nonlocal frequencies, the result of the proposed technique is very close to the exact solution [13], but there is a critical difference between the present and exact procedures. Wang et al [13] used the constitutive relation to the shear stress and strain as the same as in the local beam theory, however, in the present method according to Ref. [4], the constitutive relation for the shear stress and strain are considered as the nonlocal beam theory. Hence the present method is more complete and more accurate than Ref. [13].
Finally, the first four frequency parameters of the present method and those obtained by Farchaly and Shebl [33] and Chen [34] are shown in Table 5. The rotary inertia parameter r^{2}=I/AL^{2}=0.01 and shear deformation parameter s^{2}=Er^{2}/Gk=0.03 are considered, and also three values of compressive axial load parameter PL^{2}/EI, 0, 1 and 10, are considered. As it can be observed from the Table, the present results yield excellent agreement with those of [33,34]. The difference between the two sets of results is small and well within 0.0280%. As the compressive axial load is increased, natural frequencies of all modes are reduced.
6.2. The Numerical Examples
In this section, the bending vibration characteristics of the NUTNB will be examined. The effect of the varying crosssection area, small scale, shear deformation, rotational moment of inertia, acceleration gravity and the selfweight of the nanobeam on the nondimensional natural frequencies and normalized mode shapes is demonstrated. The following parameters, material and geometric properties used in computing the numerical values are E = 30 MPa [12], ρ = 2300 kg/m^{3}, L = 10 nm, h = varied, ν = 0.3, K_{s} = 5/6 and g = 9.81 m/s^{2}.
The problem can be solved for each arbitrary A(x). In this paper, we assume the A(x) as,
(47)
To study the effect of nonlocal parameter, frequency ratio is defined as,
(48)
6.2.1. The Convergence of N in GDQM
Minimum number of grid points is performed by the convergence test in GDQM [14] using the following equation required to obtain stable and accurate results. The error percent vs. number of grid points for the fundamental frequency (μ = 0.1) for a PinnedPinned beam is shown in Fig. 1. It can be noted form Fig. 1, that eleven number grid points are sufficient in resulting converged solution.
(49)
Figure 1. Theconvergence of nonlocal results by GDQM
Table 4. First five natural frequencies of uniform beam with L/d = 10
The nondimensional local and nonlocal frequencies are computed by using the present GDQM for the NUTNB with L/h=100 and various values of small scale for six boundary conditions and the results are listed in Table 6. The effect of the nonlocal parameter α is to reduce the natural frequencies, as it can be seen from the results, which are presented in Table 6.
Hence, as the small scale coefficient increases, the frequencies obtained for the nonlocal beam become smaller the local counterpart, and so the frequency ratio is always smaller than unity. This reduction is especially significant for the higher vibration modes. Based on the results in Table 6, it is found that for constant nonlocal parameter α, among all of the boundary conditions, clampedclamped contain largest frequency and pinnedslide contain smallest frequency. From Table 6, it is observed that fundamental frequency for clampedfree approximately remains unchanged with the increase in the nonlocal parameters. Table 6 implies that the frequencies for nonuniform (tapered) nanobeam are smaller than the same frequencies of uniform nanobeam. The above result can be demonstrated in Figs. 2 to 4.
To demonstrate the effect of higher modes, frequency ratio for various nonlocal parameters is plotted in Figs. 3 and 4, for nonuniform and uniform crosssection, respectively. From these figures, it can be obviously observed that as the nonlocal parameter value increases, the frequency ratio decreases. In addition, at higher mode numbers, all results converge to the local frequency at the higher lengths. This phenomenon is because of the increase in the interactions between atoms at small wavelengths at higher wave numbers [1].
Table 5. Thecomparison of first four dimensionless natural frequencies for a clamped–free uniform beam
In Table 7, the nondimensional first five frequency ratioes of a standing cantilever nanobeam are listed for uniform & nonuniform crosssection with A(x) = 1 0.05x^{2} and without & under gravity acceleration with gravity parameter ε = 0.9025 and L/h = 100.
As Table 7 shows quite clearly, in the case under gravity acceleration, the nondimensional frequencies are less than the counterpart frequencies in the case without gravity acceleration. Hence the frequencies decrease due to the selfweighting for a standing cantilever beam. Based on the results in this Table, the fundamental frequencies at uniform crosssection are less than the same frequencies at a nonuniform crosssection for both cases under & without gravity (see Fig. 5).
It can be seen from Fig. 6 that the frequency ratio for uniform beam decreases more with increasing nonlocal parameters with respect to the nonuniform beam, and also the frequency ratio for uniform and nonuniform crosssection trends together with the increase of the mode number in standing cantilever nanobeam under gravity.
6.2.4. The Effect of Length
Figs. 7 and 8 show the variation of nondimensional fundamental frequency ratio with the length of the beam for pinnedpinned and clampedclamped boundary conditions with nonuniform crosssection A(x) = 1 0.05x^{2}.
From Fig. 7(a), it can be comprehended that the space between the curves gradually decreases with the increase in the length of nanobeam. Further, for all curves, frequency ratio converges to one by increasing the length. It is apparent due to the fact that with the increase of length the effect of the nonlocal parameter decreases and therefore the frequency ratio tends to unity. As expected, this convergence is more rapid for the small values of nonlocal parameters. These results are similarly found from Fig. 7(b) for clampedclamped boundary condition.
Fig. 8 shows the smallscale effect on the dimensionless fundamental natural frequency of NUTNB with lengths of L = 10, 15 and 20 nm [35]. As it is shown in Fig. 8, the smallscale effects decrease with an increase in the length of a nanobeam.
Table 6. First five nondimensional frequencies of NUTNB with L/h = 100
BCs.
Mode
number
α
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
Pinned
Pinned
1
3.1300
3.1292
3.1267
3.1227
3.1171
3.1097
3.1010
3.0908
3.0793
2
6.2556
6.2495
6.2312
6.2014
6.1608
6.1105
6.0517
5.9857
5.9138
3
9.3787
9.3578
9.2970
9.2017
9.0736
8.9197
8.7498
8.5681
8.3800
4
12.4976
12.4479
12.3071
12.0934
11.8168
11.5001
11.1677
10.8296
10.4956
5
15.5993
15.5358
15.2626
14.9386
14.4850
13.8580
13.3321
12.8209
12.3355
Clamped
Pinned
1
3.9164
3.9152
3.9117
3.9058
3.8976
3.8872
3.8747
3.8602
3.8438
2
7.0372
7.0297
7.0074
6.9709
6.9215
6.8603
6.7892
6.7096
6.6233
3
10.1577
10.1338
10.0659
9.9531
9.8074
9.6346
9.4426
9.2384
9.0278
4
13.2729
13.2193
13.0652
12.8220
12.5187
12.1761
11.8141
11.4476
11.0871
5
16.3769
16.2981
16.1023
15.5693
15.0497
14.4919
13.9305
13.3870
12.8727
Clamped
Clamped
1
4.7003
4.6988
4.6943
4.6868
4.6765
4.6633
4.6475
4.6291
4.6083
2
7.8072
7.7982
7.7717
7.7285
7.6699
7.5976
7.5136
7.4200
7.3186
3
10.9286
10.9018
10.8232
10.6981
10.5345
10.3411
10.1269
9.8999
9.6668
4
14.0427
13.9832
13.8119
13.5482
13.2174
12.8452
12.4535
12.0585
11.6714
5
17.1634
17.0522
16.7389
16.2743
15.7188
15.1248
14.5292
13.9546
13.4127
Clamped
Free
1
1.8883
1.8882
1.8882
1.8882
1.8882
1.8882
1.8882
1.8882
1.8882
2
4.6907
4.6892
4.6846
4.6770
4.6663
4.6528
4.6365
4.6175
4.5959
3
7.8269
7.8179
7.7913
7.7479
7.6892
7.6167
7.5325
7.4385
7.3370
4
10.9434
10.9166
10.8380
10.7128
10.5491
10.3555
10.1411
9.9139
9.6805
5
14.0593
13.9965
13.8387
13.5618
13.2189
12.8532
12.4620
12.0664
11.6784
Clamped
Slide
1
2.3645
2.3643
2.3635
2.3623
2.3607
2.3586
2.3561
2.3530
2.3501
2
5.4739
5.4705
5.4605
5.4439
5.4212
5.3926
5.3587
5.3199
5.2768
3
8.5966
8.5823
8.5405
8.4731
8.3830
8.2739
8.1495
8.0139
7.8706
4
11.7161
11.6783
11.5708
11.4025
11.1860
10.9350
10.6631
10.3810
10.097
5
14.8176
14.7396
14.5473
14.2067
13.7950
13.3543
12.8958
12.4416
12.0178
Pinned
Slide
1
1.5599
1.5598
1.5594
1.5586
1.5577
1.5565
1.5550
1.5533
1.5512
2
4.6904
4.6878
4.6801
4.6674
4.6498
4.6278
4.6015
4.5714
4.5379
3
7.8161
7.8040
7.7685
7.7111
7.6342
7.5407
7.4337
7.3165
7.1920
4
10.9382
10.9047
10.8091
10.6588
10.4649
10.2392
9.9934
9.7373
9.4784
5
14.0388
13.9847
13.7849
13.4851
13.1123
12.6974
12.2717
11.8478
11.4374
Table 7. First five nondimensional frequency ratio of standing cantilever nanobeam under gravity acceleration with L/h = 100, ε = 0.9025
α
0
0.02
0.04
0.06
0.08
Mode
Type
U
NU
U
NU
U
NU
U
NU
U
NU
1
NG
1.8750
1.8883
1.8744
1.8882
1.8727
1.8882
1.8698
1.8882
1.8658
1.8882
G
1.8188
1.8320
1.8181
1.8318
1.8161
1.8313
1.8128
1.8304
1.8083
1.8291
2
NG
4.6928
4.6907
4.6847
4.6846
4.6608
4.6663
4.6219
4.6365
4.5697
4.5959
G
4.6738
4.6716
4.6655
4.6653
4.6410
4.6464
4.6013
4.6156
4.5478
4.5736
3
NG
7.8496
7.8269
7.8108
7.7913
7.6999
7.6892
7.5302
7.5325
7.3194
7.3370
G
7.8381
7.8153
7.7989
7.7792
7.6870
7.6760
7.5159
7.5175
7.3033
7.3196
4
NG
10.9821
10.9434
10.8718
10.8380
10.5703
10.5491
10.1452
10.1411
9.6656
9.6805
G
10.9737
10.9345
10.8626
10.8287
10.5600
10.5386
10.1330
10.1288
9.6509
9.6656
5
NG
14.1096
14.0593
13.8866
13.8387
13.2507
13.2189
12.4743
12.4620
11.6939
11.6784
G
14.0761
14.0240
13.8791
13.8174
13.2417
13.2198
12.4630
12.4516
11.6795
11.6644
(a)
(b)
Figure 2. Thevariation of nondimensional fundamental frequency ratio with the nonlocal parameter for various boundary conditions for (a) nonuniform nanobeam with A(x) = 1 0.05(x/l)^{2} and (b) uniform nanobeam
6.2.5. The Effect of Axial Load
Fig. 9 presents the influence of the tensional axial load on the pinnedpinned nonuniform nano beam with a variation of nonlocal parameters. The fundamental frequency ratio always decreases with the increase in nonlocal parameter; however, this decreasing of restraint will increases the tensional axial load. Further, when the axial load increases the nonlocal and local frequencies tend together, meanwhile, significantly increases the frequencies of NUTNB.
6.2.6. The Normalized Mode Shapes
Unlike for the pinnedpinned beam case where the mode shapes are not affected by the small scale effect parameter, the mode shapes of the clamped clamped and clampedpinned beam are significantly influenced by the small scale effect. The similar phenomenon is also found by Wang et al. [13] for linear vibration modes of the nonlocal Timoshenko beams and Yang et al. [36] for nonlinear vibration modes.
(a)
(b)
(c)
Figure 3. The variation of nondimensional first five frequency ratio with the nonlocal parameter for NUTNB with A(x) = 10.05(x/l)^{2} for (a) pinnedpinned, (b) clampedclamped and (c) clampedfree
(a)
(b)
(c)
Figure 4. The variation of nondimensional first five frequency ratio with the nonlocal parameter for uniform nanobeam for (a) pinnedpinned, (b) clampedclamped and (c) clampedfree
Figure 5. Thevariation of nondimensional fundamental frequency with the nonlocal parameter for standing cantilever nanobeam without and under gravity acceleration and uniform and nonuniform
Conclusions
Transverse vibration characteristics have been investigated for a nonuniform Timoshenko nanobeam based on Eringen’s nonlocal elasticity theory with axially loaded. A numerical approach (GDQM) has been used to study solving governing equations. Many typical results calculated by the presented method show excellent agreement with the exact results by other investigators. The same influence has been studied for the scale effect (nonlocal parameters), varying crosssection area, shear deformation, rotational moment of inertia, size of nanobeam, axial load, acceleration gravity and the selfweight of the NUTNB on the dimensionless natural frequencies. Based on the results, which have been discussed earlier, several conclusions can be addressed as follows:
(1) The small number grid points are sufficient in resulting converged solution and the presented work reflects the power of GDQM in solving nonuniform problems.
(2) The effect of the nonlocal parameter is to reduce the natural frequencies, hence the nonlocal frequencies are smaller than the local counterparts.
(3) The small scale effects are more significant for smaller size of nanobeam, larger nonlocal parameter and higher vibration modes.
(4) Among all of the boundary conditions (PP, CC, CP, CF, CS and PS), clampedclamped (CC) contain largest frequency and pinnedslide (PS) contains smallest frequency.
(a)
(b)
(c)
Figure 6. Thevariation of nondimensional frequency ratio with the nonlocal parameter for standing cantilever nanobeam under gravity acceleration (a) mode 1, (b) mode 2 and (c) mode 3
(a)
(b)
Figure 7. Thevariation of nondimensional fundamental frequency with length of beam with various values of the nonlocal parameter for nonuniform crosssection (a) pinnedpinned and (b) clampedclamped
(5) The frequency for nonuniform (tapered) nanobeam is smaller than the same frequencies of uniform nanobeam.
(6) Due to gravity acceleration and selfweighting (or distributed compressive axial force) for a standing cantilever beam, the nondimensional frequencies are less than the counterpart frequencies in the case without gravity acceleration. While the increase in the tensional axial load always causes an increase in the frequencies of NUTNB.
(7) In general, the frequency ratio for uniform beam decreases more by increasing nonlocal parameters with respect to the nonuniform beam for all of the boundary conditions of nanobeam.
(8) The small scale effect parameter is more significant for the mode shapes of the clampedclamped and clampedpinned unlike the mode shapes for the pinnedpinned beam case.
(a)
(b)
Figure 8. The variation of nondimensional fundamental frequency with the nonlocal parameter for various values of length of beam for nonuniform crosssection, (a) pinnedpinned and (b) clampedclamped
Figure 9. The influence of the tensional axial load on pinnedpinned nonuniform nanobeam with the variation of nonlocal parameters
Nomenclature
σ
nonlocal stress tensor
t(x´)
classical, macroscopic stress tensor
K
kernel function
τ
material constant
e_{0}
constant appropriate to each material
a
internal characteristics length
l
external characteristics length
E
Young’s modulus
G
shear modulus
γ
shear strain
μ
nonlocal parameter
M
nonlocal bending moment
Q
shear force
σ_{xx}
normal stress
σ_{xz}
transverse shear stress
A
crosssectional area of the beam
K_{s}
shear correction factor of the TBT
ρ
mass density of the nanobeam
P
distributed axial load along x axis
W
amplitude of the generalized displacements
Ф
rotation of beam
ω
vibration frequency of NUTNB
λ
natural frequency
ζ
slenderness ratio
α
scaling effect parameter of the nanobeam
Ω
shear deformation parameter
bending moment
axial forces
V
shear forces
ε
gravity parameter
coefficient of the axial force due to gravity
function of the axial force due to gravity
x_{j}
nodes coefficients.
w_{j}
weighting coefficients.
C_{mij}
weighting coefficients for mth derivative
w(x_{j})
function values at grid points
References
[1] Wang LF, Hu HY. Flexural Wave Propagation in Singlewalled Carbon Nanotubes. Phys Rev B 2005; 71: 1–7.
[2] Eringen AC. Nonlocal Polar Elastic Continua. Int J Eng Sci 1972; 10: 1–16.
[3] Eringen AC, Edelen DGB. On Nonlocal Elasticity. Int J Eng Sci 1972; 10: 233–248.
[4] Eringen AC. On Differential Equations of Nonlocal Elasticity and Solutions of Screw Dislocation and Surface Waves. J Appl Phys 1983; 54: 4703–4710.
[5] Eringen AC. Nonlocal Continuum Field Theories. SpringerVerlag; 2002.
[6] Lu P, Lee HP, Lu C, Zhang PQ. Dynamic Properties of Flexural Beams using a Nonlocal Elasticity Model. J Appl Phys 2006; 99: 073510.
[7] Peddieson J, Buchanan GG, McNitt RP. Application of Nonlocal Continuum Models to Nanotechnology. Int J Eng Sci 2003; 41: 305–312.
[8] Reddy JN, Wang CM. Deflection Relationships between Classical and Thirdorder Plate Theories. Acta Mech 1998; 130(3–4): 199–208.
[9] Wang Q. Wave Propagation in Carbon Nanotubes via Nonlocal Continuum Mechanics. J Appl Phys 2005; 98: 124301.
[10] Wang Q, Varadan VK. Vibration of Carbon Nanotubes Studied using Nonlocal Continuum Mechanics. Smart Mater Struct 2006; 15: 659–666.
[11] Wang CM, Zhang YY, Ramesh SS, Kitipornchai S. Buckling Analysis of Micro and Nanorods/tubes based on Nonlocal Timoshenko Beam Theory. J Phys D Appl Phys 2006; 39: 3904–3909.
[12] Reddy JN. Nonlocal Theories for Bending, Buckling and Vibration of Beams. Int J Eng Sci 2007; 45: 288–307.
[13] Wang CM, Zhang YY, He XQ. Vibration of Nonlocal Timoshenko Beams. Nanotechnology 2007; 18: 1–9.
[14] Murmu T, Pradhan SC. Buckling Analysis of a Singlewalled Carbon Nanotube Embedded in an Elastic Medium based on Nonlocal Elasticity and Timoshenko Beam Theory and using DQM. Physica E 2009; 41: 1232–1239.
[15] Şimşek M. Nonlocal Effects in The Forced Vibration of an Elastically Connected Doublecarbon Nanotube System under a Moving Nanoparticle. Comput Mater Sci 2011; 50: 2112–2123.
[16] Lu P, Lee HP, Lu C, Zhang PQ. Application of Nonlocal Beam Models for Carbon Nanotubes. Int J Solids Struct 2007; 44: 5289–5300.
[17] Reddy JN. Energy Principles and Variational Methods in Applied Mechanics. John Wiley & Sons; 2002.
[18] Reddy JN. Theory and Analysis of Elastic Plates and Shells. Taylor & Francis; 2007.
[19] Reddy JN, Pang SD. Nonlocal Continuum Theories of Beams for The Analysis of Carbon Nanotubes. J Appl Phys 2008; 103: 023511.
[20] Hutchinson JR. Shear Coefficients for Timoshenko Beam Theory. J Appl Mech 2001; 68: 1–6.
[21] Meirovitch L. Fundamentals of Vibrations. McGrawHill; 2001.
[22] Ke LL, Xiang Y, Yang J, Kitipornchai S. Nonlinear Free Vibration of Embedded Doublewalled Carbon Nanotubes based on Nonlocal Timoshenko Beam Theory. Comp Mater Sci 2009; 47: 409–417.
[23] Hijmissen JW, Horssen WTV. On Transverse Vibrations of a Vertical Timoshenko Beam. J Sound Vib 2008; 314: 161–179.
[24] Bellman R, Casti J. Differential Quadrature and Longterm Integration. J Math Anal Appl 1971; 34: 235–238.
[25] Bellman R, Kashef BG, Casti J. Differential Quadrature a Technique for The Rapid Solution of Nonlinear Partial Differential Equations. J Comput Phys 1972; 10: 40–52.
[27] Shu C. Differential Quadrature and Its Application in Engineering. Sprimger; 2000.
[28] Mestrovic M. Generalized Differential Quadrature Method for Timoshenko Beam. MIT Conf Comput Fluid Solid Mech 2003.
[29] Du H, Lim MK, Lin NR. Application of Generalized Differential Quadrature Method to Structural Problems. J Num Meth Engrg 1994; 37: 1881–1896.
[30] Du H, Lim MK, Lin NR. Application of Generalized Differential Quadrature to Vibration Analysis. J Sound Vib 1995; 181: 279–293.
[31] Mahmoud AA, Esmaeel RA, Nassar MM. Application of The Generalized Differential Quadrature Method to The Free Vibrations of Delaminated Beam Plates. J Eng Mech 2007; 14: 431–441.
[32] Wu T Y, Liu GR. A Differential Quadrature as a Numerical Method to Solve Differential Equations. Comput Mech 1999; 24: 197–205.
[33] Farchaly SH, Shebl MG. Exact Frequency and Mode Shape Formulae for Studying Vibration and Stability of Timoshenko Beam System. J Sound Vib 1995; 180: 205–227.
[35] Arash B, Wang Q. A Review on The Application of Nonlocal Elastic Models in Modeling of Carbon Nanotubes and Graphenes. Comput Mater Sci 2012; 51: 303–313.
[36] Yang J, Ke LL, Kitipornchai S. Nonlinear Free Vibration of Singlewalled Carbon Nanotubes using Nonlocal Timoshenko Beam Theory. Physica E 2010; 42: 1727–1735.
References
[1]Wang LF, Hu HY. Flexural Wave Propagation in Singlewalled Carbon Nanotubes. Phys Rev B 2005; 71: 1–7.
[2]Eringen AC. Nonlocal Polar Elastic Continua. Int J Eng Sci 1972; 10: 1–16.
[3]Eringen AC, Edelen DGB. On Nonlocal Elasticity. Int J Eng Sci 1972; 10: 233–248.
[4]Eringen AC. On Differential Equations of Nonlocal Elasticity and Solutions of Screw Dislocation and Surface Waves. J Appl Phys 1983; 54:4703–4710.
[5]Eringen AC. Nonlocal Continuum Field Theories. SpringerVerlag; 2002.
[6]Lu P, Lee HP, Lu C, Zhang PQ. Dynamic Properties of Flexural Beams using a Nonlocal Elasticity Model. J Appl Phys 2006; 99: 073510.
[7]Peddieson J, Buchanan GG, McNitt RP. Application of Nonlocal Continuum Models to Nanotechnology. Int J Eng Sci 2003; 41: 305–312.
[8]Reddy JN, Wang CM. Deflection Relationships between Classical and Thirdorder Plate Theories. Acta Mech 1998; 130(3–4): 199–208.
[9]Wang Q. Wave Propagation in Carbon Nanotubes via Nonlocal Continuum Mechanics. J Appl Phys 2005; 98: 124301.
[10]Wang Q, Varadan VK. Vibration of Carbon Nanotubes Studied using Nonlocal Continuum Mechanics. Smart Mater Struct 2006; 15: 659–666.
[11]Wang CM, Zhang YY, Ramesh SS, Kitipornchai S. Buckling Analysis of Micro and Nanorods/tubes based on Nonlocal Timoshenko Beam Theory. J Phys D Appl Phys 2006; 39:3904–3909.
[12]Reddy JN. Nonlocal Theories for Bending, Buckling and Vibration of Beams. Int J Eng Sci 2007; 45: 288–307.
[13]Wang CM, Zhang YY, He XQ. Vibration of Nonlocal Timoshenko Beams. Nanotechnology 2007; 18: 1–9.
[14]Murmu T, Pradhan SC. Buckling Analysis of a Singlewalled Carbon Nanotube Embedded in an Elastic Medium based on Nonlocal Elasticity and Timoshenko Beam Theory and using DQM. Physica E2009; 41: 1232–1239.
[15]Şimşek M. Nonlocal Effects in The Forced Vibration of an Elastically Connected Doublecarbon Nanotube System under a Moving Nanoparticle. Comput Mater Sci 2011; 50: 2112–2123.
[16]Lu P, Lee HP, Lu C, Zhang PQ. Application of Nonlocal Beam Models for Carbon Nanotubes. Int J Solids Struct 2007; 44: 5289–5300.
[17]Reddy JN. Energy Principles and Variational Methods in Applied Mechanics. John Wiley & Sons; 2002.
[18]Reddy JN. Theory and Analysis of Elastic Plates and Shells. Taylor & Francis; 2007.
[19]Reddy JN, Pang SD. Nonlocal Continuum Theories of Beams for The Analysis of Carbon Nanotubes. J Appl Phys 2008; 103: 023511.
[20]Hutchinson JR. Shear Coefficients for Timoshenko Beam Theory. J Appl Mech 2001; 68: 1–6.
[21]Meirovitch L. Fundamentals of Vibrations. McGrawHill; 2001.
[22]Ke LL, Xiang Y, Yang J, Kitipornchai S. Nonlinear Free Vibration of Embedded Doublewalled Carbon Nanotubes based on Nonlocal Timoshenko Beam Theory. Comp Mater Sci 2009; 47: 409–417.
[23]Hijmissen JW, Horssen WTV. On Transverse Vibrations of a Vertical Timoshenko Beam. J Sound Vib 2008; 314: 161–179.
[24]Bellman R, Casti J. Differential Quadrature and Longterm Integration. J Math Anal Appl 1971; 34: 235–238.
[25]Bellman R, Kashef BG, Casti J. Differential Quadrature a Technique for The Rapid Solution of Nonlinear Partial Differential Equations. J Comput Phys 1972; 10: 40–52.
[27]Shu C. Differential Quadrature and Its Application in Engineering. Sprimger; 2000.
[28]Mestrovic M. Generalized Differential Quadrature Method for Timoshenko Beam. MIT Conf Comput Fluid Solid Mech 2003.
[29]Du H, Lim MK, Lin NR. Application of Generalized Differential Quadrature Method to Structural Problems. J Num Meth Engrg 1994; 37: 1881–1896.
[30]Du H, Lim MK, Lin NR. Application of Generalized Differential Quadrature to Vibration Analysis. J Sound Vib 1995; 181: 279–293.
[31]Mahmoud AA, Esmaeel RA, Nassar MM. Application of The Generalized Differential Quadrature Method to The Free Vibrations of Delaminated Beam Plates. J Eng Mech 2007; 14: 431–441.
[32]Wu T Y, Liu GR. A Differential Quadrature as a Numerical Method to Solve Differential Equations. Comput Mech 1999; 24: 197–205.
[33]Farchaly SH, Shebl MG. Exact Frequency and Mode Shape Formulae for Studying Vibration and Stability of Timoshenko Beam System. J Sound Vib 1995; 180: 205–227.
[34]Chen WR. Bending Vibration of Axially Loaded Timoshenko Beams with Locally Distributed KelvinVoigt Damping. J Sound Vib 2011; 330: 3040–3056.
[35]Arash B, Wang Q. A Review on The Application of Nonlocal Elastic Models in Modeling of Carbon Nanotubes and Graphenes. Comput Mater Sci 2012; 51: 303–313.
[36]Yang J, Ke LL, Kitipornchai S. Nonlinear Free Vibration of Singlewalled Carbon Nanotubes using Nonlocal Timoshenko Beam Theory. Physica E 2010; 42: 1727–1735.
[1]Wang LF, Hu HY. Flexural Wave Propagation in Singlewalled Carbon Nanotubes. Phys Rev B 2005; 71: 1–7.
[2]Eringen AC. Nonlocal Polar Elastic Continua. Int J Eng Sci 1972; 10: 1–16.
[3]Eringen AC, Edelen DGB. On Nonlocal Elasticity. Int J Eng Sci 1972; 10: 233–248.
[4]Eringen AC. On Differential Equations of Nonlocal Elasticity and Solutions of Screw Dislocation and Surface Waves. J Appl Phys 1983; 54:4703–4710.
[5]Eringen AC. Nonlocal Continuum Field Theories. SpringerVerlag; 2002.
[6]Lu P, Lee HP, Lu C, Zhang PQ. Dynamic Properties of Flexural Beams using a Nonlocal Elasticity Model. J Appl Phys 2006; 99: 073510.
[7]Peddieson J, Buchanan GG, McNitt RP. Application of Nonlocal Continuum Models to Nanotechnology. Int J Eng Sci 2003; 41: 305–312.
[8]Reddy JN, Wang CM. Deflection Relationships between Classical and Thirdorder Plate Theories. Acta Mech 1998; 130(3–4): 199–208.
[9]Wang Q. Wave Propagation in Carbon Nanotubes via Nonlocal Continuum Mechanics. J Appl Phys 2005; 98: 124301.
[10]Wang Q, Varadan VK. Vibration of Carbon Nanotubes Studied using Nonlocal Continuum Mechanics. Smart Mater Struct 2006; 15: 659–666.
[11]Wang CM, Zhang YY, Ramesh SS, Kitipornchai S. Buckling Analysis of Micro and Nanorods/tubes based on Nonlocal Timoshenko Beam Theory. J Phys D Appl Phys 2006; 39:3904–3909.
[12]Reddy JN. Nonlocal Theories for Bending, Buckling and Vibration of Beams. Int J Eng Sci 2007; 45: 288–307.
[13]Wang CM, Zhang YY, He XQ. Vibration of Nonlocal Timoshenko Beams. Nanotechnology 2007; 18: 1–9.
[14]Murmu T, Pradhan SC. Buckling Analysis of a Singlewalled Carbon Nanotube Embedded in an Elastic Medium based on Nonlocal Elasticity and Timoshenko Beam Theory and using DQM. Physica E2009; 41: 1232–1239.
[15]Şimşek M. Nonlocal Effects in The Forced Vibration of an Elastically Connected Doublecarbon Nanotube System under a Moving Nanoparticle. Comput Mater Sci 2011; 50: 2112–2123.
[16]Lu P, Lee HP, Lu C, Zhang PQ. Application of Nonlocal Beam Models for Carbon Nanotubes. Int J Solids Struct 2007; 44: 5289–5300.
[17]Reddy JN. Energy Principles and Variational Methods in Applied Mechanics. John Wiley & Sons; 2002.
[18]Reddy JN. Theory and Analysis of Elastic Plates and Shells. Taylor & Francis; 2007.
[19]Reddy JN, Pang SD. Nonlocal Continuum Theories of Beams for The Analysis of Carbon Nanotubes. J Appl Phys 2008; 103: 023511.
[20]Hutchinson JR. Shear Coefficients for Timoshenko Beam Theory. J Appl Mech 2001; 68: 1–6.
[21]Meirovitch L. Fundamentals of Vibrations. McGrawHill; 2001.
[22]Ke LL, Xiang Y, Yang J, Kitipornchai S. Nonlinear Free Vibration of Embedded Doublewalled Carbon Nanotubes based on Nonlocal Timoshenko Beam Theory. Comp Mater Sci 2009; 47: 409–417.
[23]Hijmissen JW, Horssen WTV. On Transverse Vibrations of a Vertical Timoshenko Beam. J Sound Vib 2008; 314: 161–179.
[24]Bellman R, Casti J. Differential Quadrature and Longterm Integration. J Math Anal Appl 1971; 34: 235–238.
[25]Bellman R, Kashef BG, Casti J. Differential Quadrature a Technique for The Rapid Solution of Nonlinear Partial Differential Equations. J Comput Phys 1972; 10: 40–52.
[27]Shu C. Differential Quadrature and Its Application in Engineering. Sprimger; 2000.
[28]Mestrovic M. Generalized Differential Quadrature Method for Timoshenko Beam. MIT Conf Comput Fluid Solid Mech 2003.
[29]Du H, Lim MK, Lin NR. Application of Generalized Differential Quadrature Method to Structural Problems. J Num Meth Engrg 1994; 37: 1881–1896.
[30]Du H, Lim MK, Lin NR. Application of Generalized Differential Quadrature to Vibration Analysis. J Sound Vib 1995; 181: 279–293.
[31]Mahmoud AA, Esmaeel RA, Nassar MM. Application of The Generalized Differential Quadrature Method to The Free Vibrations of Delaminated Beam Plates. J Eng Mech 2007; 14: 431–441.
[32]Wu T Y, Liu GR. A Differential Quadrature as a Numerical Method to Solve Differential Equations. Comput Mech 1999; 24: 197–205.
[33]Farchaly SH, Shebl MG. Exact Frequency and Mode Shape Formulae for Studying Vibration and Stability of Timoshenko Beam System. J Sound Vib 1995; 180: 205–227.
[34]Chen WR. Bending Vibration of Axially Loaded Timoshenko Beams with Locally Distributed KelvinVoigt Damping. J Sound Vib 2011; 330: 3040–3056.
[35]Arash B, Wang Q. A Review on The Application of Nonlocal Elastic Models in Modeling of Carbon Nanotubes and Graphenes. Comput Mater Sci 2012; 51: 303–313.
[36]Yang J, Ke LL, Kitipornchai S. Nonlinear Free Vibration of Singlewalled Carbon Nanotubes using Nonlocal Timoshenko Beam Theory. Physica E 2010; 42: 1727–1735.