Document Type : Research Paper
Authors
Department of Solid Mechanics, Faculty of Mechanical Engineering, University of Kashan, Kashan, Iran
Abstract
Keywords

Mechanics of Advanced Composite Structures 2 (2015) 116


Semnan University 
Mechanics of Advanced Composite Structures journal homepage: http://macs.journals.semnan.ac.ir 
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. 



Keywords: Nonlocal elasticity Gravity Timoshenko Nonuniform nanobeam Generalized differential quadrature method



© 2015 Published by Semnan University Press. All rights reserved. 
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.
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.
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.
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.
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.
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
Case 
I 
II 

r^{2 }= 0.01, s^{2 }= 2.8 
r^{2 }= 0.001, s^{2 }= 0.5 

Mode number 
Ref. [23] 
Method 
Ref. [23] 
Method 
1 
3.2471 
3.2471 
3.5038 
3.5038 
2 
14.803 
14.8026 
21.519 
21.5191 
3 
32.415 
32.4153 
58.448 
58.4478 
4 
49.649 
49.6488 
109.98 
109.9815 
5 
65.263 
65.2632 
173.47 
173.4722 
6 
70.555 
70.5548 
246.27 
246.2743 
7 
84.075 
84.0753 
326.22 
326.2190 
8 
92.021 
92.0212 
411.60 
411.6015 
9 
105.87 
105.8678 
501.12 
501.1166 
10 
113.75 
113.7541 
593.77 
593.7673 
Table 3. The effect of gravity on the natural frequencies ω (Hz) of uniform CF local beam
Mode number 
The exact method [23] 
The present Method 
The error percent 
r^{2}= 6.30×10^{5}, s^{2}= 2.8, ε = 0.314 

1 
0.2284 
0.2283 
0.0384 
2 
1.4513 
1.4513 
0.0028 
3 
4.0496 
4.0496 
0.0011 
4 
7.8792 
7.8792 
0.0002 
5 
12.903 
12.9026 
0.0034 
6 
19.055 
19.0549 
0.0004 
7 
26.265 
26.2646 
0.0017 
8 
34.455 
34.4547 
0.0009 
9 
43.549 
43.5465 
0.0058 
10 
53.467 
53.4614 
0.0104 
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. The convergence of nonlocal results by GDQM
Table 4. First five natural frequencies of uniform beam with L/d = 10
BCs. 
Mode number 
α 

0 
0.1 
0.3 
0.5 
0.7 

Present 
Ref. [13] 
Present 
Ref. [13] 
Present 
Ref. [13] 
Present 
Ref. [13] 
Present 
Ref. [13] 

Pinned Pinned 
1 
3.0929 
3.0929 
3.0210 
3.0243 
2.6385 
2.6538 
2.2665 
2.2867 
1.9899 
2.0106 
2 
5.9399 
5.9399 
5.4657 
5.5304 
4.0663 
4.2058 
3.2713 
3.4037 
2.7968 
2.9159 

3 
8.4444 
8.4444 
7.2037 
7.4699 
4.8761 
5.2444 
3.8474 
4.1644 
3.2690 
3.5453 

4 
10.6262 
10.626 
8.3851 
8.9874 
5.3806 
6.0228 
4.2128 
4.7436 
3.5713 
4.0283 

5 
12.5413 
12.541 
9.1905 
10.206 
5.7140 
6.6333 
4.4571 
5.2009 
3.7743 
4.4107 

Clamped Pinned 
1 
3.7845 
3.7845 
3.6846 
3.6939 
3.1711 
3.2115 
2.6966 
2.7471 
2.3553 
2.4059 
2 
6.4728 
6.4728 
5.9353 
6.0348 
4.3904 
4.6013 
3.5310 
3.7312 
3.0198 
3.2003 

3 
8.8212 
8.1212 
7.5104 
7.8456 
5.0816 
5.5482 
4.0139 
4.4185 
3.4127 
3.7666 

4 
10.8800 
10.880 
8.5826 
9.2751 
5.5161 
6.2641 
4.3241 
4.9460 
3.6677 
4.5528 

5 
12.7075 
12.707 
9.3194 
10.433 
5.8062 
6.8277 
4.5335 
5.3640 
3.8408 
4.8326 

Clamped Clamped 
1 
4.4491 
4.4491 
4.3269 
4.3471 
3.7032 
3.7895 
3.1357 
3.2420 
2.7327 
2.8383 
2 
6.9524 
6.9524 
6.3546 
6.4952 
4.6553 
4.9428 
3.7269 
3.9940 
3.1808 
3.4192 

3 
9.1626 
9.1626 
7.7884 
8.1969 
5.2714 
5.8460 
4.1729 
4.6769 
3.5522 
3.9961 

4 
11.1126 
11.113 
8.7619 
9.5447 
5.6290 
6.4762 
4.4105 
5.1131 
3.7399 
4.3455 

5 
12.8627 
12.863 
9.4384 
10.649 
5.8924 
7.0170 
4.6077 
5.5283 
3.9066 
4.6986 

Clamped Free 
1 
1.8610 
1.8610 
1.8467 
1.8650 
1.7514 
1.8999 
1.6223 
2.0024 
1.4986 
 
2 
4.4733 
4.4733 
4.2956 
4.3506 
3.4516 
3.6594 
2.7448 
2.8903 
2.2849 
 

3 
7.1072 
7.1072 
6.4181 
6.6091 
4.6091 
5.0762 
3.6883 
 
3.1592 
 

4 
9.3813 
9.3813 
7.8500 
8.3151 
5.1542 
5.7875 
3.9894 
 
3.3525 
 

5 
11.3811 
11.381 
8.8244 
9.6705 
5.5939 
6.5843 
4.3838 
 
3.7244 
 
6.2.2. Effect of Nonlocal Parameter
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. The comparison of first four dimensionless natural frequencies for a clamped–free uniform beam
Axial load 
Method 
r^{2 }= 0.01, s^{2 }= 0.03 

Mode 1 
Mode 2 
Mode 3 
Mode 4 

p=0 
Ref. [33] 
1.799 
3.820 
5.642 
6.967 
Ref. [34] 
1.7964 
3.8047 
5.6170 
6.9305 

Present 
1.7985 
3.8199 
5.6423 
6.9671 

p=1 
Ref. [33] 
1.579 
3.715 
5.569 
6.904 
Ref. [34] 
1.5768 
3.6982 
5.5425 
6.8666 

Present 
1.5792 
3.7145 
5.5692 
6.9043 

p=10 
Ref. [33] 
2.199 
4.720 
6.171 
 
Ref. [34] 
2.2009 
4.7242 
6.1811 
 

Present 
2.1993 
4.7197 
6.1707 
7.3434 
6.2.3. The Effect of Gravity Acceleration
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. The variation 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. The variation of nondimensional fundamental frequency with the nonlocal parameter for standing cantilever nanobeam without and under gravity acceleration and uniform and nonuniform
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. The variation 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. The variation 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 
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