1877-7058 © 2011 Published by Elsevier Ltd.doi:10.1016/j.proeng.2011.04.294

Available online at www.sciencedirect.comAvailable online at www.sciencedirect.com

Procedia Engineering 00 (2011) 000–000

Procedia Engineering

www.elsevier.com/locate/procedia

ICM11

Energy approach vibration analysis of nonlocal Timoshenko beam theory

Bijan Mohammadi a*, S.A.M. Ghannadpourb, aAssisstant professor, School of Mechanical Engineering, Iran University of Science and Technology, Narmak, Tehran, Iran

bAssistant Professor, Aerospace Eng. Department, Faculty of New Technologies, Shahid Beheshti University, G.C., Tehran, Iran

Abstract

The aim of this paper is to present an efficient numerical method to analyze the vibration behavior of nano Timoshenko beams based on Eringen’s nonlocal elasticity theory. The vibration frequencies of beams are firstly obtained using the theorem of minimum total potential energy and Chebyshev polynomial functions. The present method provides an efficient and extremely accurate vibration solution. Numerical results for a variety of some micro- and nano-beams with various boundary conditions are given and compared with the available results wherever possible. The small scale effects on the vibration frequencies of beams are determined and discussed.

© 2011 Published by Elsevier Ltd. Selection and/or peer-review under responsibility of ICM11

"Keywords: Timoshenkov beam; Natural frequency; Eringen's nonlocality"

1. Introduction

In 1972, Eringen [1] pioneered the nonlocal continuum mechanics to account for the small-scale effect by specifying the stress state at a given point to be dependent on the strain states at all points in the body. Since then, many researchers have applied the nonlocal elasticity concept for the bending, buckling, and vibration analyses of beam-like elements in micro- or nano electromechanical systems.

A version of nonlocal elasticity was proposed by Peddieson et al. [2] to formulate a nonlocal version of Euler-Bernoulli beam theory. They concluded that nonlocal continuum mechanics could potentially play a

* Corresponding author. Tel.:+98-21-7724-0545; Fax: +98-21-7724-0488. E-mail address: Bijan_Mohammadi@iust.ac.ir.

Procedia Engineering 10 (2011) 1766–1771

B. Mohammadi and S. A. M. Ghannadpour / Procedia Engineering 10 (2011) 1766–1771 17672 B. Mohammadi, S.A.M. Ghannadpour / Procedia Engineering 00 (2011) 000–000

useful role in the analysis related to nanotechnology applications. Since the attempt on applying nonlocal continuous models was conducted, applications of nonlocal Euler–Bernoulli beam model have been employed in studying nano-beams mechanical behaviors. Wang et al. [3] obtained exact solutions for the elastic buckling behavior of nano-beams based on nonlocal continuum mechanics. They derived critical buckling loads for pinned, clamped–pinned, clamped and cantilever nano-beams with various length-to-diameter ratios and scale coefficients.

In the literature, the most attention has been focused on deriving the variational formulation of the governing equations and boundary conditions of nanostructures. Elishakoff et al. [4] made the first attempt to calculate approximate expression for the critical buckling load of a double-walled carbon nanotube using Bubnov-Galerkin and Petrov-Galerkin methods. Ghannadpour and Mohammadi were the first who investigated the behavior of nanostructures using Ritz methods [5]. They have studied the buckling behavior of micro- and nano-rods/tubes based on nonlocal Timoshenko beam theory using Chebyshev polynomials. The critical buckling loads have been obtained using the theorem of minimum total potential energy and Chebyshev polynomial functions. However, no work appears related to the vibration behavior of nanostructures using numerical methods. The aim of this paper is to present the vibration behavior of micro/nano Timoshenko beams based on Eringen’s nonlocal elasticity theory. The vibration frequencies of the beams are obtained using Hamilton’s principle and Chebyshev polynomial functions. The present method, which uses Rayleigh–Ritz technique in this paper, provides an efficient and extremely accurate vibration solution.

2. Non-local Timoshenko beam theory

According to the Timoshenko beam theory, the strain–displacement relations are given by

dx

dw

dx

dz xzxx +=−= ϕγ

ϕε ; (1)

where x is the longitudinal coordinate measured from the left end of the beam, z is the coordinate measured from the midplane of the beam, w is the transverse displacement, εxx is the normal strain, and γxz

is shear strain. The strain energy U and the kinetic energy T of the beam are obtained, respectively by the following equations:

2 2

0 0

1 1;

2 2

L Ld dwU M Q dx T A w dx

dx dx

ϕϕ ρ ω

= + + =

∫ ∫ (2)

where ω is the circular frequency of vibration, A the cross-sectional area of the beam, ρ the mass density of the beam material, L the length of the beam and M=∫ σxx z dA the bending moment. In the definition of the M, σxx is the normal stress, and A is the cross-sectional area of the beam. It should be noted that the bending moment for the nonlocal beam theory is different due to the nonlocal constitutive relations as will be shown. Adopting the theory of nonlocal elasticity, Hooke’s law for a uniaxial stress state in the one-dimensional case, is determined by:

( ) xxxx

xx Edx

dae ε

σ

σ =−2

22

0 (3)

1768 B. Mohammadi and S. A. M. Ghannadpour / Procedia Engineering 10 (2011) 1766–1771 B. Mohammadi, S.A.M. Ghannadpour/ Procedia Engineering 00 (2011) 000–000 3

where e0a is a scale coefficient that is appropriate to the small scale effect and E is the Young’s modulus. Multiplying Eq. (3) by zdA and integrating the result over the area A yields

( )2

2

2

22

0dx

wdEI

dx

MdaeM −=− (4)

where I=∫ y2 dA, is the moment of inertia. By defining Q=∫ σxz dA and integrating, the shear force, Q, can be calculated. Bending moment in nonlocal form, M, can also be obtained using two equilibrium equations which govern the vibration behavior of beams [3]. The shear stress, Q, and nonlocal bending mordent, M , are expressed as follows:

( )2 2 2

0;dw d d

Q Ks GA M EI e a A w Idx dx dx

ϕ ϕϕ ρ ω ρ ω

= + = − +

(5)

where Ks is the shear correction in the Timoshenko beam theory in order to compensate for the error in assuming a constant shear strain (stress) through the thickness of the beam. By substituting the Eqs. (5) into the Eqs. (2), the potential energy, U, minus the kinetic energy, T, of the beam can be obtained as follow:

( ) ( )( )2

2 22 2 20 0

0

22 2

1

2

2

L d dT U KsGA A e a w I e a EI

dx dx

dw dwA w KsGA KsGA dx

dx dx

ϕ ϕϕ ρ ω ρ ω

ρ ω ϕ ϕ

− = − + + −

+ − −

∫(6)

It is noted that the only unknown function in the above equation is the transverse displacement, w. By knowing the transverse displacement field w, the Eq. (6) can be obtained. By using the Hamilton’s principle, the difference between the kinetic energy and potential energy of the beam must be minimized. That is to say the partial differentiation of the Eq. (6) with respect to unknown parameters in turn leads an eigenvalue problem. The procedure that postulates the displacement field w will be shown as follows. A polynomial-based displacement function is used herein which it consists of a boundary polynomial, specifying support conditions, multiplied by a complete one-dimensional simple polynomial.

( ) ( ) ( )ξξϕξ ∑∞

=

=

0nnb Tw (7)

In the equation (7), the function ( )ξnT is the one-dimensional nth Chebyshev polynomial, and Lx=ξ

is non-dimensional coordinate. Also, the term ( )ξϕb is a polynomial expression describing the boundary conditions. The non-dimensional boundary polynomial expressions for a nano-beam with varying boundary conditions are shown as :

( ) ( )λγ ξξξϕ −= 1b (8)

B. Mohammadi and S. A. M. Ghannadpour / Procedia Engineering 10 (2011) 1766–1771 17694 B. Mohammadi, S.A.M. Ghannadpour / Procedia Engineering 00 (2011) 000–000

As γ and λ take values of γ , λ = 0, 1, 2, corresponding to the free, pined and clamped end conditions, respectively. It is essential to note that the classical one-dimensional Chebyshev polynomials of the first kind is ( ) ( )ξξ arccoscos nTn = , and its recursive relation is ( ) ( ) ( )ξξξξ 11 2

−+−= nnn TTT .

3. Results and Discussions

Using the Chebyshev polynomials of the first kind as basis functions in the Rayleigh-Ritz method, the natural frequency for nano-beam has been computed. The data adopted in generating the results are: Young’s modulus E=1 TPa, shear modulus G=E/[2(1+v)], Poisson’s ratio ν=0.19, rod diameter d=1nmand second moment of area I=πd4/64. Also, the following nondimensional terms are used:

42 2 ;

ALe a LoEI

ρλ ω α= =

(9)

where λ and α , are called frequency parameter and scaling effect parameter, respectively. In calculating the results of this paper, the length of the beams under consideration is assumed to be 6.78 nm.

After comparing with the analytical method [3], this proposed technique is very close to the exact solutions. Unlike the other approximated numerical methods, the obtained results and the exact natural frequencies are identical for the case of simply supported, and clamped–clampled nano-beam.

The convergence studies carried out for the frequency parameter λ of a nano beam for first mode of vibration with the scaling effect parameter α=3 are tabulated in Table 1. In this table, the two ends of the beam are of a variety of combinations, namely simply-simply (SS), clamped-clamped (CC), and clamped-simply (CS). It is seen that the frequency parameters obtained with 4 and 5 terms are identical. However, for the sake of confidence, all of the results presented for frequency parameters have been calculated using n =10. In Table 1 the frequency parameters are also compared with the results of Wang et al. [3].

In resume of generating the results, it is desired to investigate the effect of small scale on the frequency parameter. In Table 2, the frequency parameter for beam of various end conditions are presented for L/d=10.0 and various length-scale coefficients. Note that the results associated with α = 0 correspond to those of the local Timoshenko theory where the small scale effect is ignored and it can be seen again that the nonlocal results are smaller than the corresponding local results. The results of Wang et al. [3] are also incorporated in Table 2 and there is an excellent agreement between the present results and those of Wang. It is noted that in Ref. [3], the results are obtained by solving the governing equations. The governing equations and the boundary conditions related to the nano rod/tube were derived using the principle of virtual work.

In order to investigate the effect of small-scale on the frequency parameter for the simply supported and clamed beam, the relationship between the frequency parameter verses the small scale coefficient, e0afor five frequency shape modes and L/d=10 are shown in Figure 1. This figure shows that the frequency parameters with small-length scale are less than frequency parameter without small-length scale. It means that the frequency parameter is over predicted whenever the local beam model is applied for frequency analysis of rod/tube. As the scale coefficient, e0a, increases, the frequency parameter obtained from the nonlocal Timoshenko beam theory become smaller than those for its local counterpart. This reduction is especially significant for higher vibration modes, and thus the small-scale effect cannot be neglected. The reduction may be because the small-scale effect makes the rod/tube more flexible as the nonlocal model may be viewed as atoms linked by elastic springs while the local continuum model assumes the spring

1770 B. Mohammadi and S. A. M. Ghannadpour / Procedia Engineering 10 (2011) 1766–1771 B. Mohammadi, S.A.M. Ghannadpour/ Procedia Engineering 00 (2011) 000–000 5

constant to take on an infinite value. In sum, the nonlocal theory should be used if one needs accurate predictions of high natural frequency of micro- and nano-rods/tubes.

Table 1 First mode frequency parameters convergence study for a beam with α =0.3 and L/d=10 under different boundary conditions

alpha=0.3 Mode 1

� �� �� ��

� 2.784441 3.220137 3.803846

� 2.652435 3.208431 3.781479

� 2.652215 3.207711 3.780535

� 2.652066 3.20737 3.780058

�� 2.652066 3.207166 3.779767

�� 2.6521 3.2071 3.7798

�������� ������ ������ ������

������ ����� ����� �����

Convergence stydy

Table 2 First five frequency parameters �� for simply supported, clamped�simply supported, clamped and cantilever beams withL/d = 10 and various scaling effect parameters α = e0a/L.

� �� ����

���� ������� �������� ������� �������� ������� �������� ������� �������� ������� ��������

� ������ ������ ������ 3.0243 ������ 2.6538 ������ 2.2867 ������ 2.0106� ������ ������ ������ 5.5304 ������ 4.2058 ������ 3.4037 ������ 2.9159� ������ ������ ������ 7.4699 ������ 5.2444 ������ 4.1644 ������ 3.5453� ������� ������� ������ 8.9874 ������ 6.0228 ������ 4.7436 ������ 4.0283� ������� ������� ������� 10.206 ������ 6.6333 ������ 5.2009 ������ 4.4107

� ������ ������ ������ ������ ������ ������ ������ ������ ������ ������

� ������ ������ ������ ������ ������ ������ ������ ������ ������ ������

� ������ ������ ������ ������ ������ ������ ������ ������ ������ ������

� ������� ������� ������ ������ ������ ������ ������ ������ ������ ������

� ������� ������� ������� ������� ������ ������ ������ ������ ������ ������

� ������ ������ ������ ������ ������ ������ ������ ������ ������ ������

� ������ ������ ������ ������ ������ ������ ������ ������ ������ ������

� ������ ������ ������ ������ ������ ������ ������ ������ ������ ������

� ������� ������� ������ ������ ������ ������ ������ ������ ������ ������

� ������� ������� ������� ������� ������ ������ ������ ������ ������ ������

���

Simply supported ended beam

Clamped-simply supported beam

Clamped ended beam

� ��� ��� ���

B. Mohammadi and S. A. M. Ghannadpour / Procedia Engineering 10 (2011) 1766–1771 1771

6 B. Mohammadi, S.A.M. Ghannadpour / Procedia Engineering 00 (2011) 000–000

0

2

4

6

8

10

12

14

0 0.2 0.4 0.6 0.8

eoa

Nat

ura

l Fre

quen

cy

Mode 1

Mode 2

Mode 3

Mode 4

Mode 5

Simply-Supported beam

0

2

4

6

8

10

12

14

0 0.2 0.4 0.6 0.8

eoa

Nat

ura

l Fre

quen

cy

Mode 1

Mode 2

Mode 3

Mode 4

Mode 5

Clamped ended beam

Figure 1. (a) Length scale effect on the frequency parameter for a simply supported beam with L/d=10, (b) Length scale effect on the frequency parameter for a clamped ended beam with L/d=10.

4. Conclusion

In this study, the natural frequency analysis of a nano Timoshenko beam with nonlocal theory for various boundary conditions like simply supported, clamped-free and clamped-clamped are studied. In this way, a semi-analytical numerical technique via the minimum potential energy in conjunction with the Chebyshev polynomials was used in the framework of the Rayleigh-Ritz method in a simple and accurate way. The rapid convergence of this technique is remarkable because with only ten terms, expansion of Chebyshev series can be predicted the exact solutions, and even a seven-term solution would lead to good result, which is very unlikely in the case of the conventional power series. The natural frequencies have been calculated for various nonlocal parameter values. The calculated results are compared with analytical solutions, and a very good agreement is observed between the analytical and the present numerical method. From this analysis, it can be seen that, the solutions obtained for a nano-beam can be a promising technique in investigating more complicated nanostructures with nonlocal effects.

References

[1] Eringen A.C., Nonlocal polar elastic continua , International Journal of Engineering Science,1972, 10, 1–16.

[2] Peddieson J., Buchanan G.R., McNitt R.P., Application of nonlocal continuum models to nanotechnology,� Int. J. of

Engineering Science, ������41��305–312.

[3] Wang C.M., Zhang Y.Y., Vibration of nonlocal Timoshenko beams, Ramesh S.S., Kitipornchai S., Nanotechnology, 2007,

18 ,1-9.

[4] Elishakoff I., Pentaras D., Buckling of a Double-Walled Carbon Nanotube, Advanced Science Letters, 2009, 2, 372–376.

[5] Ghannadpour S.A.M. and Mohammadi B., Buckling analysis of micro- and nano-rods/tubes based on nonlocal

Timoshenko beam theory using Chebyshev polynomials Advanced Materials Research, 2010, 123-125, 619-622.