Frequency analysis of a double-nanobeam-system

FREQUENCY ANALYSIS OF A DOUBLE-NANOBEAM-SYSTEM

Anita Ciekot, Stanisław Kukla

Institute of Mathematics, Czestochowa University of Technology Częstochowa, Poland

anita.ciekot@im.pcz.pl, stanislaw.kukla@im.pcz.pl

Abstract. In this paper, a problem of transverse free vibration of a double-nanobeam- -system is considered. The nanobeams of the system are coupled by an arbitrary number of translational springs. The solution of the problem by using the Green’s functions proper- ties is obtained. A numerical example is presented.

Keywords: nanobeam system, free vibration, Green’s functions

Introduction

The vibrational behaviour of nanostructures is very important in the design of nanodevices applying in different fields of nanotechnology. The understanding of effect of small scale on vibration of the nanostructures is of great significance to the prediction of the vibrational behaviour of these nanostructures. Investigations of vibrations of the nanostructures, particularly of the nanobeams, are the subject of the papers [1-5].

Nonlocal theories for bending, buckling and vibration of nanobeams have been presented by Reddy in work [1]. The equations of motion of the nanobeam by using the nonlocal differential constitutive relations of Eringen are derived.

The Euler-Bernoulli, Thimoshenko, Reddy and Levinson beam theories were con- sidered. The paper [2] by Aydogdu is devoted to the nonlocal theories of bending, buckling and free vibrations of nanobeams. Besides the theories discussed by Reddy [1], the author presents the Aydogdu beam theory. The vibration of a nonlo- cal double-nanobeam-system is the subject of the paper [3] by Marmu and Adhikari. The nanobeams of the system are connected by distributed transverse springs. The presented investigation shows the small-scale effects in the free vibration of the double-nanobeam-system subjected to an initial compressive prestressed load.

The solution of the problems of free vibration of nanobeams can be obtained by applying methods such as in the classical beam theories. Exact solutions of bend- ing, natural vibration, and buckling of simply supported beams for the considered theories were presented in the papers [1, 2]. Exact solution is also obtained to the vibration problem of nonlocal double-nanobeam-system [3] assuming that the nanobeams of the system are simply supported. Ansari et al. in the paper [4] to

A sketch of the considered system of two nanobeams connected by n-discrete translational springs is shown in Figure 1. The transverse vibrations of the nano- beams are governed by the following equations [1, 2]:

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ^{( )} ( ) ( )

1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 1 1

1

2 2

0 1 1 2 2 1 1 0 1 1 1 1 1 1 1

1

, , , , ,

, , , ,

n

j j j j

j n

j j j j

j

E I w x t N w x t A w x t k w x t w x t x x

e a k w x t w x t x x e a A w x t N w x t

ρ δ

δ ρ

=

=

′′′′ + ′′ + = −  −  −

′′′′ 

 ′′ ′′   ′′

+  −  − +  + 

∑

∑

&&

&&

(1)

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ^{( ) ( ) ( )}

2 2 2 1 2 2 2 2 2 2 2 1 1 2 2 2 2

1

2 2

0 1 1 2 2 2 2 0 2 2 2 2 2 2 1

1

, , , , ,

, , , ,

n

j j j j

j n

j j j j

i

E I w x t N w x t A w x t k w x t w x t x x

e a k w x t w x t x x e a A w x t N w x t

ρ δ

δ ρ

=

=

′′′′ + ′′ + =  −  −

′′′′ 

 ′′ ′′   ′′

−  −  − +  + 

∑

∑

&&

&&

(2) Here w_i denotes the transverse displacement, N_i is the initial axial force, ρ_i is the mass density, E_i is the modulus of elasticity, A_i is the area of cross-section of the i-th nanobeam, δ( )⋅ denotes the Dirac delta function, x₁, x₂ are axial positions along the nanobeams, x_1j, x_2j, j 1, 2...= n are points of the nanobeams which are joined by a j-th spring, e₀ is a constant appropriate to nanobeam material and a is an internal characteristic size. Dots ( )& and primes ( )′ denote partial derivatives with respect to time t and position coordinate x, respectively. When e a = , the ₀ 0 equations (1)-(2) are reduced to equations of classical model of the beams system [3]. The functions w_i( , )x t satisfy the boundary conditions

2 2

2 2

(0, ) (0, ) 0;

( , ) ( , ) 0 1, 2

i i

i

i

i i i

w

w t t

x w

w L t L t i

x

=∂ =

∂

=∂ = =

∂

(3)

Fig. 1. A sketch of the double-nanobeam-system

2. Solution of the problem

In order to find the natural frequencies of the double-nanobeams-system, one assumes a solution of the problem in the form:

(

^,

) ( )

^{cos 1, 2}

i i

w x t =W x ⋅ ωt i= (4)

where ω is the circular frequency. Introducing new variables: _i xⁱ

ξ = L , _i ⁱ

i

W W

= L and W_i( )x_i =L W_{i i}( )ξ_i into equations (1)-(2), after transformation, the following non-dimensional equations are obtained:

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( )

4

1 1 1 1 1 1 1 1 1 2 2 1 1

1

2

1 1 2 2 1 1

1

n

j j j j

j n

j j j j

j

W F W p W p K hW W

h

p K W hW

ξ ξ ξ ξ ξ δ ξ ξ

µ ξ ξ δ ξ ξ

=

=

′′′′ + ′′ − Ω = −  −  −

 ′′ ′′ 

+  −  −

∑

∑

(5)

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( )

4 4 2

2 1 2 2 2 2 2 1 1 2 2 2 2

1

2 2

1 1 2 2 2 2

1

n

j j j j

j n

j j j j

j

W F W qr W q s K hW W

h q s K W hW

ξ ξ ξ ξ ξ δ ξ ξ

µ ξ ξ δ ξ ξ

=

=

′′′′ + ′′ − Ω =  −  −

 ′′ ′′ 

−  −  −

∑

∑

(6)

from equations (3)

(0) (0) 0;

( ) ( ) 0 1, 2

i i

i i i i

W W

W L W L i

= ′′ =

= ′′ = = (7)

The solution of the boundary problem (5)-(7) can be determined by using the Green’s function method [6]. The Green’s functions G_i, which are necessary in this problem, satisfy the differential equation

( ) ( )

4 2

4

4 2

i i

i i i i i i

i i

G G

F λ G ξ δ ξ η

ξ ξ

∂ ∂

+ − = −

∂ ∂ (8)

where λ = Ω₁ ⁴ p and λ = Ω₂ r ⁴ q. Moreover, these functions hold the boundary conditions:

( )

² ₂

0

0, 0

i i

i i

i

G G

ξ

η ξ

=

∂

= =

∂ (9)

(

^,

)

² ₂ ⁰

i i

i

i i i

i L

G

G L

ξ

η ξ

=

∂

= =

∂ (10)

The derivation of the Green’s functions is presented in section 4.

Using the properties of the Green’s functions, the solution of the boundary problem (5)-(7) can be presented in the form [7, 8]:

( ) ( ) ( ) ( )

( ) ( ) ( )

1 1 1 1 2 2 1 1 1

1

2

1 1 2 2 1 1 1

,

,

n

j j j j

j n

j j j j

W p K hW W G

h

p K W hW G

ξ ξ ξ ξ ξ

µ ξ ξ ξ ξ

=

 

= −  − 

 ′′ ′′ 

+  − 

∑

∑

(11)

( ) ( ) ( ) ( )

( ) ( ) ( )

2

2 2 1 1 2 2 2 2 2

1

2 2

1 1 2 2 2 2 2

1

,

,

n

j j j j

j n

j j j j

j

W q s K hW W G

h q s K W hW G

ξ ξ ξ ξ ξ

µ ξ ξ ξ ξ

=

=

 

=  − 

 ′′ ′′ 

−  − 

∑

∑

(12)

Substituting ξ₁=ξ_1i,ξ₂ =ξ_2i, (i=1, 2,... )n into equations (11)-(12) and in the second order derivative of the functions W₁( )ξ₁ and W₂(ξ₂), we obtain a system of equations

( ) ( ) ( ) ( )

( ) ( ) ( )

1 1 1 1 2 2 1 1 1

1

2

1 1 2 2 1 1 1

1

,

,

n

i j j j i j

j n

j j j i j

j

hW p K hW W G

h p K W hW G

ξ ξ ξ ξ ξ

µ ξ ξ ξ ξ

=

=

 

= −  − 

 ′′ ′′ 

+  − 

∑

∑

(13)

( ) ( ) ( ) ( )

( ) ( ) ( )

2

2 2 1 1 2 2 2 2 2

1

2 2

1 1 2 2 2 2 2

1

,

,

n

i j j j i j

j n

j j j i j

j

W q s K hW W G

h q s K W hW G

ξ ξ ξ ξ ξ

µ ξ ξ ξ ξ

=

=

 

=  − 

 ′′ ′′ 

−  − 

∑

∑

(14)

( ) ( ) ( ) ( )

( ) ( ) ( )

1 1 1 1 2 2 1 1 1

1

2

1 1 2 2 1 1 1

1

'' '' ,

'' ,

n

i j j j i j

j n

j j j i j

j

W p K hW W G

h

p K W hW G

ξ ξ ξ ξ ξ

µ ξ ξ ξ ξ

=

=

 

= −  − 

 ′′ ′′ 

+  − 

∑

∑

(15)

( ) ( ) ( ) ( )

( ) ( ) ( )

2

2 2 1 1 2 2 2 2 2

1

2 2 2

1 1 2 2 2 2 2

1

'' '' ,

'' ,

n

i j j j i j

j n

j j j i j

j

hW q h s K hW W G

q h s K W hW G

ξ ξ ξ ξ ξ

µ ξ ξ ξ ξ

=

=

 

=  − 

 ′′ ′′ 

−  − 

∑

∑

(16)

After substracting equations (15) and (16) from equations (13) and (14), respec- tively, we have a system

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

2

1 1 2 2 1 1 2 2 1 1 1 2 2 2

1

2 2

1 1 2 2 1 1 1 2 2 2

1

, ,

, ,

n

i i j j j i j i j

j n

j j j i j i j

j

hW W K hW W p G q s G

h K W hW p G q s G

ξ ξ ξ ξ ξ ξ ξ ξ

µ ξ ξ ξ ξ ξ ξ

=

=

   

− = −  −   + 

 ′′ ′′   

+  −   + 

∑

∑

(17)

( ) ( ) ( ) ( )

{ }

( ) ( ) ( ) ( )

2

1 1 2 2 1 1 2 2

1

2 2 2

1 1, 1 2 2 , 2 1'' 1, 1 2'' 2, 2

n

j j j j j

j

i j i j i j i j

K hW W h W hW

p G q s G h pG q h s G

h

ξ ξ µ ξ ξ

ξ ξ ξ ξ µ ξ ξ ξ ξ

=

   ′′ ′′ 

−  − −  −  ×

 + −  + 

   

 

∑

⁽¹⁹⁾

for i=1, 2,K,n. Assuming

( ) ( )

²

( ) ( )

1 1 2 2 1'' 1 2'' 2

i i i i i

U =hW ξ −W ξ −hµ W ξ −hW ξ 

( )

²

( )

²

( )

^{2 2}

( )

1 1, 1 2 2, 2 1'' 1, 1 2'' 2, 2

i j i j i j i j i j

A =p G ξ ξ +q s G ξ ξ −µ p G ξ ξ +q h s G ξ ξ  we can write the system of equations (19) in the form

1

1, 2, ,

n

i j j i j

j

U K U A i n

=

= −

∑

= ^K

This system of equations can be written in the matrix form

(M+E) U⋅ =0 (20)

where: M= K A_{j i j}, 1≤i j, ≤ , n U=[U₁ U₂ U₃ ... U_n]^T

The non-trivial solutions of equation (20) exist for these Ω , for which the determinant of the matrix (M+E) vanished. This yields the frequency equation

( )

det M E+ =0 (21)

Equation (21) is solved numerically. The roots Ω ,_k k =1, 2,Kof this equation are the nondimensional frequencies of the system.

3. The Green’s functions determination

The Green’s function G( , )ξ η , as a solution of the boundary problem (8)-(10)

1 0

( , ) ( , ) ( , ) ( )

G ξ η =G ξ η +G ξ η ⋅H ξ−η (22)

where H ξ η( − ) is the Heaviside function. It can be shown that both functions G₁ and G₀ satisfy the homogeneous differential equation:

4

0^IV 0 0 0

G −F Gi ′′+ Ω G = (23)

Moreover the function G₀ satisfies the conditions

0 0 0 0

x x x

G G G

ξ ξ ξ

= = ′ = = ′′ = = , G_{0 x} 1

=ξ

′′′ = (24) The solution of the boundary problem for G₀ is

0 2 2

1 1 1

( ) sin ( ) sh ( )

G ξ η α ξ η β ξ η

α β

α β

 

− = −+  − − −  (25)

where ^{α = 1}₂

(

^{Fi2+ Ω −4 4 Fi}

)

^{and β = 1}₂

(

^{Fi2 + Ω +4 4 Fi}

)

. It results that the general solution of differential equation (8) can be written in the form:

1 2 3 4 0

( ) cos sin ch sh ( ) ( )

Gξ −η =C αξ +C αξ+C βξ +C βξ+G ξ−η ⋅H ξ −η (26) The constants C₁,C₂,C₃ and C₄ are determined by using boundary conditions

0 0 0

G G

ξ= = ξξ ξ′′ = = (27)

and

1 1 0

G G

ξ= = ξξ ξ′′ = = (28)

Using the boundary conditions (27) we find C₁ =C₃=0. Therefore the function ( , )

G ξ η has the form

2 4 0

( ) sin sh ( ) ( )

Gξ −η =C αξ+C βξ +G ξ−η ⋅H ξ−η (29) The constantsC₂ and C₄ are determined by using boundary conditions (28). Finally, the Green’s function is given by equation (29) where

2

2 0 0

1( ( ) ( )sh )

C G L G L L

w ′′ ξ β ξ β

= − − −

2

4 0 0

1( ( ) ( )sin )

C G L G L L

w ξ α ξ α

− ′′

= − − −

2 2

( )sin sh

w= α +β αL βL

culated and these are plotted in Figure 2. The computations have been performed by using the Maple package [9].

The figure shows that as the parameter µ increases, the frequencies decrease for all values of spring stiffness considered. For the first frequency Ω the greatest ₁ dependence of the spring stiffness is observed. When the nonlocal effects are ignored (µ = ) the above considerations revert to the classical model of the beam 0 theory (the frequencies Ω , i = 1,…,4 on the _i Ω axis).

Conclusions

The Green function method was applied to solve the problem of transverse vibration of double-nanobeam coupled by translational springs. Simply-supported boundary conditions were employed in this study. It is observed that an increase of the parameter characterized the nanobeams (nanobeam material and internal char- acteristic size) causes a decrease of the frequencies of the nanobeam-system.

Although the number of coupling springs considered in the presented examples was limited to two, the approach can be used to solve the problems of vibration of systems consisting of many nanobeams and coupling springs.

References

[1] Reddy J.N., Nonlocal theories for bending, buckling and vibration of beams, International Journal of Engineering Science 2007, 45, 288-307.

[2] Aydogdu M., A general nonlocal beam theory: its application to nanobeam bending, buckling and vibration, Physica E 2009, 41, 1651-1655.

[3] Murmu T., Adhikari S., Nonlocal effects in the longitudinal vibration of double-nanorod systems, Physica E 2012, 43, 415-422.

[4] Ansari R., Hisseini K., Darvizeh A., Daneshian B., A sixth-order compact finite difference method for non-classical vibration analysis of nanobeams including surface stress effects, Applied Mathematics and Computation 2013, 219, 4977-4991.

[5] Eltaher M.A., Alshorbagy A.E., Mahmoud F.F., Vibration analysis of Euler-Bernoulli nanobeams by using finite element method, Applied Mathematical Modelling 2013, 37(7), 4787-4797.

[6] Duffy D.G., Green’s Functions with Applications, Chapman & Hall/CRC, 2001.

[7] Kukla S., Funkcje Greena i ich zastosowania, Wydawnictwo Politechniki Częstochowskiej, Częstochowa 2009.

[8] Ciekot A., Kukla S., Free longitudinal vibrations of nanorods system, Journal of Applied Mathe- matics and Computational Mechanics 2013, 2(12), 15-22.

[9] Richards D., Advanced Mathematical Methods with Maple, Cambridge University Press, 2009.