FREE LONGITUDINALVIBRATION OF A DOUBLE-NANOROD SYSTEM
Anita Ciekot, Stanisław Kukla
Institute of Mathematics, Czestochowa University of Technology Czestochowa, Poland
anita.ciekot@im.pcz.pl, stanislaw.kukla@im.pcz.pl
Abstract. In this paper a solution to the problem of the free longitudinal vibration of a double-nanorod-system (DNRS) is presented. The nanorods of the system are coupled by many translational springs. The clamped-clamped and clamped-free boundary conditions are employed. The problem of vibration is solved by using the Green’s function method.
The natural frequencies were numerically calculated.
Keywords: double-nanorod system, longitudinalvibration, Green’s function method
Introduction
The theory of nonlocal elasticity is often used for the analysis of vibration and instability of nanostructures like: nanorods, nanotubes, nanobeams, etc. This theory was introduced to nanotechnology by Peddieson et al. [1]. The vibration analysis of nanostructures has been of great interest because of their promising mechanical, chemical, electrical, optical properties and their applications, for example in nanoelectromechanical, nanodevices and nanooptomechanical systems.
The forced axial vibrations of nanorods are induced by the axial external forces.
The frequencies of the longitudinal free vibration of nanorods system are important parameters which characterize the behavior of this nanorod during the enforced vibration. The free vibrations of the complex nanorods system were studied in papers [2] by Marmu and Adhikari. The authors present an investigation on the longitudinal vibration of the two nanorods which are coupled by longitudinally directed distributed springs. The nonlocal frequencies of vibration by using an ana- lytical method have been derived. The study was an inspiration for the authors of the present paper to investigate the free vibration of a double-nanorod-system.
The consideration deals with the vibration of nanorods coupled by longitudinal directed discrete springs. In order to solve the vibration problem, the Green’s func- tion method is applied [3]. The problem of free vibration to the similar system as a classical model of a double-rod-system has been presented in reference [4].
1. Formulation of the problem
The system of two nanorods which are coupled by longitudinally directed n-discrete springs is considered. The equations of motion for the longitudinal vibra- tion of the nanorods can be written in the form [1, 2]:
( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( )
( ) ( )
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
2 2 2 2 2 2 2 2 1 1 2 2 2 2
1
2
0 1 1
, , , ,
, , ,
, , , ,
,
n
j j j j
j n
j j j j
j
n
j j j j
j
j j
E A u x t A u x t k u x t u x t x x
e a k u x t u x t x x e a A u x t
E A u x t A u x t k u x t u x t x x
e a k u x t u
ρ δ
δ ρ
ρ δ
=
=
=
′′ − = − −
′′ ′′ ′′
− − − −
′′ − = − − −
′′ ′′
+ −
∑
∑
∑
&&
&&
&&
( ) ( ) ( )2 ( )
2 2 2 2 0 2 2 2 2
1
, ,
n
j j
i
x t δ x x e a ρ A u x t
=
− − ′′
∑
&&(1)
where: ui( , )x t is the axial displacement, ρi( )x is the mass density, Ei( )x is the modulus of elasticity, A xi( )is the area of cross-section of the i-th nanorod, x1, x2 are axial positions along the nanorods, x1j, x2j, j=1, 2...n are points of the nano- rods which are joined by a j-th spring, e0 is a constant appropriate to nanorods material and a is an internal characteristic size. When e a0 =0, the equations (1) are reduced to equations of classical model of the rods system [3]. The functions
( , )
ui x t satisfies the boundary conditions
(0, ) ∂ ( , ) 0; 1, 2
= = =
∂
i
i i
u
u t L t i
x (2)
Fig. 1. Double nanorod configuration: clamped-free boundary condition
2. Solution of the problem
In order to find the natural frequencies of the double-nanorods system, one assumes a solution of the problem in the form:
(
,) ( )
cos 1, 2i i
u x t =U x ⋅ ωt i= (3)
where ω is the circular frequency. Introducing new variable i xi
ξ = L into equations (1), the following equations are obtained:
( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( )
2 2 2
1 1 1 1 1 1 2 2 1 1
1
2
1 1 2 2 1 1
1
2 2 2 2 2 2
2 2 2 2 1 1 2 2 2 2
1
2 2
1 1 2 2 2 2
1
1
1
n
j j j j
j n
j j j j
j
n
j j j j
j n
j j j j
j
U U K U U
K U U
r U r U a K U U
a K U U
µ ξ ξ ξ ξ δ ξ ξ
µ ξ ξ δ ξ ξ
µ ξ ξ ξ ξ δ ξ ξ
µ ξ ξ δ ξ ξ
=
=
=
=
− Ω ′′ + Ω = − −
′′ ′′
− − −
− Ω ′′ + Ω = − − −
′′ ′′
+ − −
∑
∑
∑
∑
(4)
where:
2
2 1 1 1 2
1 1
A L E A
ρ ω
Ω = , 1
1 1 j j
k L K
=E A , 2 1 1 2
2 2 1
A E L a
A E L
= ,
2
2 2 1 2
2
1 2 1
r E L E L ρ
= ρ , 0
1
e a µ = L
The functions U1and U2satisfy the boundary conditions which are obtained from equations (2)-(3)
(0)= ′(1)=0; =1, 2
i i
U U i (5)
The solution of the boundary problem (4)-(5) can be expressed with the aid of Green’s function and has the form:
( ) ( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( ) ( )
( ) ( ) ( )
1 1 1 1 2 2 1 1 1
1
2
1 1 2 2 1 1 1
1
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 n
j j j j
j n
j j j j
j
U K U U G
K U U G
U a K U U G
a K U U G
ξ ξ ξ ξ ξ
µ ξ ξ ξ ξ
ξ ξ ξ ξ ξ
µ ξ ξ ξ ξ
=
=
=
=
= − +
′′ ′′
− −
= − −
′′ ′′
+ −
∑
∑
∑
∑
(6)
Assuming ξ1=ξ1i, ξ2=ξ2i, (i=1, 2,... )n in the equations (6) and in the second order derivative of the functionsU1( )ξ1 andU2(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 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 n
i j j j i j
j n
j j j i j
j
U K U U G
K U U G
U a K U U G
a K U U G
ξ ξ ξ ξ ξ
µ ξ ξ ξ ξ
ξ ξ ξ ξ ξ
µ ξ ξ ξ ξ
=
=
=
=
= − +
′′ ′′
− −
= − −
′′ ′′
+ −
∑
∑
∑
∑
(7)
( ) ( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( ) ( )
( ) ( )
2 1
1 1 1 1 2 2 2 1 1
1 1
2
2 1
1 1 2 2 2 1 1
1 1
2
2 2
2 2 1 1 2 2 2 2 2
1 2
2
2 2 2
1 1 2 2
1
,
,
,
n
i j j j i j
j n
j j j i j
j n
i j j j i j
j n
j j j
j
U K U U G
G
K U U
U a K U U G
a K U U G
ξ ξ ξ ξ ξ
ξ
µ ξ ξ ξ ξ
ξ
ξ ξ ξ ξ ξ
ξ
µ ξ ξ
=
=
=
=
∂
′′ = − ∂ +
′′ ′′ ∂
− − ∂
∂
′′ = − − ∂
′′ ′′ ∂
+ − ∂
∑
∑
∑
∑
2(
2 2)
2
i, j
ξ ξ ξ
(8)
After substracting of the equations (7) and the equations (8) we have a system
2 1
2 1
( )
( ) ; 1, 2,...
n
i j j j ij
j n
i j j j ij
j
V K V W A
W K V W B i n
µ µ
=
=
= −
= − =
∑
∑
(9)
where: Vi=U1
( )
ξ1i −U2( )
ξ2i , Wi=U1′′( )
ξ1i −U2′′( )
ξ2i( )
2( )
1 ξ ξ1, 1 2 ξ2,ξ2
= +
ij i j i j
A G a G ; 2 21
(
1 1)
2 2 22(
2 2)
1 2
, ,
ξ ξ ξ ξ
ξ ξ
∂ ∂
= +
∂ ∂
ij i j i j
i i
G G
B a
The equation system (9) can be written in the matrix form
⋅ =
D Z 0 (10)
where: Z=[V V1 2 V3 ... Vn W W W1 2 3 ...Wn]T;
2
2
µ µ
−
=
A-E A
D B B-E
1 11 1
1 21 2
1 1
...
...
A
=
M M M
K
n n
n n
n n nn
K A K A
K A K A
K A K A
,
1 11 1
1 21 2
1 1
...
...
B
=
M M M
K
n n
n n
n n nn
K B K B
K B K B
K B K B
The non-trivial solutions of equation (10) exist if and only if the determinant of matrix D is zero
detD = 0 (11)
The roots Ω of equation (11) are called natural frequencies and can be determined k numerically.
3. The Green’s function determination
The Green’s function G1( ,ξ ξ1 1j) is a solution of the following boundary problem [3]:
2 2 2
1, 1 1 1
(1−µ Ω )G ξξ + Ω G =δ ξ( −η ) (12a)
1 1
1 0 1 1 0
ξ= = ξ= =
G G (12b)
We search the function G1 in the form
1( ,1 1) 10( ,1 1) 11( ,1 1) ( 1 1)
G ξ η =G ξ η +G ξ η ⋅H ξ −η (13)
where H(ξ1−η1) is the Heaviside function.
It can be shown that both functions G10 and G11 satisfy the homogeneous dif- ferential equation:
1 1
2 2 2
1 , 1
(1−µ Ω )Glξ ξ + ΩGl=0; l=1, 2 (14) Moreover, the function G11satisfies the conditions
1 1 1
11 1 1
11, 2 2
( , ) 0 1
ξ ξ η 1 η η
= µ
=
= − Ω
G
G (15)
The solution of the boundary problem for G11is
1 1 1 1
11 1 1 2 2 1 1 2 2 1 1
1 1
sin cos
( , ) cos sin
(1 ) (1 )
G ν η ν η
ξ η ν ξ ν ξ
ν µ ν µ
= − +
− Ω − Ω (16)
where
2 2
1 2 2
ν 1
µ
= Ω
− Ω . It results that the general solution of differential equation (12) can be written in the form:
1( ,1 1) 1cos 1 1 2sin 1 1 11( ,1 1) ( 1 1)
G ξ η =C ν ξ +C ν ξ +G ξ η ⋅H ξ −η (17)
The constants C1 and C2 are determined from boundary conditions (12b). Finally we have
[
]
1 1
1 1 1 2 2 1 1 1
1 1 1
1 1 1 1 1
sin
( , ) 1 sin ( )
(1 ) sin
sin( ( )) ( )
ξ η ν η ν ξ
ν µ ν
ν ξ η ξ η
= − − ⋅ +
− Ω
+ − −
G L
L H
(18)
The Green’s function G2 we find by replacing Ω by 2 r2Ω2 in Eq. (18):
[
]
2 2
2 2 2 2 2 2 2 2 2
2 2 2
2 2 2 2 2
sin
( , ) 1 sin ( )
(1 ) sin
sin( ( )) ( )
G L
r L
H
ξ η ν η ν ξ
ν µ ν
ν ξ η ξ η
= − − ⋅ +
− Ω
+ − −
where
2 2
2
2 2 2 2
1 r r ν
µ
= Ω
− Ω .
The Green’s functions G1 and G2 corresponding to a nanorod clamped at the left end and free at the right end (
1 1 1
1
1 0 0
L
G
G ξ ξ
ξ
= =
=∂ =
∂ ) are:
[
] [
]
1 1
1 1 1 2 2 1 1 1
1 1 1
1 1 1 1 1
2 2
2 2 2 2 2 2 2 2 2
2 2 2
2 2 2 2 2
sin
( , ) 1 cos ( )
(1 ) cos
sin( ( )) ( )
sin
( , ) 1 cos ( )
(1 ) cos
sin( ( )) ( )
G L
L H
G L
r L
H
ξ η ν η ν ξ
ν µ ν
ν ξ η ξ η
ξ η ν η ν ξ
ν µ ν
ν ξ η ξ η
= − − ⋅ +
− Ω
+ − −
= − − ⋅ +
− Ω
+ − −
(19)
3. Numerical example
Numerical results have been obtained for a system of two nanorods of identical length and the same physical properties. The system consists of clamped-free nanorods whose free ends are connected by one longitudinally directed spring.
Four different values of a spring stiffness coefficient in computation were assumed:
1 0.1; 1; 10; 100
K = . For such a system four dimensionless natural vibration frequencies as functions of parameter µ were calculated and these are plotted in Figure 2. The computations have been performed by using the package Maple [5].
Fig. 2. The first four dimensionless natural vibration frequencies as functions of µ µ
0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 Ω4
2 4 6 8 10 12
µ
0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 Ω3
1 2 3 4 5 6 7 8 9
µ
0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 Ω2
1 2 3 4 5 6
µ
0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 Ω1
1 1,5 2 2,5 3
The figure shows that as the parameter µ increases, the frequencies decrease for all spring stiffnesses considered. The frequencies
Ωnobtained for µ =0corre- spond to the classical model of the rod system.
Conclusions
The Green function method was applied to solve the problem of longitudinal vibration of a double-nanorod coupled by translational springs. Clamped-clamped and clamped-free boundary conditions were employed. Although the number of coupling springs considered in the presented examples was limited to one, the approach can be used to solve the problems of vibration of systems consisting of many nanorods and coupling springs.
References
[1] Peddieson J., Buchanan G. R., McNitt R. P., Application of nonlocal continuum models to nano- technology, International Journal of Engineering Science 2003, 41, 305-312.
[2] Murmu T., Adhikari S., Nonlocal effects in the longitudinal vibration of double-nanorod systems, Physica E 2012, 43, 415-422.
[3] Kukla S., Funkcje Greena i ich zastosowania, The Publishing Office of Czestochowa University of Technology, Częstochowa 2009.
[4] Kukla S., Przybylski J., Tomski L., Longitudinal vibration of rods coupled by translational springs, Journal of Sound and Vibration 1995, 185(4), 717-722.
[5] Richards D., Advanced Mathematical Methods with Maple, Cambridge University Press, 2009.
