54
V. Marinca and N. Herisanu
the von Karaman nonlinearity for mid-plane stretching, the equation of motion that
governs the transverse deflection W (x, t) of nanobeam, subjected to the Casimir force
is as follow [8, 16]:
ρbh
∂ W (x, t)
dt 2
+ E I
∂
4 W (x, t)
∂t 4
−
⎡
⎣ N +
Ebh
2l
l
0
∂ W (x, t)
∂ x
2
dx
⎤
⎦ ∂
2 W (x, t)
∂ x 2
− F c = 0
( 2 )
where N is the axial loading. The kinematic boundary conditions for the nanobeam
deflection of the double-clamped case are
W (0, t) = 0,
∂ W (0, t)
∂ x
= 0, W (l, t) = 0,
∂ W (l, t)
∂ x
= 0
( 3 )
and the initial conditions are
W (x, 0) = 0,
∂ W (x, 0)
∂t
= 0
( 4 )
The following dimensionless variables are utilized
ˆ
x =
x
l
, ˆ
t = t
E I
ρbkl 4 , ˆ
W =
W
d 0
ˆ
N =
12l
2
Ebk 3 N , α = 6
d 0
k
2
, λ 4 =
12l
4
π
2
c
240Ek 3 d
5
0
(5)
Using (2), the dimensionless equation of motion, dropping the hats, can be
expressed as
¨
W + W
(I V )
−
⎡
⎣ α
1
0
W
2 dx + N
⎤
⎦ W
−
λ 4
(1 − W ) 4 = 0
( 6 )
The boundary conditions (3) become
W (0, t) = 0 , W
(0, t) = 0 , W (1, t) = 0 , W
(1, t) = 0
(7)
where dot and prime denote derivative with respect to t and x, respectively.
By multiplying both sides of (6) by (1 − W )
4 , we obtain
(1 − W )
4
+ (1 − W )
4 W
(I V )
− (1 − W )
4
⎡
⎣ α
1
0
W
2 dx + N
⎤
⎦ W
− λ 4 = 0 (8)