where B
(1) and B
(2) are undetermined constants. B
(2) represents an arbitrary constant
in the potential which does not affect the electric field and is immaterial. Therefore it
will be dropped in the following. Denoting the four other roots of Eq. (3.139) by ζ
(3)
through ζ
(6) , we write the general solution of Eq. (3.136) as
u 3
φ
Δn
8
> <
> :
9
> =
> ;
¼ B
1
ð Þ
0
x 3
0
8
> <
> :
9
> =
> ;
þ
X 6
m¼3
B
m
ð Þ
α
m
ð Þ
1
β
m
ð Þ
8
> <
> :
9
> =
> ;
exp ζ
m
ð Þ x 3 ,
ð3:141Þ
where B
(3) through B
(6) are undetermined constants, and
α
m
ð Þ
¼ À
e 33 ζ
m
ð Þ
À
Á 2
c 33 ζ
m
ð Þ
À
Á 2 þ ρω 2
,
β
m
ð Þ
¼
n 0 μ
n
33 ζ
m
ð Þ
À
Á 2
D
n
33 ζ
m
ð Þ
À
Á 2 À iω
, m ¼ 3, 4, 5, 6:
ð3:142Þ
The boundary conditions are
u 3 0
ð Þ ¼ 0, T 3 L
ð Þ ¼
F
S
exp iω t
ð Þ,
ð3:143Þ
D 3 0
ð Þ ¼ 0, D 3 L
ð Þ ¼ 0:
ð3:144Þ
We also have the following circuit equation:
V
Z
¼ J
n L
ð ÞS,
ð3:145Þ
where V ¼ φ(L ) À φ(0) is the piezoelectrically generated output voltage on Z. The
substitution of Eq. (3.141) into Eqs. (3.143), (3.144), and (3.145) results in five
linear equations for B
(1) and B
(3) through B
(6) . They are solved on a computer using
MATLAB.
As a numerical example, consider a ZnO rod whose geometric parameters are that
the length L ¼ 600 nm, radius a ¼ 25 nm, and cross-sectional area
S ¼ πa
2
¼ 1.9625 Â 10
À15 m
2 . The magnitude of the end force F ¼ 0.5 nN. The
reference carrier concentration n 0 ¼ N
þ
D ¼ 10
22 m
‐3 . For the mechanically forced
vibration we are considering, to introduce some mechanical damping into the
structure, we use a complex elastic constant c 33 1 þ iQ
À1
À
Á
in our numerical calculation which can describe viscous damping in time-harmonic motions. Q is the
material quality factor. For convenience we introduce
3.11 Harmonic Vibration
63