is a piezoelectrically stiffened elastic constant. Using Eq. (3.151) 2 to eliminate the
potential, we write Eq. (3.151) 1 as
b c 33 u 3,33 ¼ ρ€ u 3 :
ð3:157Þ
The problem for u 3 consists of Eq. (3.157) and the following initial and boundary
conditions:
u 3 ¼ 0, x 3 ¼ 0,
b c 33 u 3,3 ¼ f , x 3 ¼ L,
ð3:158Þ
u 3 ¼ 0, _
u 3 ¼ 0, t ¼ 0:
ð3:159Þ
To homogenize the boundary conditions in Eq. (3.158), we let
u 3 ¼ u þ
f
b c 33
x 3 :
ð3:160Þ
The problem for u is
b c 33 u ,33 ¼ ρ € u,
ð3:161Þ
u ¼ 0, x 3 ¼ 0,
b c 33 u ,3 ¼ 0, x 3 ¼ L,
ð3:162Þ
u ¼ À
f
b c 33
x 3 , _
u ¼ 0, t ¼ 0:
ð3:163Þ
Mathematically, Eqs. (3.161), (3.162), and (3.163) are a rather standard problem. Its
solution can be obtained in a straightforward manner by the method of separation of
variables as
u x 3 , t
ð
Þ ¼
X 1
m¼0
A m cos
π þ 2mπ
2L
ct
sin
π þ 2mπ
2L
x 3
,
ð3:164Þ
where
A m ¼
2
L
Z L
0
À
f
b c 33
x 3 sin
π þ 2mπ
2L
x 3
dx 3 , m ¼ 0, 1, 2, Á Á Á,
c ¼
ffiffiffiffiffiffi
b c 33
ρ
r
:
ð3:165Þ
Once u is obtained, u 3 and φ are obtained from Eqs. (3.160) and (3.154), respectively. For uniqueness, we let φ ¼ 0 at x 3 ¼ 0. As an example, consider a ZnO rod
with L ¼ 600 nm and cross-sectional area A ¼ 2.598 Â 10
À14 m
2 . The end force
3.12 Transient Vibration
69