cos βL Á cosh βL þ 1 ¼ 0,
ð4:73Þ
and
ω
m
ð Þ
2 ¼
c 33 I
ρA
β
m
ð Þ
4 :
ð4:74Þ
In Eq. (4.71), A
(m) is determined from the initial conditions.
With v(x 3 , t) known, we make another approximation by neglecting n
(1) in
Eq. (4.64) and obtain the following boundary-value problem for ϕ
(1) :
ϕ
1
ð Þ
,33 À λ
2
ϕ
1
ð Þ
¼ À
e 33
ε 33
v ,333 ,
D
1
ð Þ
3 ¼ 0, x 3 ¼ 0, L,
ð4:75Þ
where
λ
2
¼
ε 11 A
ε 33 I
:
ð4:76Þ
The general solution to Eq. (4.75) 1 can be formally written as
ϕ
1
ð Þ
¼ C 1 t
ð Þ exp λx 3
ð ÞþC 2 t
ð Þ exp Àλx 3
ð
Þþϕ
p x 3 , t
ð
Þ,
ð4:77Þ
where C 1 (t) and C 2 (t) are determined from the boundary conditions in Eq. (4.75) 2 .
ϕ
p (x 3 , t) is a particular solution to Eq. (4.75) 1 in the form of a linear combination of
cosβ
(m)
x 3 , sinβ
(m) x 3 , coshβ
(m)
x 3 , and sinhβ
(m) x 3 . The coefficients of the linear combination depend on cosω
(m)
t. The expressions of C 1 (t), C 2 (t), and ϕ
p (x 3 , t) are
lengthy and are not presented here.
With ϕ
(1) (x, t) known, from Eqs. (4.65), (4.66), and (4.67), we have the following
initial-boundary-value problem for n
(1) :
Àn 0 μ
n
33 Iϕ
1
ð Þ
,33 þ D
n
33 In
1
ð Þ
,33 þ n 0 μ
n
11 Aϕ
1
ð Þ
À D
n
11 An
1
ð Þ
¼ I _
n
1
ð Þ ,
J
n 1
ð Þ
3
¼ 0, x 3 ¼ 0, L,
n
1
ð Þ
¼ 0, t ¼ 0:
ð4:78Þ
To make the boundary conditions in Eq. (4.78) homogeneous, we let
n
1
ð Þ x, t
ð Þ ¼ b n x, t
ð Þ þ
n 0 μ
n
33
D
n
33
ϕ
1
ð Þ x, t
ð Þ:
ð4:79Þ
The mathematical problem for b n is
4.5 Transient Bending
105