Civil Engineering Reference
In-Depth Information
1
2
2
1
⎡
⎤
2
⎛
n
π
⎞
⎛
n
π
⎞
2
2
x
z
⎡
⎤
k
=
⎢
+
⎥
=
k
+
k
2
.
(6.44)
⎜
⎟
⎜
⎟
⎣
⎦
nn
n
n
a
b
xz
⎢
⎥
x
z
⎝
⎠
⎝
⎠
⎣
⎦
Hence, the corresponding eigenfrequencies are given by
⎡
2
2
⎤
n
n
π
B
⎛⎞
⎛
⎞
⎢
x
z
fn n
(, )
=
+
⎥
,
(6.45)
⎜⎟⎜⎟
⎢
xy
2
ma
b
⎥
⎝⎠
⎝⎠
⎣
⎦
where
B
and
m
again is the bending stiffness per unit length and mass per unit area,
respectively.
pr j
θ
(, , )
y
r
z
ϕ
θ
a
b
x
Figure 6.12
Sketch for calculating sound radiation from a finite size plate.
Before we perform the calculation of the radiated power, we shall give some
qualitative comments on the situation. In the same manner as for the infinite plate, the
relationship between the wave number in the plate and the wave number in the
surrounding medium must be the determining factor for the radiation. In this case,
however, we have two partial wave numbers (or partial wavelengths) to consider.
Commonly, these are divided into three groups:
a) Surface modes
kk
,
<
k
λ λλ
(
,
>
)
nn
nn
x
z
x
z
b) Edge modes
<<
k
>>
k
k
or
k
>>
k
k
(
λ
<<
λλ λ λλ
or
)
n
n
n
n
n
n
n
n
x
z
z
x
x
z
z
x
c) Corner modes
kk
,
>
k
λ λλ
(
,
<
)
nn
nn
x
z
x
z
The reason for these terms should be evident from
Figure 6.13,
with sketches
showing a simply supported plate vibrating in a given corner mode and an edge mode,
respectively. The modal pattern is for simplicity indicated by alternating signs; the left-
hand sketch is equivalent to the one shown in
Figure 6.12
having modal numbers (5, 4).
