6- 45
Problems and Solutions Section 6.6 (6.48 through 6.52)
6.48
Calculate the natural frequencies of the membrane of Example 6.6.1 for the case
that one edge
x
= 1 is free.
Solution:
The equation for a square membrane is
w
w
w
tt
yy
tt
+
=
ρ
τ
with boundary condition given by
w
(0,
y
) = 0,
w
x
(
l,y
) = 0,
w
(
x
,0) = 0,
w
(
x,l
) = 0.
Assume separation of variables
w
=
X
(
x
)
Y
(
y
)
q
(
t
) which yields
′′
+
′′
=
= −
=
X
X
Y
Y
c
q
q
c
1
2
2
˙˙
/
ω
ρ τ
where
Then
˙˙
q
c
q
+
=
2
2
0
ω
is the temporal equation and
′′
= −
−
′′
= −
X
X
Y
Y
ω
α
2
2
yields
′′ +
=
′′ +
=
X
X
Y
Y
α
γ
2
2
0
0
as the spatial equation where
γ
2
=
ω
2
–
α
2
and
ω
2
=
α
2
+
γ
2
.
The separated
boundary conditions are
X
(0) = 0,
′
=
X
l
( )
0
and
Y
(0) =
Y
(
l
) = 0.
These yield
X
A
x
B
x
B
A
l
l
n
n
l
n
n
=
+
=
=
=
−
=
−
sin
cos
cos
(
)
(
)
α
α
α
α
π
α
π
0
0
2
1
2
2
1
2

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