Civil Engineering Reference
In-Depth Information
point source excitations have been presented in Chapters 7 and 8 for isotropic materials
and Chapter 10 for transversally isotropic materials. Here, we limit the presentation to
the prediction of the response of an elastic panel with an attached layered noise control
treatment with a mechanical load with random location (rain on the roof). The panel
is assumed to be embedded in a rigid baffle separating two semi-infinite fluids. More-
over, the mechanical excitation and the response of the structure are assumed harmonic.
Following the methods of Chapters 7 and 8, the load
f(x,y)
can be represented by a
superposition of plane waves, using a two-dimensional Fourier transform
$
+∞
+∞
1
4
π
2
f(x,y)
=
F(ξ
1
,ξ
2
)
exp[
−
j(ξ
1
x
+
ξ
2
y
]d
ξ
1
d
ξ
2
−∞
−∞
(12.44)
+∞
+∞
%
F(ξ
1
,ξ
2
)
=
f(x,y)
exp[
j(ξ
1
x
+
ξ
2
y)
]d
x
d
y
−∞
−∞
For each wave number component
(ξ
1
,ξ
2
)
, the transfer matrix method (Chapter 11)
is first used to solve for the various vibration and acoustic indicators of the system
(pressure, velocity, surface impedance, reflection coefficient, radiated power,
...)
.Next,
the first equation in Equation (12.44) is used to calculate the global response of the
structure. For example, consider a baffled rectangular panel (dimension,
L
x
x
L
y
;area,
S
L
x
L
y
)
with an attached layered material. The space averaged quadratic velocity of
the excited panel takes the form
=
L
x
L
y
υ
2
=
1
2
S
2
d
x
d
y
|
υ(x,y)
|
(12.45)
0
0
Since the panel is baffled, this equation can be written:
∞
∞
−∞
|
υ
2
=
1
2
S
2
d
x
d
y
υ(x,y)
|
(12.46)
−∞
with
S
=
L
x
L
y
the panel's surface. Using the first equation in Equation (12.44), one
obtains
∞
∞
υ
2
=
1
8
π
2
S
υ(x,y)
−∞
−∞
(12.47)
∞
∞
ξ
2
y
]d
ξ
1
d
ξ
2
d
x
d
y
υ
∗
(ξ
1
,ξ
2
)
exp[
j(ξ
1
x
+
−∞
−∞
Permutation of the order of integration and use of the second equation in Equation
(12.44), leads to
+∞
+∞
υ
2
=
F
2
8
π
2
S
2
d
ξ
1
d
ξ
2
|
υ(ξ
1
,ξ
2
)
|
(12.48)
−∞
−∞
with
F
2
the excitation power spectrum and
υ(ξ
1
,ξ
2
)
is approximated by the plane wave
normal (unit) velocity of the equivalent infinite extent panel with the layered material.
