Civil Engineering Reference
In-Depth Information
previous section, one possible outcome is to combine FEM and SEA to handle, approxi-
mately the energy exchange (coupled loss factor) between a trimmed structure (modelled
using finite element) and a fluid sub-system (modelled using SEA).
Appendix 12.A: An algorithm to evaluate the geometrical
radiation impedance
Start from the expression of the geometrical radiation impedance:
L
x
Ly
L
x
L
y
exp[
−
jk
t
(x
cos
ϕ
+
y
sin
ϕ)
]
G(x,y,x
,y
)
exp[
jk
t
(x
cos
ϕ
+
y
sin
ϕ)
]
d
x
d
y
d
x
d
y
jρ
0
ω
S
Z
=
0
0
0
0
(12.A.1)
where
k
t
k
0
sin
θ,k
0
is the acoustic wave number,
ρ
0
is the density of the fluid,
L
x
and
L
y
are respectively the length and the width of the structure.
Using the change of variables
α
=
=
2
y
L
y
2
x
L
x
,β
=
(α
β
)
2
1
2
L
x
2
1
r
2
(β
α
)
2
R
=
−
+
−
(12.A.2)
where
r
is defined by :
r
=
L
x
/
L
y
,and:
2
2
2
2
jρ
0
ω
L
y
16
π
F
n
(α,β,α
,β
)
K
(α,β,α
,β
)
d
α
d
β
d
α
d
β
Z
=
(12.A.3)
0
0
0
0
with
exp[
−
jk
0
R
]
K
(α,β,α
,β
)
=
(12.A.4)
(α
1
/
2
β
)
2
r
2
(β
−
−
α
)
2
+
and
exp
(α
β
)
sin
ϕ
j
k
t
L
x
2
1
r
(β
α
)
cos
ϕ
F
n
(α,β)
=
−
−
+
−
(12.A.5)
To reduce the order of integration, the following change of variables is used
u
=
α
−
α
v
=
α
u
=
β
−
β
v
=
β
and
(12.A.6)
Considering the variable
α
(the same formula can be written for
β
), a symbolic form
of this change of variable is
d
u
d
u
2
2
2
2
−
u
0
0
d
α
d
α
→
d
v
+
d
v
(12.A.7)
0
0
0
0
−
2
−
u
