Civil Engineering Reference
In-Depth Information
where
Z
s
(ω)
=
jωm
1
−
Dk
t
ω
2
m
is the mechanical impedance of the panel and
k
t
k
sin
θ
with
k
the wave number in the
free air.
σ
33
(M)
and
σ
33
(M
)
are respectively, the normal stresses just in front and just
behind the plate and
v
3
is the normal velocity of the thin plate
v
3
=
=
v
3
(M
)
.
v
3
(M)
=
=
[
σ
33
(M)v
3
(M)
]
T
to express the mechanical field in a point
M
of the plate, the transfer matrix [T
i
] relating
V
(M)
and
V
(M
)
is deduced directly from
Equation (11.56)
Using the vector
V
(M)
1
−
Z
s
(ω)
[
T
]
=
(11.57)
0
1
Note that the mechanical impedance of the panel can be written in the equivalent
form:
jωm
1
sin
4
θ
ω
ω
c
2
Z
s
(ω)
=
−
(11.58)
where
c
2
m
D
ω
c
=
is the panel's critical frequency. Damping in the panel can be accounted for by using a
complex value of the Young's modulus.
11.3.6 Impervious screens
Impervious screens are usually used to cover or protect acoustic materials. Their mod-
elling using the transfer matrix method depends on their mounting. When they are free
to move, that is when there is air on both sides of the screen, they can be simply mod-
elled as a thin plate with negligible stiffness. The mechanical impedance in Equation
(11.57) reduces to
Z(ω)
jωm
. When the screen is bonded onto a porous material, the
modelling needs to account for the interface forces.
Let
M
and
M
two points close, respectively, to the forward and the backward face of
a screen in mechanical contact. Assuming the screen to be flexible with a non-negligible
bending stiffness, Newton's law applied to the screen, leads to
=
jωmυ
3
(M
)
=
σ
33
(M
)
−
σ
33
(M)
−
D
∂
4
υ
3
(M
)
jω∂x
1
(11.59)
S
∂
2
υ
1
(M
)
jω∂x
1
jωmυ
1
(M
)
σ
13
(M
)
σ
13
(M)
=
−
+
(11.60)
In these equations,
v
1
and
v
3
are the
x
1
and
x
3
components of the velocity at point
M
,
respectively.
σ
33
and
σ
13
are the normal and tangential stresses at point
M
. The quantities
