Civil Engineering Reference
In-Depth Information
m,D
and
S
are the mass per unit area, the bending stiffness, and the membrane stiffness
of the screen, respectively.
The operators
∂
2
/∂x
1
and
∂
4
/∂x
1
can be replaced by
−
k
t
and
k
t
, respectively. In
consequence, the screen governing equations read:
jωm
1
−
mω
2
υ
3
(M
)
Dk
t
σ
33
(M
)
σ
33
(M)
=
−
(11.61)
jωm
1
mω
2
υ
1
(M
)
Sk
t
σ
13
(M
)
σ
13
(M)
−
=
−
(11.62)
(M)
is the same on both faces. Using the vector
V
s
(M)
=
[
v
1
(M) v
3
(M)
σ
33
(M) σ
13
(M)]
T
to describe the screen variables, Equations (11.61)
and (11.62) with equality of the velocity vector at
M
and
M
,leadtothefollowing4
The velocity of the screen
υ
×
4
transfer matrix for the screen:
1
0
0
0
0
1
0
0
[
T
]
=
(11.63)
−
Z
s
(ω)
10
0
Z
s
(ω)
−
0
0
1
Sk
t
/mω
2
)
.
The interface conditions described in Section 11.4.2 for a solid layer can be used
when the screen is in contact with a porous or fluid medium.
In the case of a negligible stiffness impervious screen, the latter can be modelled as
a flexible membrane. In this case Equation (11.59) is replaced by
Dk
t
/mω
2
)
and
Z
s
(ω)
where
Z
s
(ω)
=
jωm(
1
−
=
jωm(
1
−
I
∂
2
υ
3
(M
)
jω∂x
1
jωmυ
3
(M
)
σ
33
(M
)
σ
33
(M)
=
−
+
(11.64)
and in consequence, impedance
Z
in Equation (11.62) is replaced by
Z
s
(ω)
=
jωm(
1
−
Ik
t
/mω
2
)
where
I
is the tension of the screen.
11.3.7 Porous screens and perforated plates
For microporous screens a more refined model must be used to account for the added
resistance (proportional to the flow resistance of the screen). For a perforated plate, there
is also an important reactance term (Chapter 9). In both cases, when the screen is not
in mechanical contact, an equivalent fluid model (rigid or limp) may be used. When, it
is in contact, either a poroelastic model or a refined screen model (using four variables)
should be used (Ingard 1994; Atalla and Sgard 2007).
11.3.8 Other media
The transfer matrix method can be generalized to account for other domains such as thick
plates, orthotropic plates, composite and sandwich panels, transversally isotropic porous
materials, etc. In these cases the transfer matrices are formulated in terms of the wave
