Civil Engineering Reference
In-Depth Information
receiver face of the multilayer is a pressure release surface. The total modal impedance
is written
Z
mn,T
=
Z
mn(
Bare panel
)
+
Z
mn,NCT
(12.52)
Here,
Z
mn(Bare panel)
is the modal impedance of the bare panel, and
Z
mn,NCT
is calculated
by solving for the interface pressure,
P
interface
, at the panel - layered material interface
Z
mn(
Bare panel
)
υ
mn
=
F
mn
−
P
interface
(x,y)ϕ
mn
(x,y)
d
x
d
y
(12.53)
S
where
ϕ
mn
(x,y)
,
υ
mn
,
F
mn
are the mode shapes, modal velocity and the associated modal
force of the bare panel, respectively. The modal impedance
Z
mn,NCT
is related to
P
interface
via
Z
mn,NCT
V
mn
=
P
interface
φ
mn
(x,y)
d
x
d
y
(12.54)
S
In the modal basis,
P
interface
can be written in the form:
P
interface
=
m,n
P
mn
φ
mn
(x,y)
(12.55)
Since the modes are orthogonal, this simplifies to
P
mn
υ
mn
N
mn
Z
mn,NCT
=
(12.56)
where
N
mn
is the norm of the panel's modes
φ
mn
(x,y)
d
x
d
y
N
mn
=
(12.57)
S
The ratio
P
mn
υ
mn
is obtained from the solution of the transfer matrix of the layered
material following the methods of Chapter 11.
12.4.3 Examples
In the following, examples of flat plates with attached sound packages are used to demon-
strate the performance of the methods described in the previous section. The sound
package is made up of a layer of fibre or foam, with or without a cover panel. The perfor-
mance is measured by comparison with the finite element method. In the FE predictions,
the plates are modelled using Cquad4 elements while the fibre and foams are modelled
using brick8 porous elements (see Chapter 13). The plates are assumed to be baffled for
acoustic radiation. To compare various estimation of the radiated power, the calculations
are done using: (i) the geometrical correction (FTMM) in the wave approach, (ii) the
asymptotic expressions of the radiation efficiency developed by Leppington
et al
. (1982)
in the SEA and modal approaches and (iii) the Rayleigh integral in the FEM. Due to the
cost of the FE calculation, the results are averaged using five randomly selected point
force positions to approach a rain-on-the-roof type of excitation. In addition, due to the
nature of the methods used, the results are frequency band averaged using one-third octave
bands. In both the modal approach and FEM, the plate is assumed simply supported.
