Civil Engineering Reference
In-Depth Information
Using the same approach, the acoustic power radiated by the panel into the source
domain is given by
F
2
8
π
2
k
0
2
π
2
Re
Z
B,
∞
(ξ,ψ)ξ
d
ξ
d
ψ
rad
=
|
υ(ξ,ψ)
|
(12.49)
0
0
Here, integration in polar coordinates defined by
ξ
1
=
ξ
cos
ψ
and
ξ
2
=
ξ
sin
ψ
is
used and
k
0
Z
0
k
0
−
Z
B,
∞
(ξ
1
,ξ
2
)
=
(12.50)
(ξ
1
+
ξ
2
)
is the radiation impedance of the panel in air (the source domain),
k
0
being the acoustic
wave number. Other indicators such as the power radiated into the receiver domain can
be calculated using the same methodology.
Below the critical frequency of the main structure, Equation (12.49) will lead to poor
results. However, as discussed in the previous sections, the transfer matrix method can be
extended easily to take into account the panel's size and thus to correct for the radiation
efficiency at low frequencies. The application of the FTMM to the calculation of the
radiated power transforms Equation (12.49) into:
∞
2
π
F
2
rad
=
8
π
2
Z
0
σ
R
(k
r
,ϕ)
2
ξ
d
ξ
d
ψ
(12.51)
|
Z
B,
∞
(ξ,ψ)
+
Z
S,TMM
(ξ,ψ)
|
0
0
The 'finite size' radiation efficiency,
σ
R
(k
r
,φ)
is defined by Equations (12.10) and
(12.11).
Z
S
,
TMM
is the impedance of the infinite extent panel with layered material.
12.4.2 The TMM, SEA and modal methods
The TMM can also be used in combination with other methods to account for the effects
of a sound package. For instance, in combination with a statistical energy analysis (SEA)
model of the point load excited base structure, a light coupling can be assumed between
the structure and the sound package. In consequence, the effect of the sound package will
be simply represented by an equivalent damping
η
NCT
and an added mass correction. At
a given frequency, the dispersion equation of the bare panel (plate, solid, composite, etc.)
is solved for the propagating wave number and the latter is imposed on the excited face
of the sound package with a pressure-release condition on the rear face. The associated
TMM system is then solved to calculate the input power and dissipated powers and
in turn the equivalent added damping
η
NCT
. The response of the treated panel is finally
recovered from the SEA response of the bare panel using the total damping of the system
η
To t
=
η
s
+
η
NCT
+
η
rad
with
η
rad
the radiation damping of the panel and
η
s
its structural
damping.
The same methodology can be used in combination of a modal representation of the
bare panel's response. The response of the main structure can be written in terms of its
modes and the effect of the sound package on each mode
(
m,n
)
is replaced by a modal
impedance
Z
mn,NCT
, calculated using the TMM with a trace wave number
k
t
mn
,givenby
the modal wave number of the panel. Once again, it is assumed in this calculation that the
