Civil Engineering Reference
In-Depth Information
φ
2
K
eq
. Assuming a limp frame, the stress
tensor of the solid phase in vacuum is neglected
(σ
s
φ
2
ρ
eq
and R
=
Equation (6A.22) by
ρ
22
=
≈
0
)
and the first equation
in (11.52) leads to
ω
2
ρ
div
u
=−
γp
. Substitution of this equation in the second
equation of (11.52) yields the equivalent fluid equation for a limp material,
ρ
limp
K
eq
ω
2
p
=
0
p
+
(11.53)
where
ρ
limp
is an equivalent effective density accounting for the inertia of the frame,
ρ ρ
eq
ρ
limp
=
(11.54)
ρ
+
ρ
eq
γ
2
Assuming the bulk modulus of the elastic material from which the frame is made
much larger than the bulk modulus of the frame (i.e.
K
b
K
s
)
, valid for the majority
of porous materials, Panneton (2007) derives the following approximate expression for
the limp effective density:
ρ
0
ρ
t
ρ
eq
−
ρ
limp
≈
(11.55)
ρ
t
+
ρ
eq
−
2
ρ
0
where
ρ
t
φρ
0
is the apparent total density of the equivalent fluid limp medium.
Again, using these effective properties, the matrix representation of the limp frame limit
is given by Equation (11.9). It is observed from Equation (11.55) that, when the frame
is heavy, the rigid model is recovered. Moreover, at low frequencies lim
ω
=
ρ
1
+
→
0
ρ
limp
=
ρ
t
con-
trary to the rigid frame model (see chapter 5 for the low- and high-frequency limits of
the effective density; in particular Equation (5.37), giving the limit at low frequency:
lim
ω
0
ρ
eq
=
σ/(jω))
. In consequence, the difference between the two models is mainly
important at low frequencies. This is illustrated in Figure 11.2 for the material of
Table 11.2.
Note in passing, that the rigid frame model does not authorize rigid body motion of the
material. This is important, for instance in applications when the material is unconstrained
(free to move). This effect is mainly important at low frequencies. In such applications,
the limp model is preferred. This is key when impedance tube tests are used to derive the
acoustical properties of the material from measurement of either the surface impedance
or the prorogation constants (especially using transmission measurements).
The limp model is usable when the elasticity of the frame is neglected, either due to
the nature of the material (e.g. light fibreglass) or due to the mounting or excitation of the
material (e.g. transmission loss of light foam separated by a thin air gap from a supporting
plate). As a rule of thumb, when the material is bonded onto a vibrating structure, the rigid
frame model should not be used. On the other hand, the limp model can be used when the
bulk modulus of frame in vacuum is much smaller than bulk modulus of the fluid in the
pores. Beranek (1947) suggested the use of the limp approximation for porous materials
→
satisfying
K
c
/K
f
<
0
.
05 where
K
c
and
K
f
are the bulk modulus of frame in vacuum
and the bulk modulus of the fluid in the pores, respectively. Doutres
et al
. (2007) used
a numerical study comparing a full porous - elastic model with the limp model to derive a
criterion for the use of the latter. Two configurations were used: (i) sound absorption of a
porous layer backed by a rigid wall, and (ii) sound radiation from a porous layer backed by
