Civil Engineering Reference
In-Depth Information
30
Experiment
Rigid model
Limp model
20
10
Real part
0
−
10
Imaginary part
−
20
−
30
0
500
1000
1500
2000
Frequency[Hz]
Figure 11.2
Normalized effective density of the soft fibrous material (Table 11.2);
comparison between predictions using the limp frame and the rigid frame models. Mea-
surement performed by Panneton (2007) using the method of Utsono
et al
. (1989).
Table 11.2
The parameters used to predict the effective density of the material in
Figure 11.2.
Material
Thickness,
φ
σ
α
∞
ρ
1
(N s/m
4
)
(kg/m
3
)
h
(mm)
(
µ
m)
(
µ
m)
25
×
10
3
Soft fibrous
50
0.98
1.02
90
180
30
a vibrating wall. In both cases, the materials are assumed of infinite extent. The criterion,
named frame structural interaction (FSI), is derived from the displacement ratio of the
compressional frame-borne and airborne waves (Equation 6.71):
FSI
( ρ
limp
/ ρ
c
) K
c
/ K
f
=
where
ρ
c
ρ
12
/φ
. Using this frequency-dependent criterion they relaxed the range
of validity of Beranek's criterion to
=
ρ
1
−
|
K
c
/K
f
|
<
0
.
2. Approximating
K
f
by its isothermal
P
0
(
101
.
3kPa
)
, the latter criterion stipulates that the limp model
is applicable for materials having a bulk modulus lower than 20 kPa. Recall however
that this criterion does not account for the effects of boundary condition and mounting
effects. As mentioned above, the limp model can be used for materials having a higher
bulk modulus in various specific configurations. This is the case, e.g. for thin light foam
decoupled with an air gap from a vibrating structure.
value in air,
K
f
≈
11.3.5 Thin elastic plate
In the case of a thin elastic plate in bending (bending stiffness
D
; thickness
h
and mass
per unit area
m)
, the harmonic form of the equation of motion is given by:
Z
s
(ω)v
3
(M
)
σ
33
(M
)
=
−
σ
33
(M)
(11.56)
