Civil Engineering Reference
In-Depth Information
With the last limit value we have assumed that the impedance of the primary slab is
much larger than for the top slab, whereas the latter also is very much larger than the
impedance of the elastic layer. Now assuming that the radiated power from the primary
floor is proportional to the velocity squared, it follows
2
(
)
⎡
W
⎤
u
Z
rad
a
2a
1
Δ=⋅
L
10 lg
=⋅
10 lg
=⋅
20 lg
.
(8.43)
⎢
⎥
n
(
)
W
u
Z
⎢
⎥
⎣
⎦
rad
2b
d
b
Assuming that the impedance of the top slab is an ideal mass impedance and the elastic
layer is an ideal spring, i.e. |
Z
1
|
∼
ω
m
1
and |
Z
d
|
∼
s
d
/ω, we again obtain
⎛
2
⎞
⎛⎞
ω
m
f
1
Δ=⋅
L
20 lg
=⋅
40 lg
.
(8.44)
⎜
⎟
⎜
⎝⎠
n
s
f
⎝
⎠
d
0
An alternative to a continuous elastic layer is obtained by using elastic load-bearing unit
mounts as shown in
Figure 8.25
b). This type of connector may also be used to illustrate
the influence of structural connections (sound bridges), normally unintentional, between
the floating layer and the primary floor construction. Vér (1971) used a SEA model to
calculate the improvement in the impact sound insulation by such floating floor
constructions, assuming a reverberant bending wave field in the floating top slab. Other
important assumptions were e.g. that the energy transmission from the top slab to the
primary floor only takes place by way of the unit mounts, only transmitting forces and
not moments. In other words, the coupling by way of the air stiffness in the cavity is
disregarded. Furthermore, there is no correlation between the movements at the different
mounts. Indicating the floating floor and the primary floor by the indices as above and
having
N
mounts per unit area, each having a stiffness
s
, Vér gives the following result
⎡
3
⎤
Z
m
Z
η
η
ηπ
N
f
1
N
⋅
s
1
1
1
1
1
Δ=⋅
L
10 lg
+
+
⋅
,
where
f
=
.
(8.45)
⎢
⎥
n
0
4
Z
m
2
m
f
2
π
m
⎣
⎦
2
2
2
1
1
0
Above a given frequency, when the last term inside the parenthesis becomes the
dominating one, the frequency dependence of Δ
L
n
will be 9 dB per octave. Inserting for
the impedance of the floating floor slab we may use the approximate expression
η
π
⎡
2
ch N
f
3
⎤
L11
Δ≈⋅
L
10 lg
1
⋅
,
(8.46)
⎢
⎥
n
4
0
3
f
⎣
⎦
where
h
1
and
c
L1
is the thickness and the longitudinal wave speed, respectively. It should
be noted that the 9 dB per octave dependency presupposes that the loss factor of the
floating slab as well as the stiffness of the elastic units are frequency independent.
An example of the measured improvement using a heavy floating floor, a 50 mm
thick concrete slab on a 25 mm thick stiff mineral wool layer, is shown in
Figure 8.27.
Assuming that the total dynamic stiffness per unit area of the elastic layer is 8.0 MPa/m,
we get a resonance frequency of approximately 40 Hz. This total stiffness represents the
sum of the elastic stiffness of the mineral wool and the stiffness of the enclosed air (see
section
8.4.4)
.
