Civil Engineering Reference
In-Depth Information
E
E
d
η
=
.
(3.122)
2
π
m
Inserting Equation
(3.121)
into expressions for natural frequencies of an element, e.g.
into Equation
(3.109),
we shall get complex natural frequencies
η
⎛
⎞
f
′ =
f
1j
+
η
<<
≈
f
1j .
2
+
⎠
(3.123)
⎜
⎟
in
,
in
,
in
,
⎝
η
1
This is another formal way of stating that the system has internal energy losses, meaning
that the amplitude always has a finite value at resonance.
The loss factor may be determined by measuring either the quality factor
Q
, the
bandwidth Δ
f
for a given mode at resonance or the reverberation time
T
following an
excitation of the system at the given natural frequency
f
0
. The relations between these
quantities are:
Δ
== =
⋅
1
f
2.2
η
.
(3.124)
Qf
Tf
0
0
The energy losses of a given element will, however, always be caused by several
mechanisms; first, there will be inner losses in the material, where the vibration energy is
converted into heat, second, there will be energy radiated as sound. Another important
loss mechanism is “leakage” to connected structures, which we may call
edge losses.
The
total loss factor may therefore be expressed by a sum of loss factors representing these
mechanisms:
η
=
η
+
η
+
η
edges
.
(3.125)
total
internal
radiation
The crucial questions will then be: 1) which one or which ones of these are the most
important in an actual case and 2) how shall we arrive at the data, using either
calculations or measurement. The internal losses of metal elements are normally very
small; η
internal
is of the order 10
-3
-10
-4
. A producer of viscoelastic layers intended for
damping of metal panels would certainly be concerned with the question of how much
theη
internal
will increase by bounding the layer to the panel. He will then certainly apply a
measurement method where the other contributions to the losses are small, i.e. by freely
suspending the specimen sample and at the same time ensure that the amount of radiation
is small.
For common building constructions composed of materials such as concrete,
gypsum etc. we may find that the loss factor due to internal losses is of the order 0.01.
Being part of a building construction one normally finds that the edge losses tend to
dominate. This implies that, when performing measurements in the field, one can only
determine the total loss factor. This is however the important factor when it comes to
sound transmission and its estimation and measurement in the field or laboratory. Lastly,
we shall therefore give a supplementary expression for the last two terms in Equation
(3.125).
This will apply to a plate or panel element having a mass per unit area
m
, an area
S
and the length
l
of the edges. The expression, given below, is taken from the standard
EN 12354-1.
