Civil Engineering Reference
In-Depth Information
If we wish to calculate the number of modes inside a given frequency interval Δω
(or Δ
f
), we just count the number of points Δ
N
inside an area defined by two quarter
circular arcs having wave numbers
k
B
(ω) and
k
B
(ω
+
Δω), these being wave numbers
corresponding to the lower and upper cut-off frequencies for the given band pass filter.
We then get
π
ab
⋅
2
(
)
2
( )
⎡
⎤
Δ=⋅
N
k
ωω ω
π
+Δ −
k
⋅
.
(3.118)
⎣
B
B
⎦
2
4
In the case of
k
B
not being too small, an approximate expression for the modal density is
(
)
( )
⎡
2
2
⎤
2
k
ωω ω
+Δ −
k
Δ
N
S
S
k
∂
()
S
m
B
B
n
()
ω
==⋅
≈⋅
=⋅
(3.119)
⎢
⎥
Δ
ωπ
4
Δ
ω
4
πωπ
∂
4
B
⎢
⎥
⎣
⎦
or
Δ
==⋅
Δ
N
Sm
nf
()
,
(3.120)
f
2
B
where
S
is the plate area. As we can see, the modal density of thin plates is frequency
independent. This is again not the case for other types and shapes of structure; see e.g.
Blevins (1979).
Example
The bandwidth of a third-octave-band filter is approximately Δ
f
≈ 0.23⋅
f
0
,
where
f
0
is the centre frequency of the band. Choosing
f
0
= 1000 Hz and taking the
concrete slab used in
Figure 3.22
as an example we get, even at this relatively high
frequency, Δ
N
≈ 14. In contrast to this, a room above this concrete slab (taken as the 24
m
2
floor of the room) having a ceiling height 2.5 m will have approximately 4 400 modes
inside the same bandwidth! The latter number is calculated using the expression
4
π
V
2
Δ≈
N
⋅
f
⋅Δ
f
,
room
3
0
c
where
V
is the room volume. This expression is derived using an analogous procedure to
the one used above taking into account the equation for the natural frequencies of a
three-dimensional air-filled space (see section 4.4.1).
3.7.3.6
Internal energy losses in materials. Loss factor for bending waves.
In a previous chapter, when dealing with oscillations in simple mass-spring systems, we
introduced the loss factor by way of a complex stiffness. In a similar way we shall, for
bending waves in a given structure, define a complex bending stiffness
B
′ :
(
)
BB
η
'
=
1
+⋅
j
,
(3.121)
where η is the loss factor. By formal definition, as found in the literature, it is given by
the ratio of the mechanical energy
E
d
dissipated in a period of vibration to the reversible
mechanical energy
E
m
:
