Civil Engineering Reference
In-Depth Information
The terms on the diagonal represent the total loss factor of each subsystem. It should be
noted that in the literature some are using double indices on the symbol for the internal
loss factors, i.e. η
11
for η
1
etc.
7.3
SYSTEM WITH TWO SUBSYSTEMS
For the system shown in
Figure 7.2
, using Equations
(7.8),
we get
in
WE
=
ωηωη ωη
+
E
−
E
1
1
1
1
12
2
21
(7.9)
in
2
and
WE
=
ω ηωηωη
+
E
−
E
.
2
2
2
21
1
12
Driving one subsystem only, e.g. setting
W
2
in
equal zero, the last equation may be
written
E
η
n
η
2
12
2
21
=
=
⋅
.
(7.10)
E
η η
+
n
η η
+
1
2
21
1
2
21
Two comments should be made about this equation and here we may compare with the
simple thermal model that was our starting point:
a)
Assuming that the internal energy losses in subsystem no. 2 are small in
comparison with the energy transmitted back to subsystem no. 1, equipartition
of modal energy will occur. This implies
E
n
E
n
2
2
ηη
>>
resulting in
⇒
1.
21
2
1
1
In such a case there is no point in adding damping to subsystem no. 2, i.e.
increasing η
2
, unless it could be increased in size of the order of η
21
.
b)
E
2
/
n
2
will always be smaller than
E
1
/
n
1
, which means that η
21
always will have a
positive value.
7.3.1
Free hanging plate in a room
We shall give an example, used by Vér (1992), applying Equation
(7.10)
to the situation
depicted in
Figure 7.3
. A loudspeaker is creating a sound field in a room of volume
V
and we shall assume that the field is a diffuse one. The plate, having an area
S
and a mass
per unit area
m
is forced into vibration by the sound field. The task is to calculate the
mean velocity amplitude of the plate. Representing the room and the plate by subsystem
1 and 2, respectively, will be a suitable choice. The total energies may be written
2
p
2
EwV
=⋅ =
⋅
V
and
E mSu
= ⋅
,
(7.11)
1
2
2
00
ρ
