Civil Engineering Reference
In-Depth Information
where η
is the loss factor. It is assumed that the RMS-value is taken as a mean value over
the area of the plate and also over a frequency band Δω to include a number of modes
around the frequency ω
.
The model energy
E
modal
is expressed as
E
E
=
,
(7.3)
modal
n
ω
⋅Δ
and assumption no. 2 tells us that the net power
W
12
transmitted between the two
subsystems may be written
⎛
⎞
⎛
⎞
⎠
E
E
E
E
′
′
1
2
′
1
2
WWW b
=−=⋅
−
=⋅
b
−
.
(7.4)
⎜
⎟
⎜
12
12
21
n
⋅Δ
ω
n
⋅Δ
ω
n
n
⎝
⎠
⎝
1
2
1
2
The quantities
b
and
b
´ =
b/
Δω are factors of proportionality. The equation may be cast
into a more suitable form by introducing the aforementioned coupling loss factors η
ij
. We
then have
WE
WE
′ =
′
=
ωη
ωη
12
1
12
(7.5)
and
.
21
2
21
′
=
′
=
b
n
b
n
Hence,
ωη
and
ωη
, which gives
12
21
1
2
n
⋅
η
=⋅
n
η
,
(7.6)
1 2
2
which is called the
reciprocity
or the
consistency relation
. Equation
(7.4)
may then be
written as
ωη
⎛
E
⎞
n
1
W
=
E
−
⎟
.
(7.7)
⎝
12
12
1
2
n
⎠
2
The last assumption given was that the forces driving the various subsystems were
statistically independent. This implies that we may calculate the total modal energy of
each subsystem by adding the responses resulting from each force input. For system
containing k subsystems we get the following set of equations:
∑
⎡
⎤
ηη η
+
−
⋅
−
η
1
i
1
21
k
1
⎢
⎥
in
⎡
⎤
i
≠
1
W
E
E
⎡⎤
⎢
⎥
1
1
∑
⎢
⎥
⎢⎥
⎢
−
η
η η
+
⋅
⋅
⎥
in
W
12
2
i
2
⎢
⎥
⎢⎥
2
⎢
⎥
2
i
≠
2
⎢
⎥
⎢⎥
⋅
⎢
⎥
⋅
⋅
⋅
⋅
⋅
⎢
⎥
ω
⋅
⋅
⎢⎥
=
⎢
.
(7.8)
⎢
⎥
⋅
⋅
⎥
⎢⎥
⎢
⎥
⋅
⋅
⋅
⋅
⎢
⎥
⎢⎥
⎢
⎥
⋅
⋅
⎢
⎥
⋅
⋅
⋅
⋅
⎢⎥
⎢
⎥
E
⎢
⎥
in
⎣⎦
W
⎢
∑
⎥
k
⎣
⎦
−
η
⋅
⋅
η η
+
k
⎢
1
k
ki
k
⎥
⎣
⎦
ik
≠
