Civil Engineering Reference
In-Depth Information
F
M
Figure 6.18
Dynamic point force input
F
to a plate having point mobility
M
.
The mechanical power imparted to a structure by a point force may in general be written
⎧
⎫
1
2
{}
2
WF MF
=⋅
Re
=⋅
Re
,
(6.52)
⎨
⎩⎭
mec
Z
where Re{…} denotes the real part. Setting out to calculate the part of this power being
radiated as sound energy we shall need information, not only on the velocity amplitude
of the driving point but on the global velocity distribution as well (see Equation
(6.25)
).
In practice, we shall assume that the plate is driven by a dynamic force having a certain
frequency bandwidth Δω, and we shall further assume that a number of modes have their
natural frequencies inside this band. (Normally, having 5-6 eigenfrequencies inside Δω
will give reasonable estimates). It may then be shown (Cremer et al. (1988)) that the
mean square velocity amplitude may be expressed as
2
1
F
π
Δ
∫
2
2
Δ
ω
u
=
u
d
ω
=
⋅
⋅
n
( ,
ω
(6.53)
22
ω
2
ηω
Δ
ω
Sm
Δ
ω
where
S
,
m
and η are, respectively; the area, the mass per unit area and the loss factor of
the plate. The angular frequency ω will be the centre frequency in the band of width Δω.
The function
n
(ω) is the modal density of the plate, formerly derived in Chapter 3,
section 3.7.3.5 and given by
Sm
n
()
ω
=⋅
.
(6.54)
B
Inserting this expression into Equation
(6.53)
we get
2
2
2
F
π
ηω π
Sm
F
k
2
Δ
ω
Δ
ω
B
u
=
⋅
⋅
⋅
=
22
,
(6.55)
22
24
B
Sm
8
S
ηω
m
Δ
ω
where we have introduced the wave number
k
B
in the last expression. We arrive at an
alternative expression by introducing the mobility from Equation
(6.51)
. Hence
=
2
FM
2
Δ
ω
∞
u
.
(6.56)
Sm
ηω
Δ
ω
