Homework 1_Solution
Acoustics I
Fall 2007
A SDOF system shown in the figure below is operating on a foundation of impedance
F
Z
%
defined by
6
6
27,000
1.875
10
270
10
3000
,
3000
,
3000
F
Z
j
j
or
ϖ
ϖ
ϖ
ϖ
ϖ
ϖ
×
×
=



%
.
Plot the
displacement amplitude
1
X
%
of mass
M
1
when F = 1 as a function of frequency for the
range
0
500
/
rad
s
ϖ
=

for these three cases.
(Solution)
The impedance at the excitation point is given as:
1
1
1
j
t
m
m
m
j
t
m
m
F
F
s
js
R
R
f
Fe
j
Z
j
M
j
M
s
js
x
j
X e
R
R
j
Z
Z
ϖ
ϖ
ϖ
ϖ
ϖ
ϖ
ϖ
ϖ
ϖ
+

=
=
=
+
=
+
+

+
+
%
&
%
%
%
%
%
(1)
For
F
= 1,
1
1
m
X
j
Z
ϖ
=
%
%
(2)
1
27,000
/
s
N
m
=
1
180
/
m
R
N s m
=
⋅
1
x
%
( )
j
t
f t
Fe
ϖ
=
%
1
300
M
Kg
=
F
Z
%
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Therefore,
1
X
%
is obtained:
(1) Calculate
F
Z
%
as a function of
ϖ
.
(2) Calculate
m
Z
%
from Eq. (1) as a function of
ϖ
for the calculated
F
Z
%
.
(3) Calculate
1
X
%
from Eq. (2).
Then, graphically representing the complex amplitude
1
X
%
is the only remaining work.
(a)
27,000
3000
F
Z
j
ϖ
ϖ
=

%
,
300
,
27,000
/
,
180
/
m
M
kg s
N
m R
N s m
=
=
=
⋅
10
1
10
0
10
1
10
2
10
3
10
10
10
5
10
0
amplitude (m)
10
1
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 Fall '07
 Kim
 Natural Frequency, Impulse response, zf, Microphone, • Foundation

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