5- 29
Problems and Solutions Section 5.4 (5.37 through 5.52)
5.37
A machine, largely made of aluminum, is modeled as a simple mass (of 100 kg) attached
to ground through a spring of 2000 N/m.
The machine is subjected to a 100-N harmonic
force at 20 rad/s.
Design an undamped tuned absorber system (i.e., calculate
m
a
and
k
a
)
so that the machine is stationary at steady state.
Aluminum, of course, is not completely
undamped and has internal damping that gives rise to a damping ratio of about
ζ
= 0.001.
Similarly, the steel spring for the absorber gives rise to internal damping of about
ζ
a
=
0.0015.
Calculate how much this spoils the absorber design by determining the
magnitude
X
using equation (5.32).
Solution:
From Eq. (5.21), the steady-state vibration will be zero when
a
a
m
k
=
2
ω
Choosing
µ
= 0.2 yields
m
m
m
a
a
a
=
=
(
)(
)
=
=
=
(
)(
)
=
µ
ω
0 2 100
20
20
20
8000
2
2
.
kg
k
N/m
a
With damping of
ζ
= 0.001 and
ζ
a
= 0.0015, the values of
c
and
c
a
are
c
km
c
k m
a
a
a
a
=
=
(
) (
)(
)
=
=
=
(
) (
)(
)
=
2
2 0 001
2000 100
0 894
2
2 0 0015
8000 20
1 2
ζ
ζ
.
.
.
.
kg/s
kg/s
From Eq. (5.32),
X
k
m
F
c
F j
K
M
jC
a
a
a
=
−
(
)
+
−
+
(
)
ω
ω
ω
ω
2
0
0
2
det
Since
M
C
K
=
=
−
−
=
−
−
100
0
0
20
2 0944
1 2
1 2
1 2
10 000
8000
8000
8000
.
.
.
.
,
the denominator is –6.4
×
10
7
-1.104
×
10
6
j
, so the value of
X
is
X
k m
F
c
F j
K
M
jC
a
a
a
=
(
)
+
(
)
−
+
(
)
ω
ω
ω
ω
2
0
0
2
det

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