2 36
Problems and Solutions Section 2.5
(2.43 through 2.50)
2.43
A lathe can be modeled as an electric motor mounted on a steel table.
The table plus the
motor have a mass of 50 kg.
The rotating parts of the lathe have a mass of 5 kg at a
distance 0.1 m from the center.
The damping ratio of the system is measured to be
ζ
=
0.06 (viscous damping) and its natural frequency is 7.5 Hz.
Calculate the amplitude of
the steadystate displacement of the motor, assuming
r
ω
= 30 Hz.
Soltuion:
Given:
m
= 50 kg,
5
=
o
m
,
e
= 0.1m,
06
.
0
=
ζ
,
ω
n
=
7 5
.
Hz
Let
ω
r
=30 Hz
So,
r
r
n
=
=
ω
ω
4
From Equation 2.51,
2
2
2
2
2
2
2
2
)]
4
)(
06
.
0
(
2
[
)
4
1
(
4
50
)
1
.
0
)(
5
(
)
2
(
)
1
(
−
+
−
=
+
−
=
r
r
r
m
e
m
X
o
ζ
X
= 0.011m
X =
1.1 cm
2.44
The system of Figure 2.15 produces a forced oscillation of varying frequency.
As the
frequency is changed, it is noted that at resonance, the amplitude of the displacement is
10 mm.
As the frequency is increased several decades past resonance the amplitude of
the displacement remains fixed at 1 mm.
Estimate the damping ratio for the system.
Solution:
Equation 2.75 is
2
2
2
2
)
2
(
)
1
(
r
r
r
m
e
m
X
o
ζ
+
−
=
At resonance,
X
= 10 mm =
ζ
2
1
m
e
m
o
ζ
2
1
10
=
e
m
m
o
When
r
is very large,
1
=
e
m
Xm
o
and
X
= 1 mm, so
1
=
e
m
m
o
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 Spring '09
 abduljaba
 Force, Mass, Orders of magnitude

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