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5 69
Problems and Solution Section 5.7 (5.74 through 5.80)
5.74
A 100kg compressor rotor has a shaft stiffness of 1.4
×
10
7
N/m.
The compressor is
designed to operate at a speed of 6000 rpm.
The internal damping of the rotor shaft
system is measured to be
ζ
= 0.01.
(a)
If the rotor has an eccentric radius of 1 cm, what is the rotor system's critical speed?
(b)
Calculate the whirl amplitude at critical speed.
Compare your results to those of Ex.
5.7.1.
Solution:
(a)
The critical speed is the rotor's natural frequency, so
ω
c
k
m
==
×
14 10
100
374 2
7
.
. rad/s
3573 rpm
(b)
At critical speed,
r
= 1, so from Eq. (5.81),
()
m
5
.
0
01
.
0
2
01
.
0
2
=
=
=
ζ
α
X
So a system with higher eccentricity and lower damping has a greater whirl amplitude
(see Ex. 5.7.1).
5.75
Redesign the rotor system of Problem 5.74 such that the whirl amplitude at critical speed
is less than 1 cm by changing the mass of the rotor.
Solution:
From Problem 5.74,
k
= 1.4
×
10
7
N/m,
m
= 100 kg,
ζ
= 0.01, and
α
= 0.01m.
Since the whirl amplitude at critical speed must be less than 0.01 m, the value of
ζ
that
would satisfy this is, from Eq. (5.81),
5
.
0
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 Spring '09
 abduljaba

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