Hydrocarbon Processing - June 2021 - GP-25

TURBOMACHINERY
precision alignment during operation)
and secured with a fastener (which prevents
any axial displacement or loosening).
Fully assembled, the rotor weighs
163 lb and is approximately 24 in. long.
The shaft diameter at the bearing location
is approximately 3 in.
The contribution of the wheels depends
on their mass properties, namely
weight, polar inertia (around the shaft
axis), transverse inertia (across the shaft
axis) and the location of the center of gravity.
Those values are added to the " node "
or " station, " where the element is located
on the shaft (see the legend in FIG. 4).
As for the bearings, their contribution
consists of providing radial support stiffness
to the rotor and attenuation of the
vibrations. They are pictured in the model
in FIG. 4 as springs. These are mathematically
expressed as rotor speed-dependent
stiffness and damping coefficient matrices.
Undamped critical speed (UCS)
map. A UCS map is a plot of critical
speeds vs. bearing stiffness (FIG. 5). This
plot identifies the critical speeds and their
mode shapes, as well as how they vary
with different bearing stiffnesses. As such,
the UCS map provides the designer with a
starting point in the process to choose the
type of bearing needed. For example, for
a stiffer bearing, the designer may choose
fixed-geometry bearings with tighter
clearances, whereas for softer bearings,
the choice may be tilting-pad or flexiblepivot
bearings with larger clearances. The
goal in choosing a bearing is to avoid potentially
exciting a shaft's critical speed
within the range of operating speeds required
by the design of the turboexpander.
FIG. 5 shows that a relatively softer bearing
would be a better choice, considering that
the critical speeds with stiffer bearings
would be too close to the design speed.
Damped natural frequency (eigenvalue)
map. A damped natural frequency
map is a plot of natural frequencies (eigenvalues)
vs. running speed (FIG. 6).
It is produced with models of the bearings,
seals and any other excitation phenomena
incorporated. This plot is useful
for checking the likelihood of exciting a
natural frequency in the operating range,
as well as evaluating potentially unstable
operating points.
A common characteristic of turboexpanders
is that their shafts have a low
aspect ratio, meaning they are quite short
relative to their diameter. As a result, the
8
6
4
16
2
1
-2
-4
-6
-1
1
3
5
7
9
11
13
Axial location, in.
FIG. 4. Graphical depiction of the finite-element model for rotor dynamic simulations.
100,000
15
17
19
21
23
25
Tilting pad bearing
Four pads-load between pads
Clearance = 0.0064 in.-0.0068 in.
Tilting pad bearing
Four pads-load between pads
Clearance = 0.0064 in.-0.0068 in.
4
8
12
20
24
28
32
36
40
43
Component's added
mass and Ip and It
Stiffness and
damping matrices
" Station " or " nodes "
MCOS: 17,500 rpm
Rotor weight: 163.2 lb
Rotor length: 24.4 in.
KXX
KYY
17,500 rpm
17,980 rpm
17,170 rpm
10,000
Second critical speed
First critical speed
1,000
10,000
100,000
Bearing stiffness, lb/in.
FIG. 5. Undamped critical speed map. KXX and KYY refer to the direct bearing stiffness
in the horizontal and the vertical direction, respectively. In this case study, KXX
= KYY
natural frequency of the bending mode of
these short, " stubby " shafts ends up much
higher than their operating speeds or any
excitation frequency. Therefore, from a
rotor dynamics perspective, they are categorized
as " stiff-shaft " designs.
The implication for stiff-shaft designs
is that only the rigid body translational
modes of vibration are of serious concern.
These modes are greatly influenced
.
by the design of the bearings. Smaller
machines are usually equipped with
tight-clearance, fixed-geometry bearings
with high stiffness.
This yields a machine that does not
have any critical speeds within its operating
range. Larger rotors tend to utilize
tilting-pad or flexible-pivot bearings and
can be designed with larger clearances
and lower stiffness. With this configuraGas
Processing & LNG | MAY/JUNE 2021 25
1,000,000
10,000,000
Critical speed, cycles per minute
Shaft radius, in.
Aerodynamic
cross-coupling
of exp wheel
Radial bearing
Thrust washer
Expander wheel
Thrust washer
Radial bearing
Compressor wheel
Hydrocarbon Processing - June 2021

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