Hydrocarbon Processing - June 2021 - GP-26

TURBOMACHINERY
tion, the machine tends to pass through a
highly damped natural frequency at very
low speeds. Since the bearings are well
damped, these points of operation are not
considered critical speeds due to the low
amplification factor.
Two predominant modes of vibration
are observed within the speed range,
namely a cylindrical and a conical mode
(FIG. 7). These results show that the rotor
system is well designed since the natural
35,000
30,000
25,000
20,000
15,000
10,000
5,000
5,000
Cylindrical
" bouncing " mode
frequencies for both modes do not intersect
the 1X excitation line. Such a condition
would have indicated a potential
resonance situation.
Stability analysis. It is not enough
to know that damped natural frequencies
do not cross the 1X excitation line.
The designer must show that the bearing
system is robust against the presence of
destabilizing forces such as those induced
by the areodynamic components, seals or
bearings. It is also important to know how
much damping is inherent in each vibration
mode. Should insufficient damping be
identified for a given mode shape, the rotor
would be susceptible to high vibrations
due to excitations such as unbalance. As
mentioned previously, the bearings are the
source of damping. However, the amount
of " effective " damping is not the same for
every mode. This is quantified by the socalled
logarithmic decrement ( " log dec " )
or, alternatively, the damping factor. Designers
typically prefer using one of these
parameters for their analysis, but both are
representative of the same physical reality.
FIG. 7 depicts the predicted log dec
Conical mode
Cylindrical
" bouncing " mode
Conical mode
Synchronous
10,000
FIG. 6. Damped natural frequency map.
20
10
-10
-20
-30
-40
Rotor speed, rpm
FIG. 7. Log dec plot.
26 MAY/JUNE 2021 | GasProcessingNews.com
5,000
10,000
15,000
20,000
25,000
30,000
Cylindrical " bouncing " mode
Conical mode
MCOS
15,000
Rotor speed, rpm
20,000
25,000
30,000
for the two predominant modes of vibration
on this expander rotor. A minimum
log dec of 0.1 implies that the system is
sufficiently damped and can be considered
stable. The data indicates that the
log dec is positive for both modes from
zero speed until MCOS. However, the log
dec drops to negative values for speeds
above 25,000 rpm, which indicates that
the mode shape becomes unstable. This
means that the system becomes selfexciting,
and the vibrations will increase
sharply. This is due to hydrodynamic effects
in the oil film between the bearing
and the shaft. This is clearly an undesirable
condition, but it is not concerning
in this case because the control system
limits the maximum speed to protect the
mechanical integrity of the machine.
Also, note that the aerodynamic forces
experienced by the wheels are transferred
to the rotor bearing system and can sometimes
add significant excitation forces,
which must be considered in the calculation.
The mechanism of these aerodynamic
effects is such that they generate
a displacement that is orthogonal, and
not parallel as intuition might suggest,
to the aerodynamic force. This phenomenon
is modeled in mathematical terms
as cross-coupled stiffness and can have
the capacity to induce undesirable unstable
vibrations. In many turboexpander
cases, the destabilizing forces generated
in the expander wheel are more consequential
than in the compressor wheels.
FIG. 8 shows an example on a different
turboexpander in which the amount of
aerodynamic cross-coupling created a
significant instability.3
Bearing loads. Both static and dynamic
loads on bearings can be predicted
and are necessary in ensuring that the
Log dec
Natural frequency, cycles per minute
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Hydrocarbon Processing - June 2021

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