# Response of a Warped Flexible Rotor with a Fluid Bearing

Authors: Jim Meagher; Xi Wu; Chris Lencioni
Journal: International Journal of Rotating Machinery
 (2008)

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## Abstract

A  two-complex-degrees-of-freedom model is developed and compared to
experimental data for various amounts of rotor bow and its orientation to mass imbalance
of the rotor. The equation of motion is developed by adding constant forces that rotate
with the rotor to a Bently-Muszynska two-mode isotropic rotor model with a plane journal
bearing. Diagnostic information discernable from probes at the bearing is explored and
compared to midspan response, where previous research has concentrated. The model
presented also extends earlier work by representing the effect of a nonrigid bearing. Good
agreement between the analytical model and experiment demonstrates that the analysis
presented can be useful to diagnose and balance residual shaft bow from probes
located at the bearings, where vibration data are typically more available than midspan probes.


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## Body

## 1. Introduction

Rotor bow can be a
present in large horizontal rotors at rest for long periods, in misaligned
rotors, or in rotors, where rub causes a thermal bow on the shaft. This can
lead to instabilities, difficulty in balancing, and preload on bearings. Nicholas et al.
[1] modeled shaft bow with a rotating force similar to imbalance and used a
Jeffcott rotor model. Peak response and phase angle of bowed rotors were
compared to straight shafts for various amounts of bow and bow orientation. The
principal indicators of bow identified were phase angles other than 90 degrees
at resonance and phase change variations during acceleration through resonance.
Rao [2] used an equivalent analytical model and extensive simulations to
develop a list of observations to help diagnose the presence of rotor bow. Both
of these analytical studies used rigid bearings and the response at midspan,
which is normally inaccessible. Flack and   Rooke  [3] used a transfer matrix method to
study bowed rotor response for shafts with five different sets of fluid film
bearings. The transfer matrix approach allowed multimass rotors to be analyzed.
Direct comparisons to experimental data were also conducted for four different
types of fluid bearings typically used in industrial turbo machines. Good agreement
was reported between experiment and theory, but results were limited to midspan
motion which is usually not available and only presented the case of imbalance
and bow 180° apart. Model-based identification methods have increasingly
been used to identify rotor bow [4], but they are necessarily more complicated,
machine specific, and less general than the lumped parameter approach used
herein. Numerical methods such as transfer matrices or finite elements allow
individual machines to be modeled, but they do not easily provide parametric
models or predicted responses that are transferable to other machines. The
objective of this paper is two-fold: to provide a closed form solution for a
bowed shaft response that includes a fluid bearing, and to compare midplane
response to response at the bearing for the purpose of identifying shaft bow.
Diagnostic information of machinery condition will thus be related to typical transducer
measurement and location.


## 2. Analytical Approach

A Bently-Muszynska model [5] for a two mass
system with a fluid bearing was  modified to include rotor bow. The rotor model
as depicted in Figure 1 has two lumped masses, a midspan disk and outboard
journal, separated by a uniform flexible shaft. The rotor is supported by a
stiff bearing inboard and a fluid bearing at the outboard end. The journal mass,  M2, operates in
a fluid bearing characterized with lumped damping,  DB, and bearing
stiffness, KB. The average fluid circumferential velocity, λ, is the ratio of average fluid circumferential
angular velocity over shaft angular velocity. An imbalance is modeled at the
midplane mass and is at an angle ao relative to the bow plane. The
rotating forces fixed on the shaft that creates the bow effect, F1 and F2, are modeled using equilibrium corresponding to residual rotor
deflection ro at midplane. These forces rotate synchronously and create the
effect of a shaft bow amount of ro at the midspan disk, no deflection within
the fluid bearing and no moment on the rigid bearing; (1)F1=(K1+K2)r0,F2=(L1L1+L2)F1.  The corresponding
equations of motion follow from the dynamic equilibrium depicted in Figures 1
and 2: (2)M1r¨1+D1r˙1+(K1+K2)r1−K2r2=mruω2ej(ωt+δ)+F1ej(ωt+δ−a0),M2r¨2+DBr˙2+(K2+KB−jDBλω)r2−K2r1=F2ej(ωt+δ−a0+π).   Assuming
solutions of the formml: (3)r1=A1ej(ωt+α1),r2=A2ej(ωt+α2)   allows determination of the resulting
motion at each mass: (4)A1ejα1=[K2+KB−M2ω2+jDB(1−λ)][mruω2+F1e−ja0]ejδQ+K2F2ej(δ−a0+π)Q,A2ejα2=K2[mruω2+F1e−ja0]ejδQ+(K1+K2−M1ω2+jωD1)F2ej(δ−a0+π)Q,   where Q denotes  [K1+K2−M1ω2+jωD1][K2+KB−M2ω2+jDBω(1−λ)]−K22   and corresponding phase: (5)αi=arctan[Im(Aiejαi)Re(Aiejαi)],i=1,2. These equations were solved using Matlab
for varying amount of bow and various orientations of the bow plane relative to
mass unbalance. Single plane theoretical responses for a flexible rotor on
stiff bearings are well known, [1, 2] but the cases presented here were tested
experimentally and present results at the bearing, where vibration data are
more readily available.Figure 1
Two mass flexible rotor
model.Figure 2
Lumped parameter modeling of
forces in each plane.


## 3. Simulations

The equations above were solved using the
parameters listed in Table1. The values of these parameters were determined
experimentally as discussed in the following section. Numerical comparison of
the vibration amplitude at the midplane to the response at the bearing is shown
in Figure 3. At midspan the effect of self-balancing is evident. Depending upon
the ratio of bow to eccentricity, when bow and imbalance are 180° apart, there will be a speed at which
these effects cancel. For a Jeffcott rotor model the self-balancing speed
occurs at ω2=R0*ωn2 for ao = 180°. This occurs at approximately the same
speed, when a fluid bearing is present. This indicator of bow however is not
apparent at the fluid bearing. Response at the bearing appears to be solely
from eccentricity.Table 1
Simulation parameters.
parametersvaluesunitsM11.802KgM20.291KgL10.225mL20.183mD120N-s/mDb2100N-s/mε2.3×10−4mλ0.48Kb170000N/mδ0radResponse curves with bow
opposite imbalance for various amounts of bow, Ro.
(a)(b)A three-dimensional map of this effect is
shown in Figure4. The effect of imbalance alone is represented by curves along
Ro = 0. As bow is increased a self-balancing speed appears at the midplane.
This speed increases with bow and has a minimum overall response when Ro = 1,
which is the case where the eccentricity and bow are equal. The response at the
fluid bearing also has a minimum response for Ro = 1 but not self-balancing. The
associated phases for the responses in Figure 4 are shown in Figure 5. The
phase of a bowed shaft is quite different than that of a straight shaft with
imbalance. The most notable feature is the 180° phase jump that occurs when the response shifts from a low
speed response to bow to that dominated by imbalance. At the balance speed the
effects is cancelled.  Below and above this speed the phase is toward either
the bow or imbalance which is 180° apart for this
case.Response amplitudes at
midspan (a) and at the fluid bearing (b) for an imbalanced rotor with varying
amount of bow.
(a)(b)Phase at midspan (a) and at
the fluid bearing (b) for an imbalanced rotor with varying amount of bow. The
bow plane is opposite the imbalance.
(a)(b)Figure6 illustrates how the position of
the central disk along the shaft does not affect the self-balancing speed. For this model the effective shaft stiffness
remains the same. The peak response however gets smaller as the disk gets
closer to the fluid bearing. The response amplitude at the fluid bearing does
not monotonically decrease in the same way nor does the amplitude vary as much.
Increased damping is evidenced in both planes  by the broadening of the response
curve. Since an isotropic shaft was assumed variation in the parameter L1/L is
equivalent to changing the shaft segment stiffness on either side of the
central disk. Figure 7 shows a fuller description of the response amplitude
variations with the associated phase changes shown in Figure 8. The peak
midplane response occurs when the central disk is near the rigid bearing. The
effective damping is reduced and the amplitude ratio is four times that
compared to the disk near the fluid bearing. The phase jump at the self-balancing
speed is apparent for all length ratios. The response at the fluid bearing has
a maximum for the central disk to be at the center of the shaft. The phase
changes seen at the fluid bearing are more gradual.Response curves for various locations of disk.
Ro = 0.5, ao =180°.
(a)(b)Response amplitudes at midspan (a) and at the fluid bearing (b) for an
imbalanced rotor with Ro = 0.5 for various locations of the midspan disk.
(a)(b)Phase at midspan (a) and at the fluid bearing (b) corresponding to the
frequency response plots shown in Figure7.
(a)(b)Figure9 depicts the effect of the relative
orientation between the bow plane and the imbalance plane. The midplane
response amplitude plot, Figure 9(a), has zero amplitude at a speed ratio of
0.7 for ao = 180°. This self
balancing does not occur at the outboard bearing for any relative orientation
and only occurs at 180° at the midplane. At
0° difference, the
bow and imbalance act in phase to have the largest response. The low-speed
phase measured at midplane, Figure 9(b), is a good diagnostic for bow location.
At low speed the response phase is equal to bow orientation. Unfortunately, the
low amplitude of vibration makes the accurate measurement of phase difficult as
shown later in Figure 15. The phase jump evident in Figure 9(a) for the
midplane, with ao = 180°, is another
indicator of self-balancing. At low speed the phase at the fluid bearing lags
the bow plane by 90° but as the shaft speed increases the phase
changes due to the fluid tangential force, jDBλωr2. The lines in red of Figure 9 are for
imbalance without any bow. This shows the agreement of low-speed phase at
midspan and a 90° phase shift at the
fluid bearing.Amplitude (a) and phase response (b) for various bow orientations relative
to the imbalance plane. Ro = 0.5. The curve in red is for a
rotor with imbalance but no bow.
(a)(b)Figures10 and 11 depict a more complete
description of the parameter variation effects shown in Figure 9. The response
amplitudes decrease monotonically at midplane and at the bearing as the bow to
imbalance orientation increases from aligned to 180° apart. Self-balancing occurs for a very
particular case and is visible only at midplane. Phase jump at midplane is also
shown to be an isolated occurrence at midplance from Figure 11.Response amplitude at midspan (a) and at the fluid bearing (b) for various
orientations of bow with repect to imbalance. Ro = 0.5.
(a)(b)Phase at midspan (a) and at the fluid bearing (b) for various orientations
of bow with respect to imbalance. Ro = 0.5.
(a)(b)


## 4. Experimental Results

An RK4 Bently
Nevada Rotor kit shown in Figure12 was used for experimental measurements. The
rotor is driven through a flexible coupling by an electric motor with a speed
controller. The speed range of testing was from a slow roll of 250 rpm up to
approximately 3500 rpm, where whirl instability was observed. Shaft lateral
vibrations were measured with two sets of proximity probes in horizontal and
vertical orientations at the midspan disk and at the fluid journal bearing, Figure
12(b). The fluid bearing used was the standard pressurized fluid bearing that
comes with the Bently Nevada RK4 rotor kit. It has a 1 inch diameter lucite journal
and 15-mil diametral clearance. The bearing has four oil injection ports spaced
evenly around the bearing. The pump pressure was set to 10 psi.Experimental setup (a) with fluid bearing on the left and variable speed
motor on the right, (b) close-up of fluid bearing showing supply lines and
proximity probes.
(a)(b)Before testing, a
10 mm rotor was permanently bowed in a hydraulic test machine. The rotor was then
carefully balanced in order for the high speed imbalance response to equal the
low speed bow response with self-balancing at the critical speed.
Theoretically for a Jeffcott rotor model this occurs, when the bow plane and
imbalance are180° apart with R0 = 1.
As shown in Figure 13, this response is accurately modeled by the proposed
lumped parameter model except Ro was set to 0.87 due to differences in predicted
response with the current model. This case is used as a baseline for parameter
identification and is critical in establishing orientation of the imbalance
plane relative to the bow plane as well as for determining the amount of
residual imbalance. Subsequent simulations
for different bow factor and bow orientation use the same parameters as listed
in Table 1 except for imbalance orientation and eccentricity which were
experimentally modified using additional imbalance masses added to the rotor
disk.Figure 13
Bowed rotor experimental (exp.) and simulated (sim.) response.By identifying this
as a reference configuration, the imbalance plane orientation can be changed
with the addition of imbalance masses. The stiffness, mass, and damping factor
used in simulations are based on values obtained from nonsynchronous
perturbation testing [6]. Using this method requires a known radial
perturbation force. This was supplied by a free spinning perturbator located
adjacent to the rotor mass. The speed of the perturbation force is controlled
independently of the rotor speed using a separate motor speed controller to
control the speed of a freely spinning imbalanced disk mounted on the rotor. A
plot of direct dynamic stiffness and quadrature stiffness allows the
determination of modal mass, damping, and stiffness [6]. After some
modification to match the reference experimental data shown in Figure 13 these
parameters were held constant. This is justified from the experimental
frequency responses shown in Figures 14–17 which reveal a
single mode of response in the speed range tested.Synchronous response of a bowed rotor at midplane disk. Ro = .64, ao =180°. Drop out of data near 2000 rpm is due to
displacement exceeding transducer range.
Phase wrap evident in the experimental plot around 1700 rpm and above
3500 rpm is due to software that adds 360° for phase ranges
beyond 360°.
(a)(b)Synchronous response of a bowed rotor at outboard bearing. Ro = .64, ao =180°.
(a)(b)Synchronous response of a bowed rotor at midplane disk. Ro = 1.09, ao =180°.
(a)(b)Synchronous response of a bowed rotor at outboard bearing. Ro = 1.09, ao = 180
degrees.
(a)(b)The case of self-balancing
prior to the critical speed is shown in Figures14 and  15. Additional imbalance
mass was added opposite the bow plane to increase the eccentricity. This
decreased the bow factor while maintaining the 180° separation between the bow plane and the
imbalance plane. At low speeds the bow response is dominant and is
characterized by high whirl amplitude. The amplitude reaches a minimum when the
bow response and imbalance cancel. After resonance, the response is dominated
by the imbalance response and asymptotically approaches the eccentricity. This
is as predicted by earlier models [1, 2]. The limitation of earlier models is shown
in Figure 15, where the response at the bearing is depicted. The self-balancing
speed is no longer evident and the response is not easily identified as a bowed
rotor. The low-speed response asymptote does not as clearly indicate bow as does
the midplane response.Similar differences between midplane and
outboard measurements occur when masses opposite the bow plane are removed.
This increases the bow factor and moves the self-balancing speed above the
critical speed as shown in Figure16. Removing imbalance masses increases the
bow factor by decreasing the eccentricity. Once again the simulation and
experimental response are well matched in both planes, but single plane models
would not predict the presence of bow from the bearing data, which illustrates
the difficulty of diagnosing a bowed rotor from the bearing response.


## 5. Summary

A
closed-form parametric model of a flexible bowed
shaft with imbalance is presented. The mathematical model agrees with
experimental measurements and extends earlier models by including an outboard
fluid bearing with response predicted at the bearing, where response data is
more typically available. Self-balancing, as predicted by Jeffcott rotor models,
occurs as well as for a rotor with a fluid bearing, but is not apparent using
probes at the fluid bearing. Diagnostic indicators for bow such as self-balancing
or phase jump which can be seen at midspan are not apparent at the bearing. However,
low-speed runout at the fluid bearing can be used to indicate residual rotor
bow, but it is not as obvious as measurements made from midspan probes. The low-speed
phase at either the fluid bearing or midspan indicates the bow angle with
respect to the imbalance. Good agreement between the analytical model and
experiment demonstrates that the analysis presented can be useful to diagnose
and balance residual shaft bow from probes located at the bearings, where
vibration data are typically more available than midspan probes.



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*Source: 147653-2008-05-28.xml*