spiral vibrations in power units: modelling and

Spiral vibrations in power units: modeling and experimental evidences

P. Pennacchi1, A. Vania, S. Chatterton, R. Ricci Politecnico di Milano, Department of Mechanical Engineering Via La Masa,1, I-20156, Milano, Italy 1 email: [email protected]

Abstract One of the most remarkable aspects, in industrial rotating machines, of the rotor-to-stator rub at operating speed is the possibility of observing spiral vibration phenomena in the synchronous vibration, i.e. the vibration vector changes its amplitude and rotates even at the rated speed. This is due to the combined effect of the contact forces, which introduce heat, and of the resulting thermal bow. The phenomenon is also called vector turning or thermally induced vibrations. In the paper a model, developed by the authors, is presented, which is able to simulate spiral vibrations in real rotating machines. Then an interesting case history, in which spiral vibrations affected a 70 MVA steam power unit, is presented. The case is analyzed in deep detail and experimental evidences show that spiral vibrations affect practically only synchronous vibrations. A new tool for analyzing the phenomenon, the rate of change of the phase of the synchronous vibrations, is also introduced.

1 Introduction

Sometimes specific faults in rotating machinery cause 1X synchronous vibrations whose amplitude and phase are affected by continuous and monotone changes even in the case of a constant rotational speed. During time intervals of suitable length the phase shift can often exceed some radians. Often this dynamic behavior is caused by light rotor-to-stator rubs that generate friction forces. Therefore, owing to full-annular rubs or periodic partial-arc rubs some heat is transmitted to the shaft. As a consequence of this phenomenon, a time-dependent rotor thermal bow is generated. In general, this causes 1X vibrations whose amplitude changes depend of the variations of the shaft bow severity. At the same time the diametrical plane of the rotor in which the bow propagates tends to slowly rotate: this causes a progressive change of the phase of the 1X vibration originated by the bow itself. Moreover, it is necessary to consider that the 1X vibration, caused by the shaft thermal bow generated by light rotor-to-stator rubs, is superimposed to that caused by the unavoidable residual unbalance of the shaft and further possible sources of 1X excitations. When this dynamic behavior occurs, by plotting the historic trend of the 1X vibration vector in a polar plane, it is possible to obtain a spiral curve. Therefore, this kind of behavior is called spiral vibration. Other authors prefer to indicate this phenomenon as thermally induced vibrations, vector turning from synchronously modulated rub or Newkirk effect, from the author of the first literature paper in 1926 [1] (even if the phenomenon was already described in a GE report of 1924 by Taylor [2]). Experimental evidences of 1X vibrations are shown by Kellenberger [3] on a 722 MW turbo generator operating at 3600 rpm, by Schmied [4] on a 600 MW turbo generator, by Stegemann et al. [5] on a 100 MW 3-stage condensation turbine with re-heater during different output power operating, by Muzynska [6] on a turbine rotor (the same diagram is also reported in [7]), by Liebich and Gasch [8] who present the results measured by Drechsler on a combined cycle plant operating at 3000 rpm, by Bachschmid et al. [9] for a generator of a 50 MW combined cycle unit operating at 3000 rpm and by Eckert and Schmied [10] for a 450 MVA generator operating at 3600 rpm.

2905

In the case of expansive vibration amplitudes, the spiral vibration is called unstable. Conversely, when the amplitude continuously decreases the spiral vibration is called stable. In [3] some criteria are also presented to establish the relationships between stability and the critical speeds vs. operating speed.

2 Modeling of spiral vibrations

In literature, some different mathematical methods are available to study and simulate light rotor-to-stator rubs and related spiral vibrations. With very few exceptions (those of Schmeid [4], Eckert and Schmied [10], Liebich and Gasch [8] and Liebich [11], but all of them employ simplified thermal models), these papers use a simple Jeffcott rotor to model the rotating machine while a rigid stator is often considered, so that its motion can be neglected, and a restricted number of degrees of freedom are taken into account (see the models proposed by Kellenberger [3], Gruber [12], Larsson [13][14] and Sawicki et al. [15]). This kind of simplification does not allow one to model in detail the thermal phenomena that are related to rotor-to-stator contacts, although they can significantly affect the machine dynamic behavior and rub development. The consequence of the shaft heating is not negligible as the tangential force due to the friction may cause a significant thermal bow of the shaft and spiral vibrations can be engaged. Moreover, the anisotropy of oil-film bearings causes not circular orbiting of the shaft, and the condition in which the rotor comes into contact with the stator has to be carefully analyzed. An accurate model for a real rotor-bearings-foundation system, which takes into consideration the contact conditions and forces, the shaft thermal bow induced by the heat introduced by the friction and the stability analysis of the dynamical behavior, has been first proposed by the authors in [16][17] and then improved and validated by means of experimental results in [9], where it is also fully described. Here, only a brief summary is given. Figure 1 shows the flowchart of the domain swap method used in authors’ model. Starting from the consideration of a rotating machine, its fully assembled model is taken into account and the 1X dynamical behaviour in frequency domain is calculated at the operating speed on the basis of initial exciting conditions. In this phase a standard 1D finite beam element model, with gyroscopic, shear and secondary inertia effects, is used for the rotor.

Thermal modelFrequency response (1X vibrations)

Elliptical orbitsRub occurrences

Yes

No

Temperaturedistribution

Shaft strains

Heatgeneration

Figure 1: Flowchart of authors’ model for spiral vibration [9]

2906 PROCEEDINGS OF ISMA2010 INCLUDING USD2010

The rub conditions are then analyzed. If the rotor interferes with the stator, the first step of the simulation starts: the interaction causes the contact forces Fn and Ft, respectively the normal and tangential component, on the rotor and, due to the friction, power is introduced in a small part of the rotor surface:

Q

( )2,ii

n s t sk e k eπσσδ μ δ

−= − =F F (1)

2sDQ kμ δ= Ω (2)

where ks is the rotor-to-stator contact stiffness, because elastic deformation is considered, μ is the friction coefficient, δ the average deformation over a rotation, σ the phase of the hot spot (HS), i.e. of the point where contact forces are applied, Ω is the machine rotational speed and D is the shaft diameter at the cross section where the rub occurs. This determines temperature gradients in radial and axial directions in the cylindrical rotor part that is close to the rubbing section. In order to calculate the strain distribution, this rotor part is modeled by means of a 3D mesh and Fourier’s equations are applied and integrated in the time domain:

2 2

2 2 2

T K T T T Tt C r r r r zρ ϑ

⎛ ⎞∂ ∂ ∂ ∂ ∂= + + +⎜ ⎟∂ ∂ ∂ ∂ ∂⎝ ⎠

2

2 (3)

where K is the thermal conductivity coefficient, ρ the mass per unit volume, t the time, z the axial co-ordinate, r the radial co-ordinate, ϑ the angular co-ordinate, C the specific heat and T the temperature. Eq. (3) is a parabolic PDE in the unknown temperature distribution ( , , , )T T r z tϑ= . An initial uniform temperature distribution T0 as well as boundary conditions as specified below are coupled to Fourier’s equation for evaluating the temperature distribution on the external surface. Except for the HS, thermal balance requires that the thermal power, which flows into the rotor through its external surface and which diffuses by conduction inside the rotor, is equal to the thermal power exchanged by convection between the rotor and the surrounding fluid. Therefore, the heat transfer rate, due to each element of surface dS by conduction, is:

d dT T TQ K T S Kr r zθ

⎛ ⎞∂ ∂ ∂= − ∇ = − + +⎜ ⎟∂ ∂ ∂⎝ ⎠

S (4)

The heat transfer rate, due to convection on the infinitesimal section dS of the external surface, is:

( )dfQ H T T S= − (5)

where H is the thermal convective coefficient and Tf the temperature of the surrounding fluid. Considering eqs. (4) and (5), the temperature on the external surface of the rotor is given by:

( )fT T TK Hr r zθ

⎛ ⎞∂ ∂ ∂− + + = −⎜ ⎟∂ ∂ ∂⎝ ⎠

T T (6)

In the finite element having the HS on its surface, the boundary condition has to consider the thermal power introduced by the rubbing, so Fourier’s equation (3) becomes:

2 2 2

2 2 2 2

T K T T T T Qt C r r r r z Cρ ϑ ρ

⎛ ⎞∂ ∂ ∂ ∂ ∂= + + + +⎜ ⎟∂ ∂ ∂ ∂ ∂⎝ ⎠

(7)

MONITORING AND DIAGNOSTICS OF ROTATING MACHINERY 2907

where Q is the specific thermal power.

The thermal strain distribution σz can cause a thermal bow of the rotor, that can be reproduced by means of equivalent bending moments on the nodes of 1D finite beam model:

( )2 2

0 0( ) d , sin d d

R

x z x zAM z b A r r r

πσ σ ϑ ϑ= =∫ ∫ ∫ ϑ

( )2 2

0 0( ) d , cos d d

R

y z y zAM z b A r r r

πσ σ ϑ ϑ= =∫ ∫ ∫ ϑ

(8)

In this way, the equivalent excitations acting on the rotor are given by the initial exciting conditions (the original unbalance and bow of the rotor), by the contact forces determined in the current simulation step and by the equivalent bending moments that reproduce the thermal bow of the rotor in the current step:

1 1[ ] [ ( )] [ ( )]

m mi t

unb b x y n t e Ω⎛ ⎞+ Ω + Ω = + + + + +⎜ ⎟

⎝ ⎠∑ ∑M x D x K x F M M M F F (9)

Then the 1X response is again calculated in the frequency domain, the rub conditions are analyzed and the next simulation step starts. In all the following iteration steps, if the rotor does not come in contact with the stator, contact forces are equal to zero, no heat is introduced, but Fourier’s equations are solved anyway to define the new temperature and strain distribution and finally the new values of the equivalent bending moments. The procedure is valid as long as the thermal transient time constant is much higher than the vibrational transient time constant, as is also observed in [18]. This condition is usually satisfied in large rotating machines that have considerable thermal inertia. This model has been used to simulate spiral vibrations in different types of rotating machines. In particular, depending on the characteristics of the machine and of its rotational speed vs. critical speeds, unstable and stable spiral vibrations (even with limit cycles) caused by thermal bow have been obtained. As an example, the rotating machine shown in Figure 2 has a different spiral vibration behavior depending on its rotational speed during rubbing.

Brg. #1 Brg. #2 Brg. #3

Rubbing section

Brg. #4 Brg. #5

1000

2000

3000

4000

5000

30

210

60

240

90

270

120

300

150

330

180 0

3100 rpm

3200 rpm

3300 rpm

3400 rpm

Vibration vector in the sealVertical direction [µm]

Figure 2: Rotor model Figure 3: Simulated spiral vibrations in the range 3100-3400 rpm


This simulated case study is related to rub on a sealing carter, in which rather large clearances are allowed. Figure 3 shows unstable spiral vibrations for different rotational speeds. The evolution of the spiral is getting slower with the increase of the rotational speed. For the considered example, a limit cycle is reached at 3500 rpm, as shown in Figure 4. During the simulation the rotor rubs against the stator for a certain period, warms up, changes its thermal bow, looses contact with the stator, straightens itself, comes again in contact and the sequence continues in the same way. This behavior is more evident if the temperature distribution on the rotor surface corresponding with the rubbing section (Figure 5) and the equivalent bending moments (Figure 6) are considered. Spiral vibrations become stable at higher rotational speeds as it is shown in Figure 7 for 3700 rpm.

50

100

150

30

210

60

240

90

270

120

300

150

330

180 0

HorizontalVertical

Vibration vector in the sealVertical and horizontal direction [µm]

Figure 4: Limit cycle for spiral vibrations at 3500 rpm

Figure 5: Temperature on rotor surface in the rubbing section during the limit cycle

20

40

60

80

100

30

210

60

240

90

270

120

300

150

330

180 0

HorizontalVertical

Vibration vector in the sealVertical and horizontal direction [µm]

Figure 6: Equivalent bending moments acting on the rotor during the limit cycle

Figure 7: Stable spiral vibrations at 3700 rpm


3 Experimental case history

The capability of above described model to reproduce spiral vibrations has been shown in the previous section. In this one, the experimental dynamic behavior of the unit of a 70 MVA thermoelectric power plant is investigated. The shaft-train was composed of a single cylinder steam turbine (ST) and a generator. A load coupling was mounted between the shafts of these two main machines. Figure 8 shows the machine-train diagram and the bearing numbers. The steam turbine was supported on two four-pads tilting-pad journal bearings whose configuration was with the bearing load on pad, while the generator was supported on two elliptical oil-film journal bearings. The operating speed of the unit was 3000 rpm. Each bearing was equipped with a pair of XY proximity probes whose orientation is shown in Figure 8.

#2

#1 SteamTurbine

#4#3

Loadcoupling

Generator

X45°

Y45° Probe

orientation

DE NDEDENDE

Figure 8: Machine-train diagram and bearing numbers

At the end of a long planned outage, during which the machine was subjected to some maintenance activities and important overhauls, the unit was started up. Figure 9 and Figure 10 show the Bode plot of the synchronous vibration (1X) of the turbine shaft measured during the runup, in the X and Y directions, by the proximity probes mounted on bearings #1 and #2, respectively. The amplitude of these vibrations was rather low over the entire range of the rotational speed, that is even when passing through the first balance resonance of the steam turbine as well as at the operating speed. This indicates that the residual unbalance of the ST was rather small. Owing to the anisotropy of the oil-film journal bearings, the first flexural critical speed of the turbine shaft was split into the two values of 1390 rpm and 1590 rpm.

0 500 1000 1500 2000 2500 3000-180

-90

0

90

180

Rotational speed [rpm]

Pha

se [

degr

ee]

Bearing #1: 1X transient vibration

XY

0 500 1000 1500 2000 2500 30000

5

10

15

20


Am

plitu

de [

µm p

p] XY

0 500 1000 1500 2000 2500 3000-180

-90

0

90

180


Pha

se [

degr

ee]


XY

0 500 1000 1500 2000 2500 30000

5

10

15


Am

plitu

de [

µm p

p] XY

Figure 9: Bode plot of the turbine 1X vibration measured on bearing #1 during a runup

Figure 10: Bode plot of the turbine 1X vibration measured on bearing #2 during a runup


Figure 11 shows the Bode plot of the 1X vibration measured during the runup by the X and Y proximity probes mounted on the generator driven end (DE) bearing #3. Also in this case, accordingly with the very low vibration level measured over the entire range of the rotational speed, it is possible to assume that the residual unbalance of this shaft was rather small. However, the amplification of the 1X vibration caused by the passing through the two lowest flexural critical speeds of the generator, 1010 rpm and 1600 rpm, is evident. Anyhow, the machine transient vibrations pointed out that no flexural critical speed of the shaft-train was close to the operating speed. At the end of the runup, the unit was operated at the Full-Speed-No Load condition for a short time. After a partial load rise up to 22 MW the unit was operated with this constant load for about 35 minutes. Then, the load was increased to 45 MW and kept constant for about 80 minutes. During this period of observation, which started from the end of the runup, although the level of the machine vibrations was very low, a mechanical component of the supporting structure of the case that covered the load coupling suffered a partial yielding. This caused a vertical displacement of the case and a consequent significant change of the radial clearance. Therefore, shaft-to-case contacts occurred. Likely, this malfunction mainly caused light partial-arc rubs rather than severe impacts or full-annular rubs. Figure 12 shows the historic trend of the 1X vibration vector of the turbine shaft measured by the XY proximity probes mounted on the turbine bearing #2. It is possible to note that, after the machine reached the operating speed for a period of 60 minutes, the amplitude of the 1X vibration was very low (about 20 μm pp) and nearly constant. Then, the 1X vibration level started to increase with an average rate of change of about 3.7 μm pp/min and 2.8 μm pp/min, in the X and Y directions, respectively.

0 500 1000 1500 2000 2500 3000-180

-90

0

90

180


Pha

se [

degr

ee]


XY

0 500 1000 1500 2000 2500 30000

20

40

60


Am

plitu

de [

µm p

p] XY

0 15 30 45 60 75 90 105 120 135 150-180

-90

0

90

180

Time [minutes]

Pha

se [

degr

ee]

Bearing #2: 1X vibrations in operating condition

XY

0 15 30 45 60 75 90 105 120 135 1500

50

100

150

Time [minutes]

Am

plitu

de [

µm p

p] XY

Figure 11: Bode plot of the generator 1X vibration measured on bearing #3 during a

runup

Figure 12: Historic trend of the 1X vibration measured on bearing #2 in operating condition

Within 30 minutes, on bearing #2, the turbine vibration reached a maximum level of 140 μm pp. Since the alarm level of the ST was improperly set to 180 μm pp and the unit was not yet operated in commercial service, the attention of the personnel control room was mainly focused on other technical problems whose solution required the above mentioned maintenance actions. Therefore, they did not timely notice either the continuous increase of the ST vibrations and the abnormal levels that they reached at the DE bearing #2. Therefore, the unit continued to be operated with a constant load of 45 MW while the amplitude of the 1X vibration began to decrease, without doing any significant variation of the most important process parameters, with an average rate of change of about 5 μm pp/min in X and Y directions. Likely, this behavior was caused by changes in the radial position of the load coupling case induced by the progressive yielding of its supporting structure. This fault was caused by an improper reassembling performed within the maintenance actions.


About 20 minutes after the beginning of their negative trend these vibration levels reached a minimum value of nearly 25 μm pp (Figure 12). Then, during the following 35 minutes, further similar but less considerable amplifications and subsequent decreases of the 1X turbine vibration levels occurred. However, within these events a maximum amplitude of only 80 μm pp was reached. Finally, this abnormal dynamic behavior was noticed by the control room operators. It was decided to stop the unit and to perform a visual inspection of the portion of the shaft-train close to the load coupling, that is the area located between steam turbine and generator. The yielding of the supporting structure of the case that covered the load coupling was evident, therefore the cause of the abnormal vibrations that occurred in the operating condition at the drive end of the steam turbine was immediately identified. Anyhow, as shown below, the same results would have been obtained also by means of the on-line application of diagnostic techniques based on the analysis of the monitoring data. Figure 13 shows the polar plot of the 1X vibrations of the turbine shaft measured on bearing #2, in the X direction and in operating condition, during the same observation period of the historic trend illustrated in Figure 12. Therefore, the initial 1X vector plotted in this figure was measured at the end of the machine runup while the final vector data was collected after having operated the unit for 150 minutes in on-load condition. This polar diagram emphasizes that, in the time interval between the 60-th minute and the 90-th minute, the amplitude of this vibration vector significantly increased, while the phase was subjected to only minor changes. Then, the vibration amplitude decreased from 140 μm pp to 25 μm pp together with a gradual and considerable rotation of the 1X vector phase of about 160° degrees. This phenomenon caused a stable spiral vibration, with a limit cycle. Further important changes of amplitude and phase, occurred after the 115-th minute, caused inner loops in the polar diagram of the 1X vibration vector. Therefore, the polar plot of the 1X vibration of the turbine shaft shows the occurrence of spiral vibration. In general, this phenomenon is a typical symptom of light rotor-to-stator rubs.

50

100

150

30

210

60

240

90

270

120

300

150

330

180 0

[µm pp]Initial vectorFinal vector

Rotational speed: 3000 rpm

X [µm pp]

92-th minute

Figure 13: Polar plot of the 1X vibration measured on bearing #2 in on-load operating conditions

Figure 14 shows the historic trend of the 1X vibration of the generator rotor measured by the XY proximity probes mounted on the DE bearing #3. The maximum level of these vibrations (72 μm pp) was significantly lower than that of the respective 1X vibrations measured at bearing #2. Moreover, in the time interval between the 60-th and 115-th minutes, when the most important rub phenomena occurred accordingly with the considerable amplification of the 1X vibration measured at the DE bearing of the ST (Figure 12), the amplitude of the 1X harmonic order, measured at bearing #3, showed only minor changes. Its maximum peak occurred at the 115-th minute, that is nearly in coincidence with a relative minimum of the amplitude of the 1X vibration measured on bearing #2. Conversely, during the observation period the phase of the 1X vibration measured at both bearings #2 and #3 was affected by considerable continuous changes.


The noticeable differences between the amplitude curves of the 1X shaft vibrations measured on the adjacent bearings #2 and #3 was surprising since these measurement planes are very close together. However, on the DE bearing of the steam turbine the effects of the rubs on the shaft vibration were noticeable as the amplitude of the 1X vector increased from 20 μm pp to 140 μm pp. Conversely, the vibration levels showed a maximum change of only 35 μm pp on the DE bearing of the generator. Therefore, although the amplitude of the 1X vibration measured on bearing #3 before the occurrence of the rub phenomena was rather low, but not negligible, it was supposed that it significantly influenced the shape of the historic trend of the amplitude of this harmonic component of the generator vibration. It is well known that rotor-to-stator rubs in rotating machinery cause friction forces, shaft heating and non-linear effects in the machine dynamic behavior. Sometimes, depending on different factors, the amount of these non-linear effects can be not very important. Therefore, accordingly with the assumption that the non-linear behavior of the rotor system could be neglected, an attempt was made to estimate the 1X generator vibration caused on bearing #3 by the rubs only. The 1X vibrations measured at the beginning of the load rise, in a normal condition, were considered as reference vectors. Then, these vectors were subtracted from every 1X vibration data, associated with bearing #3, collected in operating condition. This strategy allowed obtaining an estimate of the compensated 1X vibrations that, on the basis of the above mentioned assumptions, can be mainly ascribed to the dynamic effects caused by the rubs. Figure 15 shows the historic trend of the 1X compensated vibration of the generator rotor measured by the XY proximity probes mounted on the DE bearing #3. The amplitude curves illustrated in this figure are rather similar to those shown in Figure 12, which are associated with the raw 1X uncompensated vibrations measured on bearing #2. This result indicates that the rubs that occurred in a shaft cross-section located near the DE support of the steam turbine caused similar 1X vibrations also at the generator bearing #3. However, the amplitude of these vibrations was nearly 40% lower than that of the vibrations that occurred on bearing #2. In order to validate the results of this qualitative analysis, the linear correlation coefficients between the amplitude of the 1X vibrations measured on bearings #2 and #3 were evaluated. With regard to the generator dynamic behavior, both uncompensated and compensated 1X vibrations have been considered within this study. Conversely, since the dynamic effects caused by the rubs at bearing #2 were considerable in comparison to those caused in normal condition by the turbine residual unbalance, only the raw uncompensated 1X vibrations measured at the DE bearing of the turbine were taken into account in this correlation analysis. Table 1 shows the results of this investigation, within which the monitoring data collected during different time intervals have been considered. The first interval, during which the most important rub phenomena occurred, is the most significant to point out the correlation between the vibrations, which can be ascribed to the rubs only occurred on bearings #2 and #3.

0 15 30 45 60 75 90 105 120 135 150-180

-90

0

90

180

Time [minutes]

Pha

se [

degr

ee]


XY

0 15 30 45 60 75 90 105 120 135 1500

20

40

60

80

Time [minutes]

Am

plitu

de [

µm p

p] XY

0 15 30 45 60 75 90 105 120 135 150-180

-90

0

90

180

Time [minutes]

Pha

se [

degr

ee]


XY

0 15 30 45 60 75 90 105 120 135 1500

20

40

60

80

Time [minutes]

Am

plitu

de [

µm p

p] XY

Figure 14: Historic trend of the raw 1X vibration measured on the generator bearing #3 in on-load

operating conditions

Figure 15: Historic trend of the compensated 1X vibration measured on the generator bearing #3

in on-load operating conditions


TIME INTERVAL LINEAR CORRELATION COEFFICIENT

[ minute ]

MEASUREMENT POINT

DIRECTION Compensated

Vibrations Uncompensated

Vibrations X 0.97069 0.70136 from 45 to 112 Y 0.91094 - 0.41579 X 0.97666 0.82499

from 15 to 112 Y 0.93346 - 0.229789 X 0.89491 0.55922

from 15 to 147 Y 0.82007 - 0.25433

Table 1: Linear Correlation Coefficient between the amplitude of the 1X vibrations measured on bearings #2 and #3, in the X and Y directions, during different time intervals

It is possible to note that, when the uncompensated vibrations at bearing #3 were considered, the linear correlation coefficients were not very high in the case of vibrations measured in the X direction, while they were even negative in the case of vibrations measured in the Y direction. Conversely, when the compensated 1X vibrations measured at bearing #3 in the X and Y directions were taken into account, both linear correlation coefficients were rather close to unity. Although rub phenomena cause non-linear effects in the dynamic behavior of rotating machines the results of this investigation show that the linear correlation between the 1X vibrations that can be mainly ascribed to this fault, measured at two different cross-sections of the shaft-train that are close to the area where the rubs occurred, is very high. Conversely, the analysis of the raw uncompensated vibrations would have given misleading diagnostic results. Figure 16 shows the polar plot of the 1X uncompensated vibrations of the generator rotor measured on bearing #3, in the X direction, in the operating conditions that occurred during the same observation period of the historic trend illustrated in Figure 12. Spiral vibrations occurred also at this measurement point. Owing to the spiral vibrations, the overall rotation of this 1X vector was nearly 810° degrees.

20

40

60

80

30

210

60

240

90

270

120

300

150

330

180 0

[µm pp]Initial vectorFinal vector

Rotational speed: 3000 rpm

X [µm pp]

92-th minute

Figure 16: Polar plot of the 1X uncompensated vibration measured on bearing #3 in on-load


Figure 17 shows the historic trend of the amplitude of the 2X shaft vibrations measured by the XY proximity probes mounted on bearings #2 and #3. These vibration data were collected in the same observation period of the historic trend illustrated in Figure 12. It is possible to note that on the generator DE bearing the amplitude of the 2X vibration was nearly constant: its value was not negligible owing to the typical anisotropy of the cross-section of the main body


of the rotor. The level of the 2X vibrations measured at the DE bearing of the steam turbine is rather low, but it is affected by evident changes that are correlated with the historic trend of the amplitude of the respective 1X vibrations. Likely, these 2X vibrations were influenced by non-linear effects in the oil-film journal bearings that may become not-negligible when high 1X vibration amplitudes occur.

0 15 30 45 60 75 90 105 120 135 1500

10

20

30

Time [minutes]

Am

plitu

de [

µm p

p]

0 15 30 45 60 75 90 105 120 135 1500

2

4

6

8

Time [minutes]

Am

plitu

de [

µm p

p] Bearing #2

XY

XY

2X vibrations in operating condition

Bearing #3

Figure 17: Polar plot of the 2X uncompensated vibration measured on bearing #2 and #3 in on-load


The changes of the shape of the 1X filtered journal orbits caused by the dynamic effects of the rubs were investigated. The upper and lower diagrams of Figure 18 show a sequence of 1X filtered orbits measured on bearing #2 at the times reported in Table 2. When approaching the 90-th minute after the end of the runup, the size of these elliptical orbits increased, while the ratio between the major and minor principal axes showed only minor changes. That is, in general, the rubs did not affect very much either the flatness of the orbits and the inclination angle of their major axis. Conversely, some important changes occurred in the angular position of the keyphasor dot of the orbits, especially after the 100-th minute of the observation period. A similar analysis was performed considering the generator vibrations. The upper and lower diagrams of Figure 19 show the sequence 1X filtered orbits measured on bearing #3 at the times reported in Table 2. In this case, the dynamic effects of the rubs, which were superimposed on those caused by the generator residual unbalance, caused important changes not only of the orbit size, but also of the ratio between the major and minor principal axes and of the inclination angle of the major axis of the 1X orbit.

ORBIT IDENTIFICATION NUMBER 1 2 3 4 5 6 7 8 9 10 11 12 13 14

Time [minute] 2 64 72 81 89 96 99 101 105 107 114 119 122 127

Table 2: Times at which the 1X filtered orbits shown in Figure 18 and Figure 19 have been evaluated


-60

-30

0

30

60

[µm

]

Bearing #2: 1X filtered orbits in operating condition

1 2 3 45 6

7

-60

-30

0

30

60

[µm

]

8 9 10

11 12 13 14

-30

-15

0

15

30

[µm

]

Bearing #3: 1X filtered orbits in operating condition

1 2 3 4 5 6 7

-30

-15

0

15

30

[µm

]

8

9 10

11

12

13

14

Figure 18: 1X filtered orbits measured on bearing #2 in on-load operating conditions at the

times reported in Table 2

Figure 19: 1X filtered orbits measured on bearing #3 in on-load operating conditions at the

times reported in Table 2

In the end, important changes occurred also in the angular position of the keyphasor dot of these orbits. Figure 20 shows the historic trend of the angular position of the keyphasor dot of the 1X orbits measured at bearings #2 and #3, during the same observation interval of the data illustrated in Figure 12 and Figure 14. It is possible to note that after the 90-th minute, that is when the amplitude of the 1X vibration on bearing #2 began to decrease, after the occurrence of the first important rub onset, the angular position of the keyphasor dot began to show significant changes. Since the rotational speed was constant, this phenomenon can be explained as the consequence of changes of the phase of the vectors associated with forces and bending moments that can be used to simulate the effects of the actual excitations acting on the shaft, caused by rubs and residual unbalance. Given that magnitude and phase of the residual unbalance can be considered constant during the observation period, the changes of the angular position of the keyphasor dot were mainly ascribed to changes of the angular position of the diametrical plane that contained the shaft thermal bow originated by the heat transmitted to the rotor as a consequence of the friction forces caused by the rubs. In Figure 20 the times at which the orbits shown in Figure 18 and Figure 19 have been considered are reported by means of vertical lines.

0 15 30 45 60 75 90 105 120 135 15090

270

450

630

Time [minutes]

[deg

ree]

Keyphasor-dot angular position

1 2 3 4 5 6

1411

Brg. #2Brg. #3

Figure 20: Historic trend of the angular position of the keyphasor dot along the 1X filtered journal

orbits measured on bearings #2 and #3 in operating condition

In general, in the case of common spiral vibrations, the changes of the phase of the 1X vibration vector should occur in the opposite direction with respect to that of the shaft rotation. Conversely, in the present


case, the direction of rotation of the shaft-train coincided with that of rotation of the 1X vibration vectors. Also the keyphasor dot showed a counterclockwise rotation along the 1X filtered journal orbit. This may be the consequence of changes in the radial clearance between shaft and load coupling case caused by a progressive yielding of the supporting structure and then by subsequent case subsidences. However, during the observation period, the changes of amplitude and phase of the 1X vibration vectors, which originated the spiral vibrations, occurred with a variable rate. Therefore, in order to obtain more detailed diagnostic information, the rate of change of the above mentioned parameters was investigated. The upper diagram of Figure 21 shows the historic trend of the Rate of Change of the Phase of the 1X Vibrations measured at bearing #2 in the X direction. For the sake of brevity, in the following this parameter that is a powerful tool for analyzing spiral vibration behavior, will be denoted as RC1XVP. During the first 65 minutes of the observation interval, that is before the first important increase of the turbine vibration levels, the RC1XVP was basically negative and rather small, with the exception of few events of very short duration. Then, the RC1XVP became positive. However, for about 25 minutes, it remained lower than 2.5 deg/min. During this period, as a consequence of the shaft-to-case rubs, the vibration levels showed a considerable rise from 20 μm pp to 140 μm pp. After the 90-th minute, when the amplitude of the 1X vibration reached the maximum value, the RC1XVP started to gradually increase. Then, after about 20 minutes, this parameter showed a noticeable peak reaching a maximum value of 39 deg/min that was associated with an important relative minimum of the amplitude of the turbine vibration occurred after the rub phenomena. Likely, the considerable increase of the rate of change of the 1X vector phase was caused by a temporary cessation of the rubs and friction forces and by a consequent partial cooling of the shaft. In accordance to this assumption, the magnitude of the shaft thermal bow decreased: this phenomenon could have caused a quick change of the HS angular position. After few minutes, the RC1XVP decreased and gradually tended to a nearly constant value of about 5 deg/min, when further light rubs occurred again.

0 15 30 45 60 75 90 105 120 135 150-40

-20

0

20

40

Time [minutes]

[deg

ree/

min

]

Rate of change of the 1X vibration vector

0 15 30 45 60 75 90 105 120 135 150-15-10

-505

1015

Time [minutes]

[µm

pp/

min

]

Rate of change of the 1X vector amplitude

Rate of change of the 1X vector phase

Figure 21: Historic trend of rate of change of the phase of the 1X vibrations measured at bearing #2

in the X direction

The lower part of Figure 21 shows the historic trend of the rate of change of the 1X vibration amplitude measured at bearing #2, in the X direction, during the same observation interval of the data illustrated in Figure 12. For the sake of brevity, in the following this parameter is denoted RC1XVA. During the first 60 minutes the absolute value of the RC1XVA was very low (below 0.7 μm pp/min). In the time interval


between the 60-th minute and the 90-th minute, when the first and most important rub phenomena occurred, the RC1XVA quickly approached a nearly constant value of about 3.7 μm pp/min. Therefore, the considerable rise of the 1X vibration levels caused by these shaft-to-case rubs occurred with a nearly constant rate of change. In the time interval between the 93-th and the 112-th minute the RC1XVA was negative and showed some significant fluctuations. This behavior could be the consequence of a temporary cessation of periodic 1X revolution partial arc rubs although occasional light rubs still occurred. Then, further irregular rub phenomena occurred, likely owing to a further yielding of the supporting structure of the load-coupling case. This caused the considerable changes of the RC1XVA parameter that can be noticed in the final part of the diagram illustrated in the lower part of Figure 21. When severe rotor-to-stator rubs occur, the frequency spectrum of the machine vibrations can contains important super-synchronous harmonic components. Conversely, light rubs like those that can originate spiral vibrations mainly show changes in the 1X synchronous vibration. In Figure 22 the frequency spectrum of the vibration measured in operating condition by the proximity probe mounted on bearing #2, in the X direction, when the maximum vibration level of the turbine shaft occurred (around the 90-th minute), is illustrated. In accordance with the presence of 1X spiral vibrations, this spectrum is dominated by the 1X harmonic component. Figure 23 shows the waterfall plot of the vibrations measured by means of the same probe, in operating condition, in the time interval between the 60-th minute and the 150-th minute. Also this plot shows that the rub evolution caused 1X vibrations only.

Figure 22: Frequency spectrum of the turbine vibration measured on bearing #2, in the X

direction, in occasion of the occurrence of the rub phenomena

Figure 23: Waterfall plot of the turbine vibration measured on bearing #2, in the X direction, in

occasion of the occurrence of the rub phenomena

4 Conclusions

Spiral vibrations are an interesting phenomenon that happens in rotating machines at rated speed as a consequence of thermal bows caused by rubs. A mathematical model able to reproduce the behavior of real machines, including the so-called stable and unstable vibrations, is presented in this paper. Then, an interesting case history, in which a 70 MVA steam power unit was affected by spiral vibration for a rather long period of observation is presented. Machine dynamic behavior is deeply analyzed and a new tool for spiral vibration analysis is introduced: the rate of change of the phase of the 1X vibrations. Experimental evidence shows that spiral vibrations affected only synchronous vibrations and that, contrary to other common cases of spiral vibrations, the direction of rotation of the shaft-train coincided with that of rotation of the 1X vibration vectors in the present case study.


References

[1] B.L. Newkirk, Shaft rubbing: relative freedom of rotor shafts from sensitiveness to rubbing contact when running above their critical speeds, Mechanical Engineering, Vol. 48, No. 8 (1926), pp. 830-832.

[2] H.D. Taylor, Rubbing shafts above and below the resonance speed (critical speed), General Electric Company, R-16709, Schenectady, NY (1924).

[3] W. Kellenberger, Spiral vibrations due to the seal rings in turbogenerators thermally induced interaction between rotor and stator, Journal of Mechanical Design, Vol. 102, No. 1 (1980), pp. 177-184.

[4] J. Schmied, Spiral Vibration of Rotors, Proc. of ASME Design Technology Conference, Boston (1987), pp. 449-456.

[5] D. Stegemann, W. Reimche, H. Beermann, U. Südmersen, Analysis of Short-Duration Rubbing Process in Steam Turbines, VGB Kraftwerkstechnik, Vol. 73, No. 10 (1993), pp. 739-745.

[6] A. Muzyńska, Thermal rub effect in rotating machines, ORBIT, Vol. 14, No. 1, Bently Nevada (1993), pp. 8-13.

[7] J.T. Sawicki, Some Advances in Diagnostics of Rotating Machinery Malfunctions, Proc. of JSME Annual Meeting 2002 - International Symposium on Machine Condition Monitoring and Diagnosis, Tokio (2002), pp. 17-24.

[8] R. Liebich, R. Gasch, Spiral vibrations - modal treatment of a rotor-rub problem based on coupled structural/thermal equations, Proc. of IMechE-6th Conference Vibrations in Rotating Machinery, C500/042/96 (1996), pp 405-413.

[9] N. Bachschmid, P. Pennacchi, A. Vania, Thermally Induced Vibrations due to Rub in Real Rotors, Journal of Sound and Vibration, Vol. 299, No. 4-5 (2007), pp. 683-719.

[10] L. Eckert, J. Schmied, Spiral Vibration of a Turbogenerator Set: Case History, Stability Analysis, Measurements and Operational Experience, Journal of Engineering for Gas Turbines and Power, Vol. 130, No. 1 (2008), pp. 012509-1-10.

[11] R. Liebich, Rub induced non-linear vibrations considering the thermo-elastic effect, Proc. of 5th IFToMM Int. Conference on Rotor Dynamics, Darmstadt (1998), pp. 802-815.

[12] J. Gruber, A Contribution to the Rotor Stator Contact, Proc. of 5th IFToMM Int. Conference on Rotor Dynamics, Darmstadt (1998), pp. 768-779.

[13] B. Larsson, Rub-heated shafts in turbines, Proc. of IMechE-7th Conference Vibrations in Rotating Machinery, Nottingham (2000), pp 269-278.

[14] B. Larsson, Heat Separation in Frictional Rotor-Seal Contact, ASME Journal of Tribology, Vol. 125 No. 3 (2003), pp. 600-607.

[15] J.T. Sawicki, A. Montilla-Bravo, Z. Gosiewski, Thermomechanical Behaviour of Rotor with Rubbing, International Journal of Rotating Machinery, Vol. 9 No. 1 (2003), pp. 41-47.

[16] N. Bachschmid, P. Pennacchi, A. Vania, Spiral Vibrations Due to Rub: Numerical Analysis and Field Experiences, Proc. of Schwingungen in Rotierenden Maschinen V, H. Irretier, R. Nordmann, H. Springer Editors, Vieweg Verlag, Braunschweig/Wiesbaden, Germany (2001), pp. 61-74.

[17] N. Bachschmid, P. Pennacchi, A. Vania, Modelling of Spiral Vibrations Due to Rub in Real Rotors, paper ISROMAC10-2004-096, Proc. of ISROMAC-10 Conference, March, Honolulu, Hawaii (2004), pp. 1-10.

[18] P. Goldman, A. Muszyńska, Rotor to stator, Rub-related, Thermal/Mechanical Effects in Rotating Machinery, Chaos, Solitons & Fractals, Vol. 5, No. 9 (1995), pp. 1579-1601.


spiral vibrations in power units: modelling and

Documents