rotordynamic modeling of centrifugal compressor rotors for use with active magnetic bearings

7/17/2019 Rotordynamic Modeling of Centrifugal Compressor Rotors for use With Active Magnetic Bearings

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Rotordynamic Modeling of Centrifugal Compressor Rotorsfor Use with Active Magnetic Bearings

Sergio E. DíazProfessor

Universidad Simón BolívarCaracas, Venezuela

[email protected]

Oscar De SantiagoProject LeaderCIATEQ, A. C.

Querétaro, Mé[email protected]

Víctor SolórzanoResearch Assistant

ITCCelaya, México.

[email protected]

ABSTRACTCurrent trends in modern compressor design are toward higher operating speeds, increasing power density for higher efficiency and oil-free operation (green technologies). One ofthese technologies is the use of magnetic bearing supports. Operation of a rotor supported by magnetic bearings requires a better knowledge of the natural frequencies and modalshapes at higher frequencies, compared with conventional, oil-lubricated hydrodynamic bearings, for application of active vibration control or to predict rotor behavior when in

contact with auxiliary bearings.This paper describes the development of a tuned rotordynamic model of an industrialcompressor rotor capable of predicting free-free modal shapes up to the 12th

KEYWORDS

mode at 3,992Hz (239 kcpm). This is accomplished by including a simplification of the impellersflexibility on the assumption that this is the origin of measured natural frequencies thatcannot be predicted by a conventional lumped-mass rotordynamic model. Experimentalmeasurements and rotordynamic model tuning are reported with details of the impellersflexibility included in the rotor finite element model. The results of this work are useful forthe design of compressor systems that operate with magnetic bearings.

Rotordynamics, magnetic bearings, turbomachinery.

1.

INTRODUCTIONDe Santiago et al. [1] recently develop an advanced bearing test rig in order to investigatethe rotordynamic effects with different rotor-bearing configurations. Initial experimentalworks are made with a real centrifugal compressor rotor in combination with ball rolling




bearing without levitation [2]. This is intended to investigate the dynamic behavior of a bearing mounted with clearance without being dropped by the magnetic bearings. Aftercharacterizing the bearings with clearance operation it is intended to levitate the rotor bymeans of active magnetic bearings and experiment with rotor drop. Rotordynamic modelingof the rotor and the auxiliary bearings is necessary in this case to predict rotor response.This is done initially following industrial practice with conventional lumped-mass transfermatrix formulation or finite elements. However a more accurate representation of the rotoris necessary when incorporating active magnetic bearings into the model in order to predictstability regimes correctly [3].

Nelson [4] develops finite elements for rotordynamic modeling that include gyroscopiceffects, rotary inertia and shear deflection based on Timoshenko beam theory. Childs [5] presents the finite element model development based on Nelson´s rotordynamic elements.

Current rotordynamic modeling techniques have proved that finite element approach withrelatively simple beam elements suffice to predict natural frequencies and modal shapes for

mostly every common application [6]. For some special applications sometimes a 3Drotordynamic model is developed [6-7], with the disadvantage of requiring more modelingand computing time (with no considerable improvement).

A rotor drop or the electronic control dynamics of magnetic bearings can induce very highfrequency excitations [8-10]. Thus in active vibration control for rotor-magnetic bearingsystems, an accurate model capable of predicting modal shapes at high frequencies (incomparison with a hydrodynamic bearing system) is necessary. In some instances, a bandwidth of up to 3 times the maximum operating speed is necessary to obtain satisfactoryresults and reliable, stable operation [6]. This is the reason why the experimental modalanalysis and a first attempt to include impellers flexibility are developed in [1] following

recommendations given by Vance [11].This work describes the engineering process developed to tune the rotordynamic model presented in [1], based in the estimation of the mass and inertial properties of the subjectrotor impellers. Mass properties of the impellers are approximated through a solid CADmodel and its dynamic behavior is approximated by a finite element model developed withcommercial software.

2.

TEST ROTOR DESCRIPTIONAs mentioned before, the subject rotor is a 57.8 kg, five-stage centrifugal compressor rotor.The rotor consists of a variable diameter steel shaft (AISI 4340), 5 impellers and a balancedrum. It also has a series of sleeves in order to guarantee the required surface finish androtor protection in a real environment. Figure 1 shows a cut view of the shaft and thelocation of the sleeves (except the sleeves between impellers).




Figure 1. Rotor shaft cut view (mm [in]).

The impellers production drawings are not available so they are measured and a CAD solidmodel estimates its properties, obtaining mass, inertia and location of the center of gravity.

The impellers are made of AISI SAE 4340 (except the first impeller of stainless steel),considered to have a 205 GPa elastic modulus, 0.29 Poisson ratio and a density of 7850Kg/m3

Figure 2. shows the axial location and a summary of properties for each of theimpellers and balance drum.

Figure 2. Impellers and balance drum properties and axial location (mm [in]).

Impellers and sleeves attach to the main rotor shaft though shrink fits. Kimball [12] reports(since the middle 1920’s) that internal damping developed possibly in shrink fittedcomponents can originate severe whirl instabilities above the first critical speed. Gunter[13] extends a Jeffcott rotor model to include rotor and foundation flexibility, internal andexternal damping, etc. and observed phenomena in agreement with Kimball. Vance [14] investigates the effects of loosening the fit on the threshold speed of instability on a testrotor. Vance [6] gives some recommendations to include the stiffening effect of a shrink

fitted disk assuming that it remains tight for all conditions. Jafri [15] investigatesexperimentally and theoretically the effects of internal friction and proposes to model theshrink fit as two separate shafts with a linear translational and torsional springs anddampers between them. The approach adopted here (explained in detail later) is similar toJafri’s approach with respect to the use of two separated shafts with linear and torsionalsprings; the main difference is that in this work the internal friction (coulomb damping) isnot included and the stiffness comes from the impellers flexibility instead of the shrinkfitted disks.




This work is focused on free-free modal shapes and tuning of the finite element model.Instability created by the internal friction (due to shrink fitted elements), is a self-exciting phenomenon caused by bending rotor modes and is not included in the present work because the test rotor is expected to work as a rigid rotor. Internal friction instability effectsare not visible in a free-free test. The stiffening effect of the shrink fitted sleeves isevaluated but it is considered to have very little influence in the predicted modal shapes.

3.

FIRST ROTORDYNAMIC MODEL AND EXPERIMENTAL

RESULTS SUMMARYThis section presents a brief summary of the work reported in [1] with respect to the initialrotordynamic modeling and experimental modal analysis.

The shaft is first modeled using the finite element method with 34 nodes (stations) and 48one-dimensional beam elements (Timoshenko beam elements). Six lumped mass and

inertia elements are added representing the impellers and balance drum. The impellers areconsidered to be rigidly coupled to the shaft as shown in Figure 3-a). Figure 3-b) showsthe discretized rotor with the added masses for the rigid impeller assumption.

a) b)

Figure 3. a) Rigid impeller model schematic. b) Discretized test rotor model for therigid impeller assumption.

To obtain actual free-free natural frequencies and mode shapes, the test rotor is hung withslings in an axial location close to the predicted vibration nodes of mostly all of the modes.Two accelerometers are used, one as a reference (mounted on the free end, maximumamplitude point for all the predicted modes) and the other being displaced over 12measurement points of the shaft. Table 1 summarizes the axial location for themeasurement points where the accelerometer is directly mounted on the shaft.

Table 1. Experimental measurement points axial location (from coupling end).

Point 1 2 3 4 5 6 7 8 9 10 11 12

Axial Location (mm) 12.70 101.60 184.15 260.35 400.05 482.60 552.45 628.65 730.25 827.53 948.18 1043.43

The excitation force is an impact on the coupling end of the rotor by means of a steel tipcovered with a cotton fabric, allowing exciting the whole desired frequency range withoutdamaging the shaft. Dynamic measurements are processed through a custom madeacquisition and processing code, capable of easily adjusting acquisition parameters (trigger,




pretrigger, sample rate, windows and gains) as well as transfer function and online averagecalculation. Ten measurements are taken for each accelerometer position and then averagedto obtain amplitude and phase for the frequencies of interest.

Table 2 shows a summary of the measured natural frequencies for the first twelfth modes.Also the predicted natural frequencies are presented, and its difference with measurements.Figure 4 shows the measured and predicted modal shapes.

Table 2. Measured and predicted (initial model) natural frequencies of test rotor infree-free condition.

Mode Measured Frequency [Hz] Predicted Frequency [Hz] Difference [%]

1 351 338.75 -3.5%

2 703 795.88 13.2%

3 1021 - -

4 1066 - -

5 1133 - -

6 1203 - -

7 1330 - -8 1468 1379.64 -6.0%

9 1881 1906.36 1.3%

10 2523 2445.55 -3.1%

11 3211 3177.37 -1.0%

12 3687 3992.16 8.3%

1 7

2 8

3 9

4 10

5 11

6 12

Figure 4. Modal shapes.

It is worth noting that from the 3rd to the 7th

Observation of the 2

measured mode there is no predicted naturalfrequency for the rotordynamic model with the rigid impellers assumption. These modesare presumed to be caused by significant impeller flexion and thus it cannot be predictedwith the described finite element model, which considers the impellers as added inertia andmass rigidly coupled to the rotor nodes (or stations).

nd

and 8th

Figure 4modes ( ) suggests that the rotor vibration at thesefrequencies is mainly related to the 3rd and 7th

Figure 5modes contribution. shows a typicalvibration spectrum for the free end of the rotor excited by an impact. This spectrum showsconsiderable amplitude for the 3rd to the 7th modes. Thus, it is expected that the non- predicted modes can influence the rest of the modal shapes and/or its natural frequencies,especially the 2nd and 8th modes.




Figure 5. Typical rotor spectrum caused by an impact excitation.

A rotor drop or the control dynamics in a system equipped with magnetic bearings couldlead to excitation of the high frequency modal shapes defined by the impellers flexibility.Thus, it is desirable to predict this effect with the rotor finite element model. Since theexperimental evidence indicates that higher modes (above the 7th

4.

ROTORDYNAMIC MODEL WITH FLEXIBLE IMPELLERS

mode) are predictedcorrectly, no further tuning is made on the rotordynamic model with the rigid impellerassumption.

In order to include the impellers flexibility, a finite element analysis of the rotor impellersis developed. The impeller is meshed with a commercial mesh generator using 20-nodeshexahedral, and 10-nodes tetrahedral elements with 3 DOF per node and quadrilateral 2Delements for meshing improvement. Having the impellers the same shape and construction(different dimensions), the expected modal shapes of them are the same but at differentnatural frequencies. The analysis is not expected to give the exact impeller naturalfrequencies but a qualitative behavior allowing to decide if its modal shapes would probably couple with the rotor vibration modes. The FEM analysis is developed only forthe middle impeller. Table 3 shows a summary of the impellers mass properties

approximated by the CAD solid model.Table 3. Impellers approximated mass properties.

Impeller Mass (kg)Ip

(kg*m 2

It

(kg*m ) 2 )

1 3.507 0.0284 0.0146

2 3.607 0.0291 0.0150

3 4.034 0.0321 0.0164

4 4.038 0.0319 0.0163

5 3.158 0.0232 0.0118

The nodes of the surface of the impellers in contact with the shaft (marked with an “A” inFigure 6) are restricted in all its degrees of freedom for the dynamic study presented next.




Figure 6. Impeller surfaces on contact with the shaft (fit surface).

Table 4 summarizes results for the first four impeller modal shapes. The 1st and 2nd modalshapes are bending modes. The 1st shape consists on bending along the axis A-A’ and the2nd shape bends along the axis B-B’. Thus it can be observed that the 1 st and 2nd

Table 4. Impeller modal shapes.

modes arethe impeller modes that would probably couple with the rotor shapes, affecting the totalrotor bending. The two impeller fundamental shapes are practically at the same naturalfrequency. The rest of the modal shapes are axysimmetric (symmetric with respect to theaxis of rotation) and unlikely to affect rotor lateral motions.

Mode

Predicted

freq. Hz)

Shape Mode

Predicted

freq. Hz)

Shape

1

897.443

1255.5

2

898.854

1657.8

Little variations on the impeller inlet channel inner radius, within the uncertainty range ofthe impeller geometry, results on natural frequencies over the whole measured range that isnot predicted by the model with rigid impellers. This confirms that the non-predictednatural frequencies could be produced by the impellers flexibility.

In this work the impeller movement in the fundamental shapes is approximated by aconcentrated mass and transversal inertia linked to the rotor by an angular stiffness, withthe appropriate values (shown later) to represent each impeller. Figure 7 shows theschematic representation of the proposed model.

A

A’B

B’




Figure 7. Flexible coupled impeller model schematic.

For the final model assembly, approximated mass and inertia values from Table 3 for theimpellers are taken as a base and then are proportionally adjusted (multiplying each valuefor the same factor) in order to obtain the total measured weight of the rotor. Table 5 givesa summary of the adjusted values of impeller mass and inertia.

Table 5. Adjusted impeller mass an inertia.

Impeller Mass (kg)Ip

(kg*m2)

It

(kg*m2)

1 3.943 0.0319 0.01642 4.056 0.0327 0.0168

3 4.533 0.0361 0.0185

4 4.542 0.0359 0.0183

5 3.553 0.0261 0.0133

Each of these impellers is then coupled to the shaft model by an elastic element, modeled asa constant stiffness internal bearing between two concentric shafts. This element has onlythe displacement (x,y) and rotational (αx, αy) direct stiffness. The crossed stiffness termsare null. The direct displacement stiffness is fixed with a value several orders of magnitudehigher than the flexural shaft stiffness (K xx=K yy=109lb/in) in order to simulate a rigid radialattachment of the impeller. The direct angular stiffness for the transversal rotation (impeller bending, K α

x, K α

y) is estimated pairing the measured natural frequencies for the 3rd to the

7th

1

mode with the impeller inertias, ordered from the heavier (lower frequency) to thelightest (higher frequency). The angular stiffness values for the impeller flexural attachmentare obtained with equation ( ) and are shown in Table 6. This procedure is shown also byVance [11] .

= (2)2 Where: is the impeller angular stiffness (=K αx=K αy

is the transversal moment of inertia [kg-m) [ N-m].

2

is the measured rotor natural frequency corresponding to the impeller [Hz].].

(1)

Table 6. Estimated impeller attachment angular stiffness from measurements of

rotor natural frequencies.

Impeller Kt (N-m)

1 9.386E+05

2 8.524E+05

3 7.598E+05

4 8.206E+05

5 9.257E+05




Figure 8 shows the rotor model from [1] with the impellers as point mass and inertiaslinked to the rotor by elastic elements. Table 7 summarizes the measured and predictedvalues for the first 12 natural frequencies of the flexible impeller model. Figure 9 summarizes predicted and measured modal shapes.

Figure 8. Flexible impellers rotor model.

Table 7. Measured and predicted natural frequencies considering flexible impellers[1].

Mode Measured freq. (Hz) Predicted freq. (Hz) Difference1 351 330.31 -5.9

2 703 752.14 7.0

3 1021 1025.45 0.4

4 1066 1041.44 -2.3

5 1133 1138.28 0.5

6 1203 1171.10 -2.7

7 1330 1282.61 -3.6

8 1468 1521.66 3.7

9 1881 1908.53 1.5

10 2523 2537.89 0.6

11 3211 3491.91 8.7

12 3687 4402.91 19.4

1 7

2 8

3 9

4 10

5 11

6 12

Figure 9. Predicted and measured modal shapes by the flexible impellers model.

The rotordynamic model with the flexible impellers assumption predicts the five naturalfrequencies that the model with rigid impellers is not capable of predicting. This is the mostnotable characteristic for the rotordynamic model with flexible impellers and confirms theeffect of the impeller inertia and attachment flexibility.

A clear tendency is observed in the higher rotor modes (8th to 12th modes). In general, thesemodes are not affected by impeller flexibility, since they are predicted by the rigid




impellers model. Nevertheless, the impellers flexibility effect, such as considered in themodel, increase the difference between measured and predicted natural frequencies. Forexample, in the 11th and 12th

Table 2

mode cases this differences are quite larger with flexible thanwith rigid impellers as can be seen in and Table 7.

It is clear that a better knowledge of the impellers geometry, that allows defining itsstiffness and inertia more precisely, should improve predictions. However, it can beobserved that in higher modes there is practically non lateral rotation of the impellers. The promising results for this model encourage a further tuning of the rotordynamic model withflexible impellers.

5. TUNED ROTORDYNAMIC MODEL WITH FLEXIBLE

IMPELLERSThe impellers fundamental modal shape is such that part of the impeller inertia

(corresponding to the impeller hub) has no significant movement relative to the shaft, andthe modal angular equivalent stiffness only acts on the impeller external section. Figure 10 shows a cut view of the fundamental modal shape of a typical impeller and its associatedrelative displacements. A hub zone (colored in blue) with displacements lower than 5% ofthe maximum displacement (colored in red) is clearly observed.

Figure 10. Impeller fundamental vibration shape cut view.

This more detailed analysis allows representing more accurately the impellers movementaccording to its modal shape. Figure 11 shows the tuned impeller model, which considersthe impeller as two sections: a central section (blue colored), that stays rigidly attached tothe shaft and an external section (orange colored), which moves angularly with respect tothe shaft across an elastic element. In contrast, the preceding flexible impeller model(shown in Figure 7) considers the hub and external section moving together.

Figure 11. Flexible coupled impeller tuned model schematic.

Due to geometric uncertainty, a detailed analysis for each impeller is not carried out but a

typical geometry is evaluated and then a proportional mass and inertia associated to therigid section and the mobile section is used in the model. Taking the impeller hub section asrigidly coupled to the shaft; and considering the rest of the impeller as a rigid body withangular movement (relative to the shaft) results in two impeller sections as shown in Figure

12. Table 8 summarizes the mass and inertia distribution considered as rigid or as angularlyflexible. Angular stiffness (K T) is then estimated with the same procedure described before,




using equation (1) with the transverse inertia value for each impeller moving section. Table

9 shows the newly calculated angular stiffness values.

a) Rigid impeller section

b) Flexible impeller section

Figure 12. Impeller sections for mass and inertia estimation.

Table 8. Impellers mass and inertias distribution for the tuned flexible-impellerrotordynamic model.

Mass (%) Polar Inertia (%) Transverse Inertia (%)

Rigid section 28.5 9.8 10.1

Flexible section 71.5 90.2 89.9

Table 9. Estimated impellers angular stiffness for the tuned rotordynamic modelwith flexible impellers.

Impeller Kt (N-m)

1 6.760E+05

2 6.139E+05

3 5.473E+05

4 5.910E+05

5 6.667E+05

Table 9 shows that taking just the impeller moving section inertia when calculating angularstiffness results in lower stiffness values, close to 80% the values used in the first modelwith flexible impellers.

Figure 13 shows the discretized rotor model considering the impellers divided in twosections, one rigidly coupled to the shaft and the other connected by elastic elements. Table

10 shows a comparison of the first 12 natural frequencies for the measured and the predicted values by the rotordynamic model with impellers as two separated sections (huband external section). Figure 14 summarizes the measured and predicted modal shapes forthe tuned rotordynamic model.

Figure 13. Tuned rotor model with flexible impellers.




Table 10. Measured and predicted natural frequencies of test rotor consideringsectioned flexible impellers.

Mode Measured freq. (Hz)Predicted freq.

(Hz)Difference

1 351 334.48 -4.7%

2 703 763.66 8.6%

3 1021 1025.45 0.4%

4 1066 1048.46 -1.6%

5 1133 1138.55 0.5%

6 1203 1174.17 -2.4%

7 1330 1285.43 -3.4%

8 1468 1558.79 6.2%

9 1881 1981.50 5.3%

10 2523 2617.92 3.8%

11 3211 3575.22 11.3%

12 3687 4604.96 24.9%

1 7

2 8

3 9

4 10

5 11

6 12

Figure 14. Predicted and measured modal shapes of test rotor for the flexibleimpellers tuned model.

With the divided impeller inertias in the rotordynamic model, the differences in the predicted values for the frequencies of the modes 1 to 7 stays practically unaltered. On theother hand, as expected, the predicted frequencies for the 8 th to 12th modes improvesignificantly, although they keep an overestimating bias.

The predicted modal shapes keep showing good correlation with the measured modalshapes for the entire frequency range. The major differences are observed in modes directlydefined by the impellers flexibility (3rd to 7th

6.

FINAL COMMENTS ABOUT THE ROTORDYNAMIC MODEL

mode), mainly where the uncertainty of themodel is concentrated due to the difficulty of measuring the internal impeller geometry.The predicted modal shapes with the proposed model allows to know, with an acceptableuncertainty, the system natural frequencies, the associated modal shapes and the nodal points location, providing the required information for locating the magnetic bearing and

tuning its control algorithm.

Figure 15 summarizes the three impeller models with a logical progression. In the 1 st

[16

model with the rigid impeller geometry, the impeller is considered as mass and inertiasrigidly coupled to the shaft. This is the traditional rotor modeling technique widely spreadin the compressor industry -17].




In the 2nd model there is a lateral view of the fundamental impeller mode and its shape isapproximated by an angular movement of the impeller considered as a rigid body with anangular stiffness spring connecting it to the shaft. A more detailed study, including thetransversal section of the deformed impeller, shows that the central section of the impellerdoes not have significant relative movement with respect to the shaft. This behavior issimulated by dividing the impeller in two sections, the first one (hub) rigidly coupled to theshaft and the second section connected to the shaft through an angular stiffness.

R i g i d m o d e l

F l e x i b l e

i m

p e l l e r

F l e x i b l e

I m p e l l e r

( T u n e d )

Figure 15. Impeller models comparison.

Table 11 allows comparing the predicted natural frequencies for each model. The meanquadratic error (rms ERROR) and the arithmetic error (mean ERROR) of the first 12natural frequencies are included for comparison. The quadratic mean gives an amplitude

measure of the differences between the measured and predicted data. The arithmetic meangives a measure of the prediction bias.

Table 11. Natural frequencies comparison for the three rotor models.

MeasuredRotordynamic model with

rigid impellers.Rotordynamic model with

flexible impellers.Tuned rotordynamic model

with flexible impellers.

ModeFreq.(Hz)

Freq. (Hz) Difference % Freq. (Hz) Difference % Freq. (Hz) Difference %

1 351 338.75 -3.5 334.25 -4.8 330.31 -5.9

2 703 795.88 13.2 758.27 7.9 752.14 7.0

3 1021 - -100 1023.72 0.3 1025.45 0.4

4 1066 - -100 1047.90 -1.7 1041.44 -2.3

5 1133 - -100 1144.12 1.0 1138.28 0.5

6 1203 - -100 1173.03 -2.5 1171.10 -2.7

7 1330 - -100 1285.22 -3.4 1282.61 -3.68 1468 1379.64 -6.0 1585.46 8.0 1521.66 3.7

9 1881 1906.36 1.3 1997.72 6.2 1908.53 1.5

10 2523 2445.55 -3.1 2652.66 5.1 2537.89 0.6

11 3211 3177.37 -1.0 3675.57 14.5 3491.91 8.7

12 3687 3992.16 8.3 4738.35 28.5 4402.91 19.4

rms ERROR 64.7% 10.2% 8.9%

mean ERROR -40.9% 4.9% 4.1%




It is worth mentioning that if the 3rd to the 7th

If all the modes of interest are considered,

modes which are not predicted by the rigidimpeller model are not included for the first model, an rms ERRROR of 6.6% would beobtained and a mean ERROR of 1.3%. This excellent prediction confirms the validity ofmodels used for traditional rotordynamic studies. However, these models, such as are usedcommonly (i. e. considering rigid impeller), have limitations in a rotordynamic study for amagnetic bearing system. In these applications, knowledge of the high frequency modes isrequired because they affect significantly the magnetic bearing control tuning and the rotorresponse when in contact with auxiliary bearings.

Table 11 shows that the first model gives thelargest global errors by not predicting 5 modes associated with impeller flexibility. Theimpeller flexibility inclusion in the 2nd

[3

model allows predicting modes associated with thisflexibility. This greatly minimizes the errors and provides information for the magnetic bearing control tuning; without this information the control would be tuned by trial anderror, considering the resonances that are not predicted as (large) model uncertainty ].

Finally, the third model shows a significant bias in the higher modes (8 th to 12th

ACKNOWLEDGMENTS

modes),where all the frequencies are overestimated. Better predictions are expected with a moreaccurate knowledge of the impellers geometry. However it is clear that even withsignificant uncertainty on the impellers geometry, the simple modification introduced in thelast model to include the impellers flexibility gives satisfactory results for a rotor to besupported by magnetic bearings.

The authors are thankful to CIATEQ A. C. for providing funds for this work. Thanks alsoto the Mexico National Council for Science and Technology (CONACYT), Universidad

Simón Bolívar, Instituto Tecnológico de Celaya, and to the following persons: Eloy E.Rodríguez, Fernando Aboites Dávila, Ramón Rodríguez Castro and Álvaro SánchezRodríguez.

REFERENCES[1] De Santiago, O., Díaz, S., Solórzano, V., Arcega, C., 2011, “Diseño de un banco de pruebas para cojinetes magnéticos y auxiliares a escala real,” XII Congreso y exposiciónlatinoamericana de turbomaquinaria, 21-25 febrero, Querétaro, Qro.[2] De Santiago, O., Solórzano, V., Rodríguez, E., Arcega, C., De Santiago, O., 2012,“Initial testing of auxiliary bearing for machines with active magnetic bearings,” XIIICongreso y exposición latinoamericana de turbomaquinaria, 12-14 marzo, Querétaro, Qro.

[3] Li, G., 2007, “Robust Stabilization of Rotor-Active Magnetic Bearing Systems,” Ph. D.Dissertation, University of Virginia, VA, USA, Jan.[4] Nelson, H. D., 1980, "Finite Rotating Element Using Timoshenko Beam Theory,”ASME Journal of Mechanical Design, Vol. 102, pp. 793.[5] Childs, D., 1993, “Turbomachinery Rotordynamics Phenomena, Modeling, &Analysis,” John Wiley and Sons, New York, USA.




[6] Vance, J., Zeidan, F., Murphy, B., 2010, “Machinery vibration and rotordynamics,”John Wiley & Sons, New Jersey, USA.[7] Rao, J. S., 2011, "History of Rotating Machinery Dynamics,” Springer, Bangalore,India.[8] Mushi, S., Lin, Z., Allaire, P., 2010, “Design, control and modeling of a flexible rotoractive magnetic bearing test ring,” Proceedings of ASME Turbo Expo 2010, GT2010-23619, Glasgow, UK.[9] Sahinkaya, M., Hadi, A., Abulrub, G., Clifford R., 2011, “An adaptive multi-objectivecontroller for flexible rotor and magnetic bearing systems,” Journal on Dynamics andSystems, Meas., Control Vol. 133, DOI:10.1115/1.4003421.[10] Polajzer, B., Ritonja, J., Stumberger, G., Dolinar, D., Lecointe, J., 2006,“Decentralized PI/PD position control for active magnetic bearings,” ElectricalEngineering, Vol. 89, DOI: 10.1007/s00202-005-0315-1.[11] Vance, J., 1988, "Rotordynamics of Turbomachinery," John Wiley and Sons, USA.[12] Kimball, A. L., 1924, “Internal Friction Theory of Shaft Whirling,” General ElectricReview,Vol. 27 (4), pp. 244-251.[13] Gunter, E. J., 1966, “Dynamic Stability of Rotor-Bearing Systems,” NASA TechnicalReport, SP-113.[14] Vance, J., Ying, D., 2001, "Effects of Interference Fits on Thresold Speeds ofRotordynamic Instability," Paper No. 2007, Proc. of the International Symposium onStability Control of Rotating Machinery, August 20-24, Tahoe, California, U. S. A.[15] Jafri, S., 2007, “Shrink Fit Effects on Rotordynamic Stability: Experimental andTheoretical Study,” PhD thesis, Texas A&M University, College Station, Texas, U. S. A.[16] API Standard 684, 2005, “Tutorial on Rotordynamics: Lateral Critical, UnbalanceResponse, Stability, Train Torsional and Rotor Balancing,” Second Edition, AmericanPetroleum Institute, Washington, D. C.

[17] API Standard 617, 2002, “Axial and Centrifugal Compressors and Expander-compressors for Petroleum, Chemical and Gas Industry Services,” Seventh Edition,American Petroleum Institute, Washington, D. C.

rotordynamic modeling of centrifugal compressor rotors for use with active magnetic bearings

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