INSTITUTE OF PHYSICS PUBLISHING JOURNAL OF MICROMECHANICS AND MICROENGINEERING

J. Micromech. Microeng. 17 (2007) 297–303 doi:10.1088/0960-1317/17/2/016

Rotordynamic characteristics of a microturbo generator supported by air foilbearingsYong-Bok Lee1, Dong-Jin Park1, Chang-Ho Kim1 and Keun Ryu2

1 Tribology Research Center, Korea Institute of Science and Technology,Seoul 136-791, Korea2 Department of Mechanical Engineering, Texas A&M University, College Station,TX 77843–2123, USA

E-mail: lyb@kist.re.kr

Received 4 July 2006, in final form 19 October 2006Published 9 January 2007Online at stacks.iop.org/JMM/17/297

AbstractThe rotordynamic characteristics of a micro power system supported by airfoil bearings were investigated. Stability analysis was performed by a finiteelement method with the predicted dynamic coefficients of the foil bearings.A preliminary test rig was developed to simulate the operating characteristicsof the micro power system. It consisted of a rotor supported by two air foiljournal bearings and two air foil thrust bearings, and an impulse driventurbine. The foil journal bearings had a diameter of 7 mm and a length of7 mm (L/D = 1). The test rig was operated stably under various situationsand speeded up to 300 000 rpm. The main portion of the rotor response wassynchronous and the amplitude of synchronous vibration was about5–20 µm. Further, theoretical and experimental results for the unbalanceresponse were compared. From this study, we showed the possibility ofstable performance for the micro power system supported by air foilbearings.

(Some figures in this article are in colour only in the electronic version)

Nomenclature

c bearing clearance (m)E elastic modulus for foil material (N m−2)h bump foil height (m)L bearing axial length (m)l0 bump foil half length (m)w bump deflection (m)Ke stiffness coefficient of bump foil (N m−1)Ce damping coefficient of bump foil (Ns m−1)Ki,j stiffness coefficient of overall air foil bearings (N m−1)Ci,j damping coefficient of overall air foil bearings (Ns m−1)Kn Knudsen number (= λ/h)Pa ambient pressure (N m−2)R bearing radius (m)S pitch between bumps (m)t foil thickness (m)U journal linear velocity (m s−1)

α compliance of bump foilε eccentric ratioλ molecular mean free path of air (m)ν Poisson ratio of air foil bearing materialϕp rarefaction coefficients

1. Introduction

The micro power system is a newly portable power source,based on the Brayton cycle, which consists of a compressor,a turbine, a generator and a combustion chamber. Air iscompressed when it passes through the compressor and istransferred to the combustion chamber. Here, the combustionprocess occurs and the chemical energy of the fuel istransferred into the air. The high-temperature gas which isproduced by combustion has a high specific enthalpy. Theturbine extracts a portion of the energy of the high-temperaturegas and generates electrical energy with the generator. The

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Y-B Lee et al

R

H

e

Hmin

φ0

X

Y

θ

(a)

(b)

(c)

(d)

(e)

Figure 1. Air foil journal bearing schematic. (a) Housing, (b) bumpfoil, (c) top foil, (d) thin air film, (e) journal.

micro power system requires an extremely high rotating speedto generate a sufficient compression ratio due to its small size.A small rotor of millimeter-scale diameter was also designedto operate at a surface speed of 500 m s−1 [1] and the temper-ature of turbine inlet component is very high (about 1000 ◦C).Existing rolling element bearings and conventional lubricatedbearings have performance limits in this extreme environmentof high operating speeds and temperatures. Gas bearings suchas the externally pressurized gas bearing [2] or hydrodynamicherringbone groove and spiral groove bearings also operate athigher surface speeds and higher operating temperatures [3, 4].

However, they require additional air supply systems orextreme precision manufacturing procedures by the MEMSfabrication process. And the whirl instability due to too muchthin gas film (3–10 µm) makes the rotating speed have somelimit values. To overcome the above-mentioned problems,air foil bearings were selected. Foil bearings (see figure 1)are self-acting compliant-surface hydrodynamic bearings thatuse ambient air or any process gas as a lubricating fluid [5].The hydrodynamic film pressure builds up in the small gap(30–40 µm) between the rotating shaft and the smooth topfoil. The top foil provides a smooth bearing surface andis often supported by a series of bump foils that act as flatsprings to make the air foil bearings compliant. Because of thecompliant bearing surface, there are certain advantages overthe traditional rigid bearings including a higher load capacityfor a given minimum film thickness, a lower power loss andan increased stability. Air foil bearings are less susceptibleto damage due to large foreign particles in the lubricant flow,as foils can deform instead of seizing up. The compliantair foil bearings are also more tolerant of misalignmentand centrifugal/thermal growth, since compliant foils canaccommodate these changes in shaft diameter and bearingclearance. With the development of a new air foil bearingdesign and coating materials, air foil bearings are now goodcandidates for high-temperature environments [6].

This paper is organized as follows: section 2 presents theair foil bearings model and analysis theory; section 3 is forthe rotordynamic analysis of this micro turbo rotor with airfoil bearings; section 4 briefly describes the components (see

Foil Journal Bearing

Foil Journal Bearing

Foil Thrust Bearing

Foil Thrust Bearing

Rotor

Figure 2. The components of the micro power system.

figure 2) and test rig; section 5 is for experiment results andsection 6 for some concluding remarks.

2. Analysis of air foil bearings

The micro power system is operated in a high-temperatureenvironment with high eccentricity, so the rarefaction effect onthe air foil bearing is considered. The rarefaction coefficientis expressed in terms of the Knudsen number. If Kn < 10−2,the fluid is considered a continuum, and the compressibleReynolds equation is used to describe the gas flow. If 10−2 <

Kn < 10, the fluid is considered a rarified gas, and the Reynoldsequation with rarefaction coefficients should be used. IfKn > 10, the fluid is considered a free molecular flow[7, 8]. When a high load is applied to the air foil bearing at ahigh temperature, the local Knudsen number of the minimumfilm thickness may be greater than 0.01. In such a case, theslip flow effect becomes especially large as the molecular meanfree path increases with temperature.

In this study, the slip flow effect was considered inestimating the dynamic coefficients of an elastically-supportedair foil bearing when a high or allowable load was applied. Thepressure distribution within the clearance of the air foil bearingfor the coordinate system in figure 1 is determined using thefollowing modified Reynolds equation:

∇ ·(

− 1

12µϕpph3∇p +

⇀

U

2ph

)+

∂

∂t(ph) = 0 (1)

where ϕp denotes the molecular rarefaction coefficient givenas

ϕp = 1 + 6λ

h. (2)

In the model of a air foil bearing, the smooth top foil issupported by an elastic foundation (see figure 3), the localdeflection which depended only on the pressure at the pointof application. Table 1 shows the parameters of the elasticfoundation. When the bending and membrane stresses onthe foil were neglected, the elastic foundation includingthe equivalent viscous damping and the film thickness wasrepresented by the following equations, respectively:

h = c − e cos(θ − φ) + w (3)

ke · w + ce · dw

dt= p − pa. (4)

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Rotordynamic characteristics of a micro turbo generator supported by air foil bearings

ρ θ

h

S

L

t

Figure 3. Bump foil configuration for the micro power system.

Table 1. Specification of the bump air foil bearing.

Parameters Values

Foil thickness, t 75 µmBearing length, L 7 mmBump height, h 0.4 mmBump pitch, S 1.25 mBump half length, l0 0.5 mmNumber of bumps 17

Table 2. Specification of the micro power system.

Parameters Values

Journal diameter 7 mmJournal length 7 mmThrust collar diameter 15 mmOverall length 26 mmOverall weight 14.74 gClearance 30 µm

The variables of ke and ce are, respectively, the foil structuralstiffness and damping per unit area. And the variable w

expresses the bump deflection due to the area force of thefilm. The film thickness h can be normalized as follows:

h = h/c = 1 − ε cos(θ − φ0) + α(p − 1). (5)

α is the compliance of bump foil, the reciprocal of the bumpstiffness, and is as follows:

α = 2paS

CE

(l0

t

)3

(1 − ν2). (6)

Applying the perturbation method, the dynamic characteristicsof air foil bearings were calculated. With the solutions forsteady-state and perturbed pressure, the stiffness and dampingcoefficients were readily calculated as:[

Kxx Kxy

Kyx Kyy

]= c

paR2

[Kxx Kxy

Kyx Kyy

]

=∫ L/D

−L/D

∫ 2π

0

[px sin θ py sin θ

−px cos θ −py cos θ

]dθ dz

(7)[Cxx Cxy

Cyx Cyy

]= cω

paR2

[Cxx Cxy

Cyx Cyy

]

=∫ L/D

−L/D

∫ 2π

0

[px sin θ py sin θ

−px cos θ −py cos θ

]dθ dz.

(8)

Figures 4 and 5 show the numerical results of the dynamiccharacteristics for the air foil bearings. The temperatureenvironment was room temperature, 300 K and each valueis applied to both right and left bearing elements. The resultsrepresent that the direct terms of the stiffness coefficients(Kxx and Kyy) increase as the rotating speed increases becausethe normal force of the pressure distribution increases andafter some speed limits, they converge as the film thicknessconverges. In the case of the damping coefficients, Cyy termsincrease initially and converge, and Cxx terms decrease andlater converge. It is because the eccentricity and the filmthickness are same as the case of the stiffness. At othertemperatures (300–1300 K), the dynamic characteristics havethe same trends.

3. Rotordynamic analysis

The rotordynamic model of the micro power system wasestablished using a finite element analysis. The bearing androtor specifications for prediction are presented in tables 1 and2. Figure 6 shows the rotor divided into finite elements. Bothdisc elements for the turbine and compressor are D1 and D2,respectively, and D3 represents the thrust collar. The staticequilibrium position, where the air foil bearing load capacityis equal to the rotor weight, was found using the Newton–Raphson method. The stiffness and damping coefficients ofthe air foil bearings at the static equilibrium position werecalculated by solving four first-order equations. Using thedynamic coefficients of air foil bearings and the rotordynamicmodel of the micro power system, the vibration orbit waspredicted through time integration. In figure 6, the BRG.1 and2 are the grid points applied for each air foil bearing.

Figure 7 is a Campbell diagram for the micro powersystem. While the operating environment was at roomtemperature, 300 K, the first rigid mode (the translatory mode)and the second rigid mode (the conical mode) were predicted at10 100 rpm and 12 300 rpm, respectively. On the other hand,the first rigid mode and the second rigid mode consideringthe turbine inlet temperature of 1300 K were predicted at13 750 rpm and 25 700 rpm, respectively. In the regionbetween 300 K and 1300 K, each mode is included betweenboth values. As the temperature increases, the predicted modevalues increase slightly. It is the cause that the viscosityof lubricating air increases as the temperature increases. Inany case, the first bending mode is always predicted over1000 000 rpm regardless of temperature conditions. The micro

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Y-B Lee et al

(a)

(b)

Figure 4. Non-dimensional stiffness coefficients of foil journalbearings versus rotating speed (rpm). (a) Room temperature(300 K), (b) high temperature (1300 K).

power system supported by air foil bearings operates underconditions within the range of 100 000–700 000 rpm. Thisindicates that the operating speed of the micro power systemis beyond the first and second critical speed (i.e., the translatorymode and the conical mode) and is much lower than that ofthe third mode, the first bending mode.

Generally, the damping characteristics of air foil bearingsare lower than those of oil-lubricated bearings. Thus, theassurance of stability is an essential consideration. Figure 8shows logarithmic decrement versus rotating speeds. It showsthat the stability of the rotor-bearing system is expected to bepositive over the entire operating speed range, which meansthat the micro power system can be operated at a regularrotating speed, 700 000 rpm, with dynamic stability. Actually,air foil bearings are tolerant of external impacts and transientdynamic conditions.

4. Preliminary test rig

The components for the preliminary test rig of the micropower system (see figure 2) were developed to simulatethe operating characteristics of the micro power system. Itconsisted of a rotor supported by two foil journal bearingsand two foil thrust bearings, and an impulse drive turbine.The rotor was made of Inconel 718 for high-temperatureoperating, had a length of 26 mm and a weight of 14.74 g.

(a)

(b)

Figure 5. Non-dimensional damping coefficients of foil journalbearings versus rotating speed (rpm). (a) Room temperature(300 K), (b) high temperature (1300 K).

D1 D2 D3

BRG. 1 BRG. 2

Figure 6. Finite element model for the rotor (D1, D2, D3: discelements /BRG.1, 2: air foil bearing elements).

The residual mass unbalance was 0.0035 g mm. Dummy discswere located at each end of the rotor. These were used tosimulate the turbine and compressor. This preliminary testrig could be operated at 300 000 rpm given the limit of therotating force based on the impulse turbine configuration andstructure.

The foil journal bearings had a diameter of 7 mm and alength of 7 mm (L/D = 1). The foil thrust bearings had aninner diameter of 10 mm and an outer diameter of 15 mm.A verified high-reliability coating was applied to the smoothtop foil. To measure the vibrations of the rotor, fiber opticdisplacement sensors were positioned in the horizontal andvertical directions at each end of the rotor. Figure 9 showsdetected points while test operating. The flow diagram of

300

Rotordynamic characteristics of a micro turbo generator supported by air foil bearings

Figure 7. Campbell diagram of the micro power system.

Figure 8. Logarithmic decrement versus rotating speed.

Figure 9. Detected points while test operating.

the inlet air into the impulse driven turbine is presented infigure 10. Using the pressure regulator, the inlet air pressureand flow rate are controlled and then the vibration and rotatingspeed are measured using the fiber optic gap sensors mounted

RPM meter

Flowmeter

Pressure transducer

Air compressor

Exhaust air

Displacement sensor

DataAcquisition

Ai

3

r input

4

5

2

16

Figure 10. Configuration of the micro power system test rig.① Impulse turbine, ② air foil journal bearing, ③ air foil thrustbearing, ④ millimeter scale rotor, ⑤ disc for compressor part, ⑥ discfor turbine part.

Air input

Fiber optic displacement sensor

RPM meter

Air output

Figure 11. Preliminary micro power system test rig.

into both the horizontal and vertical directions. Figure 11shows the test rig installed.

5. Test results and discussion

The main purpose of the experiments was to verify the stabilityof the micro power system supported by air foil bearings, andto identify the prediction of the vibration orbit by analyticalresults compared to those of the experiments. The micro powersystem was tested at horizontal and vertical attitudes. Thesetests showed that the performance, while dynamic conditionswere imposed on the micro power system, adapted to dynamicapplications such as those of miniature aerial vehicles (MAVs),mobile robots for dangerous work and others. Figures 12and 13 show the frequency spectra waterfall plots and vibrationorbits measured from fiber optic displacement sensors for each

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(a)

(b)

(c)

Figure 12. Waterfall plot and vibration orbit for horizontaloperation. (a) Compressor part (X1) waterfall plot, (b) turbine part(X2) waterfall plot, (c) compressor part (X1–Y1) vibration orbit.

of the operating tests to validate the running characteristics ofthe micro power system. The detected points (X1, X2 and Y1)indicated in figures 12 and 13 are defined in figure 9.

Considering these test results, the main portion ofthe rotor response was synchronous and the amplitude ofsynchronous vibration in all directions was about 5–20 µm.No subsynchronous or other responses appeared. Theseclear frequency spectra indicated a well-damped rotor-bearingsystem. Especially, the rotor was mainly supported by foilthrust bearings while operating in a vertical attitude. As shownin figure 13, not only the foil journal bearings but also thefoil thrust bearings had sufficient load capacity, stiffness anddamping characteristics to maintain the system with stability.

(a)

(b)

(c)

Figure 13. Waterfall plot and vibration orbit for vertical operation.(a) Compressor part (X1) waterfall plot, (b) turbine part (X2)waterfall plot, (c) compressor part (X1–Y1) vibration orbit.

Figure 14 compares experimental and theoretical orbitsat the end of the rotor. As shown in this figure, theexperimental vibration orbit had decreasing trends while therotating speed was increasing, and the theoretical vibrationorbit had increasing trend while the rotating speed wasincreasing. These results explained the experimental trendthat an increased rotating speed could increase the stiffness ofthe elastic foundation (bump foil) in the air foil bearing. It alsoexplained that the vibration amplitude grew by an unbalancemass while the rotating speed was increasing. Consideringthese results, it was expected that the actual stiffness ofair foil bearings would be a little larger than the predicted

302

Rotordynamic characteristics of a micro turbo generator supported by air foil bearings

(c)(a) (b)

Figure 14. Comparison of vibration orbit between the analysis results and the experimental results: (a) at 150 000 rpm, (b) at 200 000 rpmand (c) at 250 000 rpm.

stiffness, according to an analysis in the high-rotating-speedregion.

6. Conclusion

In this paper, rotordynamic analysis and the stability of amicro power system supported by air foil bearings wereinvestigated. How numerical predictions can effectivelyexplain the experimental observations was also investigated bycomparing the experimental and theoretical research findingson the unbalance response of the rotor-bearing system for themicro power system with air foil bearings. The rotor-bearingelements were established using finite element analysis, andthe behavior of the system was predicted. The micropower system had first and second critical speeds (i.e., thetranslatory mode and the conical mode) in the range of 10 000–30 000 rpm according to various environmental temperatures,and a third critical speed (the first being mode) over1000 000 rpm. These showed that there was no resonancein the regular operating speed region.

Also, the operating characteristics were simulated usinga test rig of the micro power system with air foil bearings.The test rig was operated stably at various speeds andunder various situations. The foil journal bearings and thefoil thrust bearings had enough load capacity, stiffness anddamping characteristics to maintain the stability of the system.According to a comparison of experimental and theoreticalvibration orbits, the actual stiffness of the air foil bearings

was a little larger than that shown by analysis. This limitationwill be addressed by advanced air foil bearing analysis and anextensive structural dynamic test of the elastic foundation.

Acknowledgments

This work was supported by a grant from ‘The developmentof intelligent sensors & actuators for high-speed rotatingmachinery’ project of the Korea Institute of Science andTechnology, ‘The standardization of test method for airfoil bearing and turbo blower’ project of the Ministry ofCommerce, Industry and Energy, and ‘The development ofa high-speed motor system’ project of the Korea EnergyManagement Corporation, Korea. The authors would liketo thank KIST, MOCIE and KEMCO.

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Technol. 105.4 647–55[6] Mohawk Innovative Technology Inc. 2003 Dev. Newsl. 17 1–4[7] Lee Y B, Kwak H D, Kim C H and Lee N S 2005 Tribol. Int.

38 89–96[8] Lee N S, Choi D H, Lee Y B, Kim T H and Kim C H 2002

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