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Tribiol Int (2

Tribology International ] (]]]]) ]]]–]]]

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Theoretical considerations of static and dynamic characteristics of airfoil thrust bearing with tilt and slip flow

Dong-Jin Parka, Chang-Ho Kima, Gun-Hee Jangb, Yong-Bok Leea,�

aKorea Institute of Science and Technology, 39-1 Hawolgok-dong, Sungbuk-gu, Seoul 136-791, Republic of KoreabHanyang University, Hengdang-dong, Sungdong-gu, Seoul 133-791, Republic of Korea

Received 2 April 2007; received in revised form 14 June 2007; accepted 6 August 2007

Abstract

The thrust pad of the rotor is used to sustain the axial force generated due to the pressure difference between the compressor and

turbine sides of turbomachinery such as gas turbines, compressors, and turbochargers. Furthermore, this thrust pad has a role to

maintain and determines the attitude of the rotor. In a real system, it also helps reinforce the stiffness and damping of the journal

bearing. This study was performed for the purpose of analyzing the characteristics of the air foil thrust bearing. The model for the air foil

thrust bearing used in this study is composed of two parts: one is an inclined plane, which plays a role in increasing the load carrying

capacity using the physical wedge effect, and the other is a flat plane. This study mainly consists of three parts. First, the static

characteristics were obtained over the region of the thin air film using the finite-difference method (FDM) and the bump foil

characteristics using the finite-element method (FEM). Second, the analysis of the dynamic characteristics was conducted by

perturbation method. For more exact calculation, the rarefaction gas coefficients perturbed about the pressure and film thickness were

taken into consideration. At last, the static and dynamic characteristics of the tilting condition of the thrust pad were obtained.

Furthermore, the load carrying capacity and torque were calculated for both tilting and nontilting conditions. From this study, several

results were presented: (1) the stiffness and damping of the bump foil under the condition of the various bump parameters, (2) the load

carrying capacity and bearing torque at the tilting state, (3) the bearing performance for various bearing parameters, and (4) the effects

considering the rarefaction gas coefficients.

r 2007 Elsevier Ltd. All rights reserved.

Keywords: Air foil bearings; Thrust pad; Tilting condition; Perturbation method

1. Introduction

The operation of the rolling element and oil-lubricatedsliding bearings is limited at high temperatures and speeds;therefore, the tension dominated foil bearing was proposedas an alternative [1]. But it had the shortcoming of lowload carrying capacity; therefore in order to increase the loadcarrying capacity, several types of foil bearing have beendevised. The first generation air foil bearing which isanalyzed in this study mainly consists of corrugated bumpfoil with flexibility and flat top foil with a role to generatethe lubricated air film. This flexible structure settles criticalproblems due to sudden disturbance, thermal expansion of

ee front matter r 2007 Elsevier Ltd. All rights reserved.

iboint.2007.08.001

ing author. Tel.: +822 958 5663; fax.:+822 958 5659.

ess: [email protected] (Y.-B. Lee).

s article as: Park D-J, et al. Theoretical considerations of static a

007), doi:10.1016/j.triboint.2007.08.001

the rotor, and large foreign particles and provides stabilitywith sufficient damping capacity. With these advantages,many analytical and experimental studies to apply the airfoil bearing to turbomachinery have been executed.First, Walowit and Anno [2] analytically calculated the

structural stiffness of the bump using the circular beamequation concerning the inner mounted bump by assumingthe plane strain theory. The bump stiffness equationsuggested in that study did not contain the friction effectbetween the bumps and housing or bumps and top foil.Heshmat et al. [3] numerically analyzed the bump foilbearing by using Walowit’s equation for bump stiffness. Inthat study, it was assumed that the bump foil is an elasticfoundation and used the compressible Reynolds equationabout thin air film between the journal and the top foil.In addition, the load carrying capacity and bearing loss

nd dynamic characteristics of air foil thrust bearing with tilt and slip flow.

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Nomenclature

bi bump width at ith bump (m)c thrust pad initial clearance (m)e thrust pad eccentricity (m)h air film thickness (m)ri bearing inner radius (m)ro bearing outer radius (m)ur linear velocity in the radial direction (m/s)uy linear velocity in the angular direction (m/s)u whirl frequency (1/s)w angular velocity (rad/s)Bn the number of bumpsCe bump damping (N s/m)C the damping coefficient of the air foil bearing

(N s/m)Dgas the diameter of the gas molecule �0.3 nmFi concentrated force on the top of the bump (N)Fa,i vertical reacting force on the left side of ith

bump (N)Fb,i vertical reacting force on the right side of ith

bump (N)H bump foil height (m)H1 height difference between inclined and flat

planes (m)K the stiffness of the air foil bearing (N/m)Ke bump stiffness (N/m)

Lb length of a bump (m)NA Avogadro’s number �6.02� 1023

P pressure (N/m2)Pa ambient pressure �1.014� 105 (N/m)Pb peach of a bump (m)T bearing torque (Nm)W bearing load (N)r,y cylindrical coordinatesX, Y, Z rectangular coordinatesDP perturbed pressure (N/m2)Do perturbed bump deflection (m)Kn Knudsen numbera bump foil complianceb the ratio of the inclined plane to flat planee thrust pad eccentricityd bump foil deflection (m)m viscosity of air (N s/m2)ye one pad angle (1)ye,Z the tilting angle about the Z-axis (1)ye,X0 the tilting angle about the X0-axis (1)Z friction coefficient between bump and bearing

housingc friction coefficient between bump and top foill the mean free path of the gas (m)jp rarefaction gas constantg whirl frequency ratioL bearing number

D.-J. Park et al. / Tribology International ] (]]]]) ]]]–]]]2

torque were calculated using the finite-difference method(FDM). The authors verified that bump foil bearings havegreater load carrying capacity than air bearings under thesame air film thickness. Carpino and Peng investigatedthe static characteristics of bump foil bearing by using thefinite-element method (FEM) [4]. The authors comparedboth results using FEM and FDM and verified that FEMresults were identical to FDM results. They also calculatedthe damping coefficient of the bump foil using equivalentviscous damping derived from energy dissipation principles[5]. Furthermore, the authors calculated the dynamiccharacteristics using these damping components. In thatstudy, the equivalent viscous damping for the Coulombdamping was calculated by equating the energy dissipatedin a cycle of journal excitation. The paper verified that thisequivalent viscous damping component increases theoverall stiffness and damping of the bump foil bearing.Heshmat and Ku calculated structural stiffness usinganalytical methods and compared them with the experi-mental results. From this examination, the authors verifiedthat the stiffness of each bump is different due to theinteraction between the bumps [6]. Lee et al. [7] calculatedthe characteristics of the air foil journal bearing consider-ing the slip flow effect and verified that the load carryingcapacity decreases compared with the no-slip boundarycondition. San Andres and Kim analyzed air foil bear-ing considering a structural loss factor, exemplifying the

Please cite this article as: Park D-J, et al. Theoretical considerations of static a

Tribiol Int (2007), doi:10.1016/j.triboint.2007.08.001

dry-friction damping capacity of the underlying foil-bumpstrip structure, and verified that it enhances the dampingability of air foil bearings [8].Despite the existence of many studies on the analysis of

air foil journal bearing, the study of air foil thrust bearingis out of fashion. Yet as the operating speed of theturbomachinery and the axial force due to the pressuredifference between the turbine and compressor sidesgradually have increased, studies with the purpose ofincreasing the load carrying capacity of the air foil thrustbearing are in progress. Bruckner [9] designed the top foilas the shell model using bi-harmonic function and obtainedthe static characteristics considering the thermal effects.Heshmat et al. [10] calculated the displacement anddeformation of the bump foils using FEA modeling forthe foil thrust bearing. The authors also analyzed the airfoil thrust bearing using the FDM and showed that theoptimized geometry for a bearing is as follows: ye ¼ 451,b ¼ 1 [11]. In addition, it was presented that the inclinedplane of the thrust bearing (see Fig. 1) plays an importantrole in generating the pressure gradient. Iordanoff [12,13]proposed the equation for the local bump compliance.Using those results, he calculated the overall staticcharacteristics. Recently, experimental research has mainlybeen undertaken [14,15]. But those former works did notconsider the tilting condition and axial movement of therotor. Also, the important parameters needed to obtain the



ARTICLE IN PRESS

Top foilBump foil

Bearing Housing

Inclined plane Flat plane

Spot welding

Fig. 1. The configuration of the air foil thrust bearing.

10-5 10-4 10-3 10-2 10-1 100 101 102 103 104 105 10610-610-510-410-310-210-1100101102103104105106107

Knu

dsen

Num

ber

h = 0.1μmh = 1 μmh = 10 μmh = 100 μm

108

Pressure [KPa]

Fig. 2. Knudsen number of thin air film at room temperature, 300K.

D.-J. Park et al. / Tribology International ] (]]]]) ]]]–]]] 3

rotor characteristics versus the rotating speed, the dynamiccharacteristics, were not referred to in previous works.Thus, this paper was written with the purpose of obtainingthe dynamic characteristics and the bearing performancevariation at tilting condition.

This analysis was based on the FDM using the cylin-drical coordinates. For more exact solutions, the bump foilwas calculated by the FEM suitable to the structureanalysis. Because the position of the bump foil is notsuitable for the cylindrical coordinates, the process to fitthe FEM grids to the FDM grids was preceded. One bumpfoil acts under the correlation among the several bumpsand the static and dynamic characteristics are determinedby the pressure distribution and bump number [16]. Also,in the bump foil FEM analysis, the reacting force at bothsides of a bump was taken into consideration, calculated,and introduced to the overall air foil thrust bearinganalysis. The rarefaction coefficients are dependent onthe temperature, pressure, and film thickness. The rarefac-tion coefficients perturbed about the pressure and filmthickness were calculated using the equation proposed inthe literature [17]. Finally, those coefficients were used tocalculate the bearing static and dynamic characteristics.

2. Thin air film analysis theory

2.1. Modified Reynolds equation

When the Knudsen number is between 0.01 and 10, itcan be assumed that the flow is the slip flow and themedium gas is the rarefaction gas. The film thickness of theair foil bearing is normally in the 10–50 mm range as can be



seen in the analysis of the rarefaction gas from Fig. 2.The modified Reynolds equation can be derived fromthe continuity equation and Navier–Stokes equation usingthe first order slip boundary condition as follows:

urðz ¼ 0Þ ¼ lqur

qz

��z¼0

, (1)

urðz ¼ hÞ ¼ �lqur

qz

��z¼h

, (2)

uyðz ¼ 0Þ ¼ wrþ lqur

qz

��z¼0

, (3)

uyðz ¼ hÞ ¼ �lqur

qz

��z¼h

. (4)

The obtained radial and angular velocity of the flow areas follows:

ur ¼ �1

2mqP

qrðhlþ hz� z2Þ, (5)

uy ¼ �1

2mr

qP

qyðhlþ hz� z2Þ þ wr 1�

zþ lhþ 2l

� �. (6)

Introducing both velocity terms to the continuityequation, the modified Reynolds equation can be obtainedfrom the following equations:

qqr

rrh3

12mjP qP

qr

� �þ

1

r

qqy

rh3

12mjP qP

qy�

rhðwr2Þ

2

� �¼

qqtðrhrÞ.

(7)

Here, jP represents the rarefaction coefficients and is asfollows:

jP ¼ 1þ 6lh¼ 1þ 6Kn. (8)

Each variable can be normalized using typical para-meters. The normalized form of the modified Reynolds



ARTICLE IN PRESS

X

Y

Z

�e,z

�e,x'

X

Y

Z

X’

Y’

Fig. 4. Tilting condition of thrust pad due to the eccentricity of the

rotating journal.


equations is as follows:

qqr̄

rPh3ð1þ 6KnÞ

qP

qr̄

� �þ

1

r̄

qqy

Ph3ð1þ 6KnÞ

qP

qy

� �

¼ 2LcqðPhrÞ

qt̄þ L

qqyðPhrÞ, ð9Þ

where r̄ ¼ r=ro; h̄ ¼ h=c; t̄ ¼ tu, and P ¼ P=Pa. Also, gis the whirl frequency ratio (u/w) and L the bearingnumber:

L ¼6mwR2

Pac2. (10)

2.2. Air film thickness calculation

The simplified 1 DOF system with stiffness and dampingis used for calculating the film thickness. The magnitude ofthe bump deflection changes with the magnitude of thepressure, and the related equation is as follows:

P̄� 1 ¼ K̄edþ C̄egdd̄dt̄

. (11)

Here, K̄e and C̄e represent normalized stiffness anddamping coefficients of the bump foil as follows: Kec/Pa

and Cecw/Pa, respectively.The normalized film thickness can be calculated using

the above equation of the bump foil deflection and is asfollows:

h ¼ 1� �þ ðr=eÞ cosðj� ye;ZÞ sinðye;X 0 Þ þ d̄

when � p=2þ ye;Zojpp=2þ ye;Z, ð12Þ

h ¼ 1� �� ðr=eÞ cosðj� ye;ZÞ sinðye;X 0 Þ þ d̄

when p=2þ ye;Zojp3p=2þ ye;Z, ð13Þ

A thrust bearing

r0

ri

X

Y

�e,z

�

Fig. 3. Brief configuration of the air foil thrust bearing.



where ye,Z and ye,X0 represent the tilting angle about theZ- and X0-axes, respectively. The angles are determinedfrom the eccentricities at both journals. Figs. 3 and 4 showthose variables. Also, d̄ is the normalized bump foildeflection under pressure which acts on the top foil:

d̄ ¼ aðP̄� 1Þ. (14)

Here a represents the compliance of the bump foil,inversely proportional to the bump foil stiffness.

2.3. Tilting pad condition

When the thrust pad tilts, more load acts on the area atwhich the film thickness decreases. Moreover, it affects theattitude of the rotating journal with a nonsymmetric shapeof both sides. So, the load calculation at each degree ofangle is necessary to calculate the eccentricity until itconverges to any value satisfying the force and momentumequilibrium state. Fig. 4 shows the thrust pad at steadystate and tilting condition. When the eccentricity isgenerated at both journal parts nonuniformly, tilting isgenerated. The ye,Z is the equivalent eccentricity line andye,X0 presents the tilting angle.

2.4. Perturbation method

The dynamic operating regime is traditionally describedas a small amplitude motion of the rotor about anequilibrium position. For displacement perturbation,the film thickness, h, is described by the followingequations:

h̄0 þ h̄zDzþ h̄_zD_z ¼ 1� �� Dz� D_zþr

ccosðj� ye;ZÞ

� sinðye;X 0 Þ þ ðd̄0 þ d̄zDzþ d̄_zD_zÞ

when � p=2þ ye;Zoj

pp=2þ ye;Z, ð15Þ



ARTICLE IN PRESSD.-J. Park et al. / Tribology International ] (]]]]) ]]]–]]] 5

h̄0 þ h̄zDzþ h̄_zD_z ¼ 1� �� Dz� D_z�r

ccosðj� ye;ZÞ

� sinðye;X 0 Þ þ ðd̄0 þ d̄zDzþ d̄_zD_zÞ

when p=2þ ye;Zoj

p3p=2þ ye;Z, ð16Þ

and the perturbed pressure is as follows:

P̄o þ P̄zDzþ P̄_zD_z� 1 ¼ K̄eðd̄0 þ d̄zDzþ d̄_zD_zÞ

þ C̄egd

dtðd̄0 þ d̄zDzþ d̄_zD_zÞ, ð17Þ

where DP and Do represent the perturbed pressure anddeflection, respectively, and are as follows:

DP ¼ PzDzþ P_zD_z, (18)

Dd ¼ dzDzþ d_zD_z. (19)

To apply the perturbation method to the Knudsennumber, the following equations regarding the Knudsennumber were used [16]:

Kn ¼RT

pffiffiffi2p

d2gasNAPh

. (20)

The Knudsen number is a variable related with pressure,film thickness, and environment temperature. In this study,a fixed temperature, 300K, was assumed. Hence, it wasregarded as a variable related to pressure and filmthickness. The equation is as follows:

KnðP; hÞ ¼C1

Ph, (21)

where C1 is 6.7� 10�2. The rarefaction coefficient is asfollows:

jP ¼ jP0 þ

qjP

qP

� �0

DpþqjP

qh

� �0

Dh. (22)

Here the perturbed rarefaction coefficients in terms ofpressure and film height can be expressed as follows:

qjP

qP

� �¼ �

0:402

P2h;

qjP

qh

� �¼ �

0:402

Ph2. (23)

Using the perturbed variables, the modified Reynoldsequation can be divided into a zeroth-order equation andtwo first-order equations (Dz̄ and D _̄z terms). These are asfollows:

Zeroth-order equation

qqr̄

rP̄0h̄3

0jP0

qP̄0

qr̄

� �þ

1

r̄

qqy

P̄0h̄3

0jP0

qP̄0

qy

� �¼ L

qqyðP̄0h̄0rÞ.

(24)



First-order equationDz̄ term:

qqr̄

r̄P̄0h̄3

0jP0qP̄z

qr̄þ r̄P̄zh̄

3

0jP0qP̄0

qr̄

þ3r̄P̄0h̄2

0h̄zjP0qP̄0

qrþ r̄P̄0h̄

3

0qjP

qP

� 0P̄z

qP̄0

qr̄

þr̄P̄0h̄3

0qjP

qh

� 0hz

qP̄0

qr̄

8>>>>><>>>>>:

9>>>>>=>>>>>;

þ1

r̄

qqy

P̄0h̄3

0jP0qP̄z

qy þ P̄zh̄3

0jP0qP̄0

qy

þ3P̄0h̄2

0hzjP0qP̄0

qy þ P̄0h̄3

0qjP

qP

� 0P̄z

qp̄0qy

þP̄0h̄3

0qjP

qh

� 0h̄z

qP̄0

qy

8>>>>><>>>>>:

9>>>>>=>>>>>;

¼ Lr̄qqyðP̄zh̄0 þ P̄0h̄zÞ � 2Lgr̄ðP̄_zh̄0 þ P̄0h̄_zÞ. ð25Þ

D _̄z term:

qqr̄

r̄P̄0h̄3

0jP0qP̄_zqr̄þ r̄P̄_zh̄

3

0jP0qP̄0

qr̄

þ3r̄P̄0h̄2

0h̄_zjP0qP̄0

qrþ r̄P̄0h̄

3

0qjP

qP

� 0P̄_z

qP̄0

qr̄

þr̄P̄0h̄3

0qjP

qh

� 0h̄_z

qP̄0

qr̄

8>>>>><>>>>>:

9>>>>>=>>>>>;

þ1

r̄

qqy

P̄0h̄3

0jP0qP̄_zqy þ P̄_zh̄

3

0jP0qP̄0

qy

þ3P̄0h̄2

0h_zjP0qP̄0

qy þ P̄0h̄3

0qjP

qP

� 0P̄z

qp̄0qy

þP̄0h̄3

0qjP

qh

� 0h̄_z

qP̄0

qy

8>>>>><>>>>>:

9>>>>>=>>>>>;

¼ Lr̄qqyðP̄_zh̄0 þ P̄0h̄_zÞ � 2Lgr̄ðP̄zh̄0 þ P̄0h̄zÞ. ð26Þ

Here, perturbed film thickness can be derived fromEqs. (13)–(15) and those are as follows:

h̄z ¼K̄eðP̄x � K̄eÞ þ C̄egðP̄ _x � C̄egÞ

K̄2e þ ðC̄egÞ

2,

h̄_z ¼K̄eðP̄x � K̄eÞ � C̄egðP̄ _x þ C̄egÞ

K̄2e þ ðC̄egÞ

2. ð27Þ

Note that the bump foil stiffness and damping decrease theperturbed film thickness.

2.5. Static characteristics analysis

The static pressure and film thickness are calculatedusing the FDM with 105 grids to the angular direction and45 grids to the radial direction until the values of thepressure at all grid points converge using the successiveunder relaxation (SUR) method. The load and torque canbe obtained along with the pressure and film thickness.Those can be expressed as the following equations:

W ¼

ZZA

P0r̄dydz̄;W ¼W=Par20, (28)



ARTICLE IN PRESS

0.1 1 10

H1=4.0

Non

dim

ensi

onal

Loa

d

No slip condition Slip condition

H1=1.0

0.1 1 10

No slip conditionSlip condition

Non

dim

ensi

onal

Bea

ring

Tor

que

H1=4.0

H1=1.0

0.3

0.6

0.9

1.2

1.5

0.0

Bearing Number, �

0.00

0.05

0.10

0.15

0.20

Bearing Number, �

Fig. 5. Nondimensional bearing static characteristics considering slip flow

and no slip flow condition at bump foil compliance; a ¼ 0.1 and

eccentricity e ¼ 0.6.0.1 1 10

0.05

No slip conditionSlip condition

Non

dim

ensi

onal

Stif

fnes

s

H1=4.0

H1=1.0

0.04

0.08

0.12

0.16

Non

dim

ensi

onal

Dam

ping

Coe

ffic

ient

s

No slip condition Slip condition

0.30

0.25

0.20

0.15

0.10

0.00

0.20

H1=4.0

H1=1.0

Bearing Number, �


T ¼

ZZA

r̄h̄

2

qP̄0

qy

� �þ

mr̄3w

h̄

�dydz̄; T ¼ T=cPar20. (29)

Fig. 5 shows the analysis results of the static character-istics. Two conditions, slip flow and no slip flow, weretaken into consideration to confirm the slip flow effect. Inthe case of load carrying capacity, the results with the slipflow condition have low values at overall bearing numbers.This is because the hydrodynamic pressure decreases due todecreased linear velocity at both walls with the slip flowcondition. The bearing torque also had the same trends butthe difference magnitude was less than those of the loadcarrying capacity. Further, as the gradient of the inclinedplane decreases, the static characteristics increase.

1 100.00

0.1

Bearing Number, �

Fig. 6. Nondimensional bearing dynamic characteristics considering slip

flow and no slip flow conditions with bump foil compliance; a ¼ 0.1,

eccentricity e ¼ 0.6, bump foil stiffness K̄e ¼ 25, and bump foil damping

coefficients C̄e=w ¼ 2:5� 10�2.

2.6. Dynamic characteristics analysis

The dynamic characteristics, stiffness, and dampingcoefficients are calculated from the perturbed pressure.Similarly with static characteristics analysis, the SUR



method was used for stable convergence. In this study,only the coefficients in the radial direction under theperturbation in the radial direction, direct term, were takeninto consideration. These can be expressed as the followingequations:

K̄zz ¼

ZZA

P̄z dydz̄; K̄zz ¼ Kzz=PaR2. (30)

C̄zz ¼

ZZA

P̄_z dydz̄; C̄zz ¼ Czz=PaR2. (31)

Fig. 6 present the stiffness and damping coefficientsunder two conditions. Similarly with the static character-istics, the results with slip flow condition were lesser thanthose with no slip flow condition. Furthermore, trendsindicating that both results decrease at high bearingnumbers were confirmed. Fig. 7 shows the stiffness anddamping coefficients of the air foil thrust bearing withvarious structural stiffness coefficients of the bump foil.



ARTICLE IN PRESS

1 10

Non

dim

ensi

onal

Stif

fnes

s

Ke =250

Ke =50

Ke =25

1 10

Non

dim

ensi

onal

Dam

ping

Ke = 250

Ke =50

Ke =25

Bearing Number, �

Bearing Number, �

0.1

0.1

0.18

0.16

0.14

0.12

0.10

0.08

0.06

0.04

0.02

0.00

0.00

0.05

0.10

0.15

0.20

0.25

Fig. 7. Dynamic characteristics of bump foil thrust bearing at various

bump stiffnesses when e is 0.6 and C̄e=w ¼ 2:5� 10�2.

1 10

Non

dim

ensi

onal

Stif

fnes

s

Ce / w = 2.5 x 10-4

Ce / w = 2.5 x10-3

Ce / w = 2.5 x10-2

1 10

0.055

Non

dim

ensi

onal

Dam

ping

Ce / w = 2.5 x 10-4

Ce / w = 2.5 x 10-3

Ce / w = 2.5 x 10-2

0.07

0.06

0.05

0.04

0.03

0.02

0.060

0.045

0.040

0.035

0.030

0.025

0.020

0.015

0.0100.1

Bearing Number, �

Bearing Number, �

0.050

0.1

Fig. 8. Dynamic characteristics of bump foil thrust bearing at various

bump damping coefficients when e is 0.6 and K̄e ¼ 25.

D.-J. Park et al. / Tribology International ] (]]]]) ]]]–]]] 7

When the structural stiffness increases, both the dynamiccoefficients increase. Yet in the case of Fig. 8, the dampingcoefficients of the air foil thrust bearing decrease as thestructural damping increases. It is because h̄z increasesregardless of whether h̄_z decreases when the bump foilstructural damping increases. There results can be checkedfrom Eq. (27) for the perturbed film thickness.

3. Bump foil analysis

Heshmat et al. [10] verified that the case with eight pads,where each pad consists of 451, has optimum performance.Here, the inclined plane of each pad generates morepressure. Recently, the air foil thrust bearing divided intoeight thrust pads, at 901, has been studied. Fig. 9 shows onethrust bearing configuration. The inclined part height canbe adjusted by putting several foil sheets between bumpand top foil or using the various bump foil heights. Similarto the effect of the height difference between inclined and



flat planes, the angle ratio (the angle of inclined plane/theflat plane) is an important parameter. The air foil thrustbearing with a configuration where the angle ratio issmaller can achieve greater then more load carryingcapacity (see the results of Figs. 5 and 6). In this chapter,analysis of the air foil thrust bearing, on the bump foil,which has 8 pads and an angle ratio of one was executed.As well, the boundary with the spot welding point of Fig. 1is regarded as a fixed end and the other boundary as a freeend.

3.1. Bump foil modeling

Fig. 10 shows one bump foil model with two cases ofboundary conditions. The deformations due to the normalforce acting on the top touched with the top foil varieswith the constraint condition at both ends of the bump.Fig. 10(a) presents the bump at the fixed end side. The leftend plays the role of preventing displacements to both thevertical and horizontal directions, and the displacement of



ARTICLE IN PRESSD.-J. Park et al. / Tribology International ] (]]]]) ]]]–]]]8

the other end is determined by the friction coefficient andnormal force, Fb. On the contrary, Fig. 10(b) shows thebump model with both free ends, which are constrained bythe state of related bumps. The methods to analyze theinteraction among bumps are divided into two cases: one isdesigned to analyze each bump and couple all the analysisresults using the interacting force [15], and the other isdesigned to calculate the overall bumps simultaneously.

h

H1

�e

��e

w

ro

ri

Boundary 2Boundary 1

Boundary 3

�

Fig. 9. A thrust bearing schematics, H1 is the height of the inclined plane

and bye the angle of the inclined plane.

spot welded

Free end

Pressure (Fi)

�Fb,i

�Fi

�Fi

Fixed end

Pressure (Fi)

�Fb,iFb,i�Fa,i Fa,i

Fb,iFa,i

Fig. 10. One bump model configuration.



The former method is suitable to the bump foil analysiswith more than 20 bumps for time saving regardingnumerical scheme. But in order to obtain the internalforces for the purpose of comparison with the frictionforces at both points of a bump, the stiffness matrix has tobe transformed to an inverse matrix form twice. Thus,methods which calculate the displacements of overall bumpelements are adaptable to bump foil analysis with bumpsunder 15 for a simple numerical scheme because they donot need to obtain the internal forces. The number of

1 3 4 5 7 8

Stru

ctur

al S

tiffn

ess

[N/m

m]

Bump length: 3.2 mm Bump length: 3.0 mm Bump length: 2.8 mm

1

0

5x100

Bump length: 3.2 mm Bump length: 3.0 mm Bump length: 2.8 mm

Stru

ctur

al D

ampi

ng [[

N.s

/mm

]]

2.5x104

2.0x104

1.5x104

1.0x104

5.0x103

0.0

4x100

3x100

2x100

1x100

2 3 4 5 6 7 8

Bump Position

Bump Position

2 6

Fig. 11. Structural stiffness and damping of bump foil with various bump

lengths under uniform loads.

Table 1

The parameters for the bump analysis

Parameters Values

Outer radius (ro) 0.05m

Inner radius (ri) 0.025m

Foil thickness (t) 0.1mm

Bump length (Lb) 3.0mm

Bump height (H) 0.5mm

Friction coefficients (c,Z) 0.2

Exciting frequency 1000Hz




bumps for the thrust bearing analysis approximatelyranged from 6 to 12 according to the bump foil and thrustpad sizes. So, the latter method, the stiffness matrixmeasuring 1536� 1536 (8 bumps; 64 elements for eachbump; transverse and horizontal deflections, and bending),was selected.

0 2 4 6 8

1.5x104

Bump Height: 0.1 mm Bump Height: 0.2 mm Bump Height: 0.3 mm Bump Height: 0.4 mm Bump Height: 0.5 mm

Stru

ctru

al S

tiffn

ess

[N/m

m]

1 3 5 7

Stru

ctur

al D

ampi

ng [[

N.s

/mm

]]

Bump Height: 0.1 mm Bump Height: 0.2 mm Bump Height: 0.3 mm Bump Height: 0.4 mm Bump Height: 0.5 mm

2.5x104

2.0x104

1.0x104

5.0x103

0.0

4.0

3.5

3.0

2.5

2.0

1.5

1.0

0.5

0.0

Bump Position

2 4 6 8

Bump Position

3 5 7 91

Fig. 12. Structural stiffness and damping of bump foil with various bump

heights under uniform loads.

Fig. 13. Compliance of a thrust bearing pad.



As the preceding process to calculate the structuralstiffness and damping of bumps demonstrated, the bumpwidths which are important parameters that determine thevalue of the structural characteristics were obtained. It isexpected that the characteristics increase as the bumpwidth increases. The bump widths can be calculated from

Fig. 14. Nondimensional stiffness coefficients.

Fig. 15. Nondimensional damping coefficients.

Table 2

The parameters of the bump foil used for the air foil thrust bearing

analysis

Parameters Values

Outer radius (ro) 0.1m

Inner radius (ri) 0.05m

Foil thickness (t) 0.1mm

Bump length (Lb) 3.0mm

Bump height (H) 0.5mm

Bump pitch (Pb) 7.0mm

The number of bumps 11

Friction coefficients (c,m) 0.2

Angle of inclined part 22.51

Total pad number 8




the following equations:

bi ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffir2o � ðPbiÞ2

q�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffir2i � ðPbiÞ2

qat i ¼ 12Bn1, (32)

bi ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffir2o � ðPbiÞ2

q� Pbi= sin ye at i ¼ Bn1þ 12Bn.

(33)

Here i and Bn1 represent the ith bump and the number ofbumps which meet at boundary 1 in Fig. 9.

3.2. Numerical scheme for bump foil analysis

For the analysis method, the FEM suitable to structureanalysis was conducted. A bump model was divided into

Fig. 16. Pressure distribution over the 8 pad thrust pad.

Fig. 17. Nondimensional pressure distribution versus the



two-dimensional 64 elements with widths variable to theelement position to save calculating time. Because thenumber of bumps is determined from the bump foilparameters, bump pitch, bearing inner radius, outer radius,and one pad angle, an analysis to determine the numberwas conducted. Additionally, bearing performances undervarious loads were calculated. For damping coefficients,the energy dissipation equation using the dry frictionwas used [5]. The equivalent Coulomb damping can beexpressed as follows:

Ceq ¼4ZFL;i

podL;xþ

4ZFR;i

podR;x. (34)

Here, dL,x, and dR,x represent the horizontal deflections ofthe bump foil at the point at which the bump is touchedwith top foil and bearing housing, respectively.

3.3. Results of numerical analysis

When the friction forces, ZFL,i, ZFR,I, and cFi (seeFig. 10) are generated, they accumulate to the fixed end.When the reacting force at the end of the ith bump is largerthan the internal force by which the bump deforms, thebump can be supposed to have fixed boundaries at bothsides. At that time, the bump has large stiffness and nodamping due to no-relative displacement between bumpfoil and bearing housing. Fig. 11 shows the stiffness anddamping of the bumps in contrast with the bump length.Here, the bump pitch is the same in all cases and the bump

eccentricity of thrust pad at L ¼ 10 and H1/c ¼ 1.0.




position represents the bumps from fixed end (1) to free end(8), respectively. The variables for this analysis are presentedin Table 1. The bump at the side of the fixed end has thelargest stiffness. Nevertheless, the damping coefficient is zerodue to no-relative displacement. In this case, both the ends ofthe bump can be supposed to have fixed ends because theinteracting force by the friction force is larger than theinternal force so that the bump is deflected by the load. Also,as the bump position moves to the free end, the stiffness anddamping coefficients decrease. This is because the bumpwidth and the interacting force decrease. The results alsoreveal that the bump performances are inversely proportionalto the bump length and the damping coefficient is propor-tional to the stiffness only in cases where both the ends of abump are not fixed. Fig. 12 presents the results for the bumpperformance versus the bump height. As the bump heightincreases, the stiffness and damping increase. The overalltrends are similar to the results of Fig. 11.

1 10

1

Non

dim

ensi

onal

Loa

d

H1=1

H1=2

H1=3

H1=4

1

0.1

1

Non

dim

ensi

onal

Loa

d

ε = 0

ε = 08.

0.01

0.1

0.01

1E-30.1

Bearing Number, �10

Bearing Number, �0.1

1E-3

ε = 0.6

ε = 0.4

ε = 0.2

Fig. 18. Nondimensional load versus bearing number.



3.4. Calculation of air foil thrust bearing characteristics

using bump analysis results

The data of the bump foil analysis are used to obtain thepressure distribution of the air film over the top foil anddynamic characteristics using the perturbation method.Because these are discontinuous at the boundaries betweenbumps, the stiffness and damping coefficients were curvefitted using a polynomial equation. Figs. 13–15 show thenondimensional coefficient curves fitted using 9–25 points.Similar to the results of bump analysis, the stiffnessdecreases from the fixed end to free end. The dampingcoefficients also followed the same trends. The coefficientsare changed by the bearing parameters and the loadcondition, and the calculation and curve fitting processhave to be undertaken before air film analysis. For air filmanalysis, the perturbation method was applied to therarefaction coefficients.

1

1

Non

dim

ensi

onal

Bea

ring

Tor

que

H1=1

H1=2

H1=3

H1=4

1 101E-3

0.01

0.1

1

10

ε = 0

ε = 0.2

ε = 0.4

ε = 0.6

ε = 0.8

Non

dim

ensi

onal

Bea

ring

Tor

que

0.1

0.01

0.1

Bearing Number, �

Bearing Number, �

10

0.1

Fig. 19. Nondimensional bearing torque versus bearing number.




4. Analysis results and discussion

The numerical results obtained using the methodsintroduced above were based on the normalized variables.The parameters of the bump foil model used for bearinganalysis are presented in Table 2. The compliance, stiffness,and damping data at all grids are the same as those ofFigs. 13–15. Note that those values change as the bump foilparameters and pressure profiles change. Thus, in everynumerical iteration process to calculate pressure of air thinfilm, the datum representing the bump foil characteristicshave to be calculated and curve fitted. For the dampingcoefficients, the magnitude varies with the variation of thebearing number.

The zeroth-order equation based on the perturbationmethod was used for calculating the pressure distributionof thin air film, as shown in Fig. 16. Here, the z-axispresents the normalized pressure, where the eccentricitywas 0.2 and the ratio of the clearance, c, to H1 was 1. Alsothe tilting angle was zero. Fig. 17 shows the nondimen-

1 10

0.24H1=1

H1=2

H1=3

H1=4

Non

dim

ensi

onal

Stif

fnes

s

1 10

Non

dim

ensi

onal

Stif

fnes

s

ε=0ε=0.2ε=0.4ε=0.6ε=0.8

0.22

0.20

0.18

0.16

0.14

0.12

0.10

0.08

0.06

0.04

0.02

0.00

0.30

0.25

0.15

0.20

0.10

0.05

0.000.1

Bearing Number, �

Bearing Number, �

0.1

Fig. 20. Nondimensional stiffness versus bearing number.



sional pressure distribution versus the eccentricity of thethrust pad. As the eccentricity increases, a high pressureregion moves to the free end of the thrust pad and isbroadened to both sides, where the pressure totallyincreases. The pressure distribution in cases where theeccentric ratio becomes 0.87 has a discontinuous regionand the numerical iteration diverges. It can be supposedthat the eccentric ratio is the limit for the load carryingcapacity of the air foil thrust bearing. Fig. 18 presentsnondimensional load versus bearing number. When thegradient of the inclined part decreases (H1/c; i.e. H̄1,decreases), the load which the bearing can supportincreases. It is because the pressure gradient increases asthe film thickness gradient decreases at the same bearingnumber. Fig. 18(b) shows that the load increases as theeccentricity increases for the same reason as the results ofFig. 17. Generally, as the eccentric ratio increases, the loadgradient versus bearing number increases.Fig. 19 shows the bearing torque versus bearing number.

Because the torque is proportional to the pressure gradientand angular velocity, it increases as the bearing numberincreases. As was also demonstrated in Fig. 18, as the

1 10

Non

dim

ensi

onal

Dam

ping

H1=1

H1=2

H1=3

H1=4

0.1 1 10

Non

dim

ensi

onal

Dam

ping

ε=0.4ε=0.6ε=0.8

0.30

0.25

0.20

0.15

0.10

0.05

0.000.1

Bearing Number, �

Bearing Number, �

0.40

0.35

0.30

0.25

0.20

0.15

0.10

0.05

0.00

ε=0ε=0.2

Fig. 21. Nondimensional damping coefficients versus bearing number.




gradient of the inclined part decreases, the load which thebearing can support increases. However, the effects werelesser than those of the load. It is because the loadmagnitude is proportional to the pressure magnitude. Also,the torque is similar to the pressure gradient. As the

0 20 60 80 120 160

0.40

0.45

Non

dim

ensi

onal

Loa

d

θeX'= π x 10-5

θeX'=2π x 10-5

θeX'=3π x 10-5

θeX'=4π x 10-5

θeX'=5π x 10-5

θeX'= π x 10-5

θeX'=2π x 10-5

θeX'=3π x 10-5

θeX'=4π x 10-5

θeX'=5π x 10-5

0 40 60 80 140

1.6

Non

dim

ensi

onal

Tor

que

The tilting angle about the Z axes [[Deg]]

The tilting angle about the Z axes [[Deg]

0.8

0.6

0.4

1.0

1.2

1.4

1.8

-20 100 140 180400.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

16012010020

Fig. 22. Nondimensional performance versus bearing circumference.

Fig. 23. Pressure distribu



eccentric ratio increases, the pressure is almost uniformover the post-half part of the thrust pad.The dynamic characteristics of the air foil thrust bearing

were calculated using the two first-order equations,Eqs. (23) and (24), with the perturbation terms of themodified Reynolds equation. Both the equations arecoupled with each other through the perturbed pressure(Pz and Pz) and perturbed film thickness (hz and hz). Thedifference equations are calculated until each pressure termconverges using the SUR method.The results are presented in Figs. 20 and 21. Fig. 20

shows the nondimensional stiffness for various bearingnumbers, the gradients of the inclined part, and theeccentric ratios. Those effects are the same as the case ofload results except for the magnitudes. The damping resultsin Fig. 21 generally have the same trends with the stiffness.Both the coefficients had a limit at some bearing number.This is because the air foil bearing has a vibration systemwhere two springs and two dampers (one is bump foil andthe other is thin air film) are connected in series. Here, theangular velocity characteristics are dependent on those ofthe bump foil when the air film becomes stiff at a highbearing number. So, it can be supposed to approach theresults of the values from Figs. 11–15.Fig. 22 shows the load and bearing torque at the

condition of L ¼ 10, H1/c ¼ 1.0, and e ¼ 0.6. When thethrust pad tilts, the pressure distribution is changed due tothe variation of the eccentricity. Both load and bearingtorque show the trends of harmonic function versus thetilting angle about the Z-axis. Also, the magnitudeincreases as the tilting angle about the Y0-axis increasesdue to the increase of the film thickness variation and largeeccentricity over the total pad area. In particular, at theboundary between the inclined plane and flat plane the filmthickness increases (the bearing number increases), theoverall dynamic gradient decreases, and high load andbearing torque are generated. Fig. 23 shows the pressuredistribution over a pad under the tilting condition. As thetilting angle increases, high pressure is generated. Thedistribution center moves to the part at which the filmthickness is the least; i.e., the outer circumference.

tion at tilting state.




5. Conclusions

In this study, an analysis of the air foil thrust bearingwas conducted. First, the characteristics of the bump foilwere analyzed using the FEM, considering the frictionforce between the top and bump foil, as well as the bumpfoil and bearing housing. In addition, using the deflectionat each bump under a uniform load, the stiffness anddamping were calculated. The results show that thestiffness and damping are proportional and the magnitudedecreases from the fixed end to the free end. The data fromthe bump analysis were curve fitted at several grid points toapply the characteristics of the bump foil to the overall airfoil bearing analysis.

The overall air foil bearing analysis was conducted usinga modified Reynolds equation with a rarefaction coeffi-cient. Moreover, using the perturbation method, a zeroth-order and two first-order equations were obtained. Thestatic pressure and perturbed pressure about the perturbedmotion to the axial direction were calculated using theSUR method. The numerical results were divided intostatic characteristics, dynamic characteristics, and thecharacteristics under tilting conditions. In the case of staticcharacteristics, as the bearing number and eccentric ratioincrease, the load and bearing torque increase. Also, as thegradient of the inclined part decreases, load and bearingtorque increase. The dynamic characteristics, stiffness, anddamping coefficients have the same trends but had a limitat some bearing number due to the characteristics ofdependency to the bump foil at a high bearing number.Finally, the static characteristics under tilting conditionswere analyzed. It was found that the more the rotor tilts,the more load and bearing torque is generated due to thedecreased film thickness gradient. These analysis methodsand results might be used for coupling the thrust with thejournal bearing.

Acknowledgments

This work was performed by KIST and supported by agrant from the ‘‘Development of intelligent sensors andactuators for high speed rotating machinery’’ project ofKorea Institute of Science and Technology, ‘‘Standardiza-tion of test method for air foil bearing and turbo-blower’’project of Ministry of Commerce, Industry and Energy,



and ‘‘The development of a high-speed motor system’’project of Korea Energy Management Corporation,Korea. The authors thank KIST, MOCIE, and KEMCO.

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