The comparison in stability of rotor-aerostatic bearing system compensated by orifices and inherences

ARTICLE IN PRESS

Tribology International 43 (2010) 1360–1373

Contents lists available at ScienceDirect

Tribology International

0301-67

doi:10.1

� Corr

E-m

journal homepage: www.elsevier.com/locate/triboint

The comparison in stability of rotor-aerostatic bearing system compensatedby orifices and inherences

Cheng-Hsien Chen a, Te-Hui Tsai b, Ding-Wen Yang c, Yuan Kang d,�, Juhn-Horng Chen e

a Department of Vehicle Engineering, Army Academy, Taiwan, ROCb Department of Mechanical Engineering, Army Academy, Taiwan, ROCc Department of Commercial Design, Vanung University, Taiwan, ROCd Department of Mechanical Engineering, Chung Yuan Christian University, Chung-Li 320, Taiwan, ROCe Department of Mechanical Engineering, Chung Hua University, Taiwan, ROC

a r t i c l e i n f o

Article history:

Received 12 May 2009

Received in revised form

8 January 2010

Accepted 15 January 2010Available online 20 January 2010

Keywords:

Aerostatic bearing

Orifice

Inherence

Critical inertial force

9X/$ - see front matter & 2010 Elsevier Ltd. A

016/j.triboint.2010.01.006

esponding author. Tel.: +886 3 2654315; fax

ail address: [email protected] (Y. Kang).

a b s t r a c t

This paper studies static and dynamic characteristics of a rigid spindle supported by aerostatic bearings

which are compensated by orifices and inherences. Eigen-solution method is used to determine the

stability thresholds of both critical inertial force and critical whirl ratio belong to the motion equation of

the rigid rotor-aerobearing system which is obtained by perturbation method. Various eccentricities,

rotor speeds and restriction parameters are taken into considerations for the comparisons of restriction

effect between both orifice and inherence. The influences of the number and the locations of air entries

on the aerostatic bearing are also estimated.

& 2010 Elsevier Ltd. All rights reserved.

1. Introduction

Due to the unique and absolute advantages of low friction andhigh precision, aerostatic bearing has been recognized as one veryimportant component of super-precision and ultra-high speedspindles, and it also has been extensively applied in severaldomains of machine tools, measurement equipments and labora-tory instruments. However these bearings tend to have someweaknesses occurring at high speed or low speed such as:pneumatic phenomenon and unstable whirl, which has led to alimitation in using the aerostatic bearings, thus, to analyze thesaid bearings’ static, dynamic characteristics and stability hasbecome a critical procedure while designing the gas bearings.

Up to now, many researchers have adopted both the analyticaland experimental methods to broadly probe into various types ofaerostatic bearings. Lund [1–3] analyzed the four shoe tilting padand three-lobe air-journal bearings and acquired data related tothe load capacities as well as the stiffness and dampingcoefficients that can be used to verify the whirl instability ofaerostatic bearings, wherein, these analyses were performedbased on first-order small-amplitude perturbation and foundout relevant instability threshold by scanning the range of whirlfrequency. Leonard and Rowe [4] utilized mathematical relations

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to compute the dynamic coefficient and whirl mechanism ofaerostatic bearing, thereto, the prediction of the occurrence ofinstability conformed to the experiment results, but such resultscould be applied in the context of small eccentricity. Wadhwaet al. [5] adopted the finite element method and modifiedperturbation formula to calculate the dynamic coefficient andwhirl instability resulted from the orifice restriction gas bearing,thereto, the relevant results indicated that while having a higherrestriction parameter, the change of pressure supply would havean obvious influence on the bearing performance, in addition,they also proved that aerostatic bearing would have a larger loadcapacity and higher stability than the plane journal bearing withsame geometric and operating parameters.

Han et al. [6] used a direct-numerical method to obtain theexternal pressurized air bearing’s static and dynamic character-istics, and meanwhile observing the uneven movement trackoccurring at rotor center through experiments. Zhang and Chang[7] proved that with a small eccentricity, both the orificerestriction and narrow oil-pocket type hybrid air-journal bearingswould have a higher stability, which was superior to one 8-orificegas bearing. Talukder and Stowell [8] studied the occurrence andcharacteristics of the external pressurized orifice-restriction gasbearing, they proposed that oil-pocket orifice with a sufficientdiameter and depth could provide larger oil-film stiffness,however, it would tend to generate the pneumatic phenomenonthan the peripheral orifice. Su and Lie [9] probed into the dynamicinstability of the rotor being supported by hybrid gas bearings

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ARTICLE IN PRESS

Nomenclature

a land width of axial flow (m)Bij;Bij damping coefficients, Bij ¼ Bijco=PaLD, (i,j=e,f and

x,y)b pocket depthc radial clearance of bearing (m)cb decaying exponent of damping whirlCDo discharge coefficient of orificeCDh discharge coefficient of inherence restrictorD diameter of bearing (m)do diameter of orifice (m)dh diameter of inherence (m)dr inlet diametere, e eccentricity (m), e=e/ce0, e0 steady-state eccentricity, e0=e0/cep, ep perturbed eccentricity, ep=ep/cFx, Fy, F x, F y force components (N) due to air film in the x and y

directions, ðF x; F Þ ¼ ðFx; FÞ=PaLD

h0;h0 static film thickness (m), h0 ¼ h0=c

h;h film thickness (m), h ¼ h=c

hr ;hr film thickness at the rth feeding hole (m), hr ¼ hr=c

Ic critical inertial force Ic ¼ ðMg2Þc

Kij;K ij stiffness coefficients, K ij ¼ Kijc=PaLD, (i,j=e,f and x,y)L axial length of bearing (m)l supply orifice length (m)M non-dimensional mass parameter, M ¼mco2=ð2PaLDÞ

Mc critical mass parameter, Mc ¼mco2c =ð2PaLDÞ

m rotor mass (kg)Nx, Nz circumferential and axial index for gridsn number of entriesOb, Oj bearing center, spindle or journal centerP ; Pk non-dimensional pressure, Pk ¼ Pk=Pa, subscript k

denoted for 0, e and fPa atmospheric pressure (N/m2)P0 static pressure (N/m2)Pr ;P rk non-dimensional pressure at the rth inlet, subscript k

denoted for 0, e and fPr0 static pressure at the rth inlet (N/m2)Pe, Pf dynamic pressure (N/m2)Pre, Prf dynamic pressure at the rth inlet (N/m2)Ps; Ps supply pressure (N/m2), Ps ¼ Ps=Pa

Q r ;Q rk non-dimensional rate of mass flow at the rth inlet,Q rk ¼ 24mRTQrk=P2

a c3, subscript k denoted for 0, eand f

Qr0 static mass flow rate (m3/s) at the rth inletQre, Qrf dynamic mass flow rate (m3/s) at the rth inletReo Reynolds number of air flowing through orifice,

Reo=ruh/ms characteristic frequency ratio of the journal motion,

s=cb/O7 i

S; S transformed variable for the transformation ofReynolds equation, S¼ ðPhÞ2, S0 ¼ ðP0h0Þ

2, Se ¼

P0h0ðPeh0þP0 cosyÞ, Sf ¼ P0h0ðPfh0þP0 sinyÞt time (sec)u velocity of air (m/s)T absolute temperature of air film (K)v specific volume of air (m3/kg)W ;W load capacity (N), W ¼W=PsLD

x, y Cartesian coordinates of air filmx; y displacement of journal center, x ¼ Xest, y ¼ Y est

X ;Y initial constant amplitudez; z axial coordinate of bearing, z ¼ z=ðL=2ÞL bearing number, L¼ 6mo=Paðc=RÞ2

G restriction parameterGo restriction parameter of orifice, Go ¼ CDo

ð6mpd2o

ffiffiffiffiffiffiffiRTp

=Pac3Þ

Gh restriction parameter of inherence, Gh ¼ CDh

ð24mpdh


=Pac2Þ

R universal gas constant (8.314 kJ/kg K)O whirl frequency of journal center about the equili-

brium axisc circumferential angle between adjacent restrictions,

c¼ 2p=n

f0 steady-state attitude anglefp perturbed attitude anglem dynamic viscosity of air (N s/m2)k isotropic expansion indexy angular coordinate, y=x/Ryr circumferential position of the rth feeding holer air density (kg/m3)t non-dimensional time, t=Ot

o spindle speed (rad/s)oc critical spindle speed (rad/s)d substitution variableg whirl ratio, g=O/ogc critical whirl ratio, gc=O/oc

C.-H. Chen et al. / Tribology International 43 (2010) 1360–1373 1361

which were consisted of multi-array entries or porous typebearings, and the research results revealed that load capacitywould increase while increasing the numbers of inlet array, andthe rotor system supported by the multi-array orifices were morestable at high speed.

Pink and Stout [10] analyzed the experiment data relating to theorifice restriction air-journal bearings and proposed that inletnumber of each array should be larger than 6, and they suggested, ifthere were insufficient number of entries locating between twoadjacent orifices, it would then lead to a smaller loading capabilitydue to the dispersion of pressure. A general computing method wasused by Kazimierski and Trojnarski [11] to study the externalpressurized air bearings used in different feeding systems, wherein,this method recognized the mass flow rate of Reynolds equation ofeach feed orifice as one source, and their research results were ableto identify that all theoretical results were validated with theexperimental results. Kassab et al. [12] proposed that when

enlarged the ratio of the orifice’s diameter to its length, it couldhence enhance its load capacity due to the increment of pocketpressure thereof. Kwan and Post [13] proposed one designingprocedure which was applicable to the aerostatic bearing with oneinherence or orifice restriction, and for meeting the requirements inthe designing of load capacity and stiffness, it would be able toreduce the production errors by controlling its tolerance. Renn andHsiao [14] adopted the CFD method to modify the critical pressureratio of the nozzle restrictor. Lo et al. [15] and Belforte et al. [16]utilized the numerical and experimental method respectively toobtain the static and dynamic characteristics of high-speed rotorthat was supported by the aerostatic bearings, however, theirresearch results disregarded the influence of numbers and positionsof feeding entries. Moreover, Belforte et al. [17] proposedapproximation functions to determine the values of dischargecoefficients which are functions of the Reynolds number and thefeeding systems geometric parameters for both annular orifice and

ARTICLE IN PRESS

planesymmetric

Oj

Ob

Pn

Pr

W

φ0

θ1

e0

Fx

Fy

x

y

ψ

L

A

A

D

a

P1

�

�

C.-H. Chen et al. / Tribology International 43 (2010) 1360–13731362

simple orifice with feed pocket. The data obtained with approxima-tion functions show a fairly fit with experimental results. Yang et al.[18] studied the influence of the number and the locations oforifices related to the stability of rotor-aerostatic bearing system toindicate that the different effects due to various designs.

Although both the static and dynamic characteristics ofaerostatic bearing as well as the stability of the rotor-bearingsystem were studied successfully, however, very few literatureshad ever looked into the influence of the number and locations ofair entries on bearing characteristics and stability of rotor-bearingsystem. Thus this study aims at rotor-aerostatic bearing systemwith double-arrays restriction due to orifices and inherences toacquire the static and dynamic characteristics of aerostaticbearing and the stability thresholds of rotor-bearing system. Bothcritical inertial force and whirl ratio under different situations dueto various rotor speeds, journal eccentricity, and restrictionparameter will be determined in this study in order to comparethe differences of bearing performances between two kinds ofrestrictors.

sectionAA−

h

dr

Ps

bearing

shaft journal

lb

do

h

dr

Psbearing

shaft journal

l

dh

�r

Fig. 1. Configurations of (a) an aerostatic bearing with double-array entries

compensated by (b) orifice restriction (c) inherence restriction.

2. Lubrication analysis

2.1. Lubrication equations of aerostatic bearing

For an aerostatic journal bearing as shown in Fig. 1(a), airsupplied from an externally pressurized source is passing throughentries with orifice or inherent compensation as shown inFig. 1(b) and (c), respectively, which are located double-rowsabout a symmetric plane and evenly around the circumference ofthe bearing. With the assumptions of compressible, isothermallaminar flow situated in bearing clearance and perfect gas for air,the non-dimensional Reynolds equation of this air film which isderived from Navier–Stokes and continuity equations can beexpressed in the two dimensional Cartesian coordinates system asshown

@

@yh

3 @P2

@y

" #þ

D

L

� �2 @

@zh

3 @P2

@z

" #¼ 2L

@

@yðPhÞþ4gL @ðPhÞ

@tð1Þ

where D and L are bearing diameter and length, P and h are non-dimensional pressure and thickness of film, y and z are angularand axial coordinates of bearing, respectively, t is non-dimensional time, L is bearing number and g is whirl ratio.

The journal center rotates about its steady-state position (e0,f0) with a small whirl as shown in Fig. 2 which can bedecomposed into radial and tangential components representedby ReðepeitÞ and Reðe0fpeitÞ, respectively. This small journal whirlwill induce small perturbations of the film pressure and filmthickness which are composed of steady-state part subscripted by0 and dynamic parts subscripted by e and f. Thus, the non-dimensional film thickness and pressure distributions can beexpressed by

h ¼ h0þepeit cosyþe0fpeit siny ð2aÞ

where h0 is the static film thickness and defined by

h0ðy; zÞ ¼ 1þe0 cosðy�f0Þ ð2bÞ

where f0 is steady-state attitude angle, and

P ¼ P0þepeitPeþe0fpeitPf ð3Þ

Furthermore, at the rth inlet the non-dimensional perturbedmass flow rate and downstream pressure of entries can beexpressed by

Q r ¼ Q r0þepeitQ reþe0fpeitQ rf ð4Þ

Pr ¼ Pr0þepeitPreþe0fpeitPrf ð5Þ

The appropriate boundary conditions of Eq. (1) with P and h

perturbations are described by P0ðy; z ¼ 1Þ ¼ 1 and Pe ¼ Pf ¼ 0for the both open axial ends, ð@Pk=@zÞðy; z ¼ 0Þ ¼ 0 for thesymmetric axis of bearing, Pkðy; zÞ ¼ Pkðyþ2p; zÞ for the continu-ity of an intact film, where k denoted for 0, e and f. Moreover, themethod described in Kazimierski and Trojnarski [11] is utilized,substituting the variation of the supply mass flow rate of adiabaticprocess into Eq. (1) to form the non-dimensional continuityequation which is the boundary condition at the position of eachinlet as

@

@yh

3 @P2

r

@y

" #þ

D

L

� �2 @

@zh

3 @P2

r

@z

" #¼ 2L

@

@yðPrhÞþ4gL @ðPrhÞ

@t �D

LQ r

ð6Þ

ARTICLE IN PRESS

φp

φ

ε0φpc

εpc

ε

Oj

Ob

Fig. 2. Perturbed orbit of journal center whirling.


the mass flow rate Q r for orifice restriction can be determined by

Q r ¼GoPsd; r¼ 1;2; . . . ; n ð6aÞ

where Go ¼ CDoð6mpd2o


=Pac3Þ is restriction parameter oforifice, and the discharge coefficient of orifice CDo which dependsonly on Reynolds number can be determined by utilizing theformula proposed by Belforte et al. [17] as

CDo ¼ 1:5ð1�e�0:005Reo Þ ð6bÞ

where, Reo ¼ ruh=m is Reynolds number for orifice. And, the massflow rate Q r for inherence restriction can be determined by

Q r ¼GhhrPsd; r¼ 1;2; . . . ; n ð6cÞ

for restriction, respectively, where hr ¼ hr=c is film thickness atthe rth feeding hole, Gh ¼ CDhð24mpdh


=Pac2Þ is inherencerestriction parameter. Similarly, the discharge coefficient ofinherence CDh can be determined by

CDh ¼ 0:85ð1�e�8:2h=dh Þð1�e�0:001Reo Þ ð6dÞ

and d can be described respectively by

d¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2kk�1

Pr

Ps

!2=k

�Pr

Ps

!ðkþ1Þ=k24

35

vuuut asPr0

Ps

42

kþ1

� �k=ðk�1Þ

of subsonic condition ð6eÞ

and

d¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik 2

kþ1

� �ðkþ1Þ=ðk�1Þs

asPr0

Ps

r2

kþ1

� �k=ðk�1Þ

of supersonic condition: ð6fÞ

2.2. Solution method

The perturbation method suggested by Lund [3] is thatsubstituting Eqs. (2) and (3) into Eq. (1) with neglecting higher

order terms gives the perturbed Reynolds equations of static anddynamic pressures. A new variable is introduced into theseperturbed equations, which is defined by

S¼ ðPhÞ2 ¼ S0þ2Seepeitþ2Sfe0fpeit ð7Þ

where S0 ¼ ðP0h0Þ2, Se ¼ P0h0ðPeh0þP0 cosyÞ and Sf ¼ P0h0ðPfh0

þP0 sinyÞ. Introducing these variables into the perturbed Rey-nolds equations gives a transformed form in coupling the staticand dynamic parts as shown

@2S0

@y2þ

D

L

� �2 @2S0

@z2�X0

@S0

@y�

2

h0

@2h0

@y2S0 ¼ 0 ð8aÞ

@2Se

@y2þ

D

L

� �2 @2Se

@z2�X0

@Se@y�Y0Se

¼�1

2h0

ðX0 cosyþsinyÞ@S0

@y�

2 cosyh0

1

h0

@2h0

@y2þ1

!S0

" #ð8bÞ

@2Sf

@y2þ

D

L

� �2 @2Sf

@z2�X0

@Sf

@y�Y0Sf

¼�1

2h0

ðX0 sin yþcosyÞ@S0

@y�

2 sinyh0

1

h0

@2h0

@y2þ1

!S0

" #ð8cÞ

where X0 ¼ 1=h0

�ð@h0=@yÞþðL=

ffiffiffiffiffiS0

qÞ

�and Y0 ¼ 1=h0

�2ð@2h0=@y

2Þ

�ðL=2S3=2

0 Þð@S0=@yÞþ i2gðL=ffiffiffiffiffiS0

qÞ

�. Since the variable Sk is even

function in z-axis, the boundary conditions for Eqs. (8a)–(8c) can

be transformed accordingly by S0 ¼ ðP0h0Þ2, Se ¼ P

2

0h0 cosy and

Sf ¼ P2

0h0 siny at z ¼ 1 and 0ryr2p, ð@Sk=@zÞ ¼ 0 at z ¼ 0 and

0ryr2p, and SkðyÞ ¼ Skðyþ2pÞ at 0rzr1, where k denoted for

0, e and f.Substituting Eqs. (2), (4) and (5) into Eq. (6) gives the

perturbed and transformed form of the non-dimensional staticand dynamic continuity equations at the orifice entry down-stream individually. Thus, the non-dimensional static and dy-namic mass flow rates restricted through orifice supplied from therth inlet during adiabatic process can be expressed by

Q r0 ¼GoPs

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2kk�1

Pr0

Ps

!2=k

�Pr0

Ps

!ðkþ1Þ=k24

35

vuuut ð9aÞ

Q re ¼Go

ffiffiffiffiffiffiffiffiffiffi2kk�1

r2

kPr0

Ps

!ð2�kÞ=k�ðkþ1Þ

kPr0

Ps

!1=k24

35 Pr0

Ps

!2=k24

�Pr0

Ps

!ðkþ1Þ=k35�ð1=2Þ

Pre ð9bÞ

Q rf ¼Go


r2

kPr0

Ps


kPr0

Ps

!1=k24

35 Pr0

Ps

!2=k24

�Pr0

Ps


Prf ð9cÞ

for ðPr0=PsÞ4�

2=ðkþ1Þ�k=ðk�1Þ

4 , and

Q r0 ¼GoPs


kþ1

� �ðkþ1Þ=ðk�1Þs

ð9dÞ

ARTICLE IN PRESS


Q re ¼ 0 ð9eÞ

Q rf ¼ 0 ð9fÞ

for Pr0=Psr�


, respectively.And, the non-dimensional static and dynamic mass flow rates

restricted through inherence supplied from the rth inlet duringadiabatic process can be expressed by

Q r0 ¼ hrGhPs


Pr0

Ps

!2=k

�Pr0

Ps

!ðkþ1Þ=k24

35

vuuut ð10aÞ

Q re ¼ hrGh


r2

kPr0

Ps


kPr0

Ps

!1=k24

35 Pr0

Ps

!2=k24

�Pr0

Ps


Pre

þGhP s


Pr0

Ps

!2=k

�Pr0

P s

!ðkþ1Þ=k24

35

vuuut cosy ð10bÞ

Q rf ¼ hrGh


r2

kPr0

Ps


kPr0

Ps

!1=k24

35

�Pr0

Ps

!2=k

�Pr0

Ps

!ðkþ1Þ=k24

35�ð1=2Þ

Prf

þGhPs


Pr0

Ps

!2=k

�Pr0

Ps

!ðkþ1Þ=k24

35

vuuut siny ð10cÞ

for ðPr0=PsÞ4ð2=kþ1Þk=ðk�1Þ, and

Q r0 ¼ h0GhPs


kþ1

� �ðkþ1Þ=ðk�1Þs0

@1A

1=2

ð10dÞ

z

j: Nz

Nz-1

43

2

11 2 3

sym

�

�r

�1

Fig. 3. Grid structure for t

Q re ¼ cosy


kþ1


@1A

1=2

ð10eÞ

Q rf ¼ siny


kþ1


@1A

1=2

ð10fÞ

for P ro=Psr�


, respectively.

The Eqs. (8a) with (9a) and (10a) or with Eqs. (9d) and (10d)are coupled nonlinear steady-state equations which can be solvedto obtain the static solution of S0 in advance. Eqs. (8b) and (8c) arelinear equations with dependent complex variables Se and Sf,which are solved numerically in finite difference form depend onsubstituting the steady state solutions. The number of totalnodes in the grid structure for numerical analis is Nx�Nz asshown in Fig. 3, for each ðSkÞi;j, k= 0, e and f, coupled equationswill be solved simultaneously with continuity Eqs. (9a)–(9f) or(10a)–(10f) at the position of each inlet and the boundaryconditions as mentioned above by using Newton–Raphsonalgorithm with a successive over-relaxation scheme asdescribed in pp. 109–129 of finite difference method [19].

For a given eccentricity the iteration process of the steady-statesolution is started with initial guesses of node pressures and anattitude angle. The iteration will be repeated until all the ðS0Þi;j

and the attitude angle converges to the error allowances. Thecorrect steady-state attitude angle f will be allowed due to theabsolute value of the ratio of non-dimensional resultant force ofstatic film pressure P0 in both coordinate axis directions jF x=F yj

being smaller than a criterion value. And then substitute staticpressure P0 and thickness h0 into Eqs. (8b) and (8c), the dynamicpressures Se and Sf for a given whirl ratio g can be solvedsimultaneously. The error allowance of the numerical convergenceis defined by

MaxjðSkÞ

mi;j�ðSkÞ

m�1i;j j

ðSkÞmi;j

!rE; k¼ 0; e;f; for 1r irNx and 1r jrNz

ð11Þ

metry plane

Nx-1 Nx :i�

feeding hole

Δz

Δ�

he numerical analysis

ARTICLE IN PRESS


where m and m�1 denoted for the orders of the current and therecently last iteration. E=10�5 is the value of error allowance inthis study.

The load capacity W , stiffness K ij and damping Bij, (i,j=e,f and

x,y) coefficients of bearing are equivalent to the resultant forcesand thus determined by pressure distributions of air film in thebearing clearance. The detailed determinations can be seen inYang et al. [18].

0 100.0

0.2

0.4

0.6

0.8

1.0

W

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

W

1 2 3 4 5 6 7 8 9�

0 101 2 3 4 5 6 7 8 9�

Fig. 5. Load capacity (W ) and attitude angle (f0) due to P s ¼ 5 with respect to restric

– – – – –for inherence and ——— for orifice. m, n:e0=0.1; ’, &:e0=0.3; K, J:e0=0.5.

(x,

equilibrium position

bearing

Ob

Oj

six entries type 1 of thre

Fig. 4. Rigid rotor and aerostatic bearing system (a) model (b) t

3. Whirl instability analysis

For a rigid rotor supported horizontally by two identicalaerostatic journal bearings on both ends as illustrated in Fig. 4(a),the rotor is assumed to consist of a massless shaft and a single,centered mass m without unbalance mass. When the bearing isoperating under steady-state condition if the gas film in aerostaticbearing is disturbed by dynamic excitations with a frequency O,

0 100

10

20

30

40

50

60

70

10

15

20

25

30

35

40

0 101 2 3 4 5 6 7 8 9�

�0

1 2 3 4 5 6 7 8 9�

�0

tion parameter (G) for bearing with type 1 of three entries, (a) L=1.0, (b) L=5.0,

y)

m

whirl orbit of

journal center

center

Ob

Oj

e entries type 2 of three entries

hree types for various numbers and locations of air entries

ARTICLE IN PRESS

0 100.0

0.2

0.4

0.6

0.8

1.0

W

0 100

10

20

30

40

50

60

70

0 100.0

0.2

0.4

0.6

0.8

1.0

W

0 1010

15

20

25

30

35

40

�1 2 3 4 5 6 7 8 9

�0

1 2 3 4 5 6 7 8 9�

1 2 3 4 5 6 7 8 9�

�0

�1 2 3 4 5 6 7 8 9

Fig. 6. Load capacity (W ) and attitude angle (f0) due to P s ¼ 5 with respect to restriction parameter (G) for bearing with type 2 of three entries, (a) L=1.0, (b)L=5.0,

– – – – – for inherence and for ——— orifice. m, n:e0=0.1; ’, &:e0=0.3; K, J:e0=0.5.

0 100.0

0.2

0.4

0.6

0.8

1.0

1.2

W

0

10

20

30

40

50

60

70

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

W

0

5

10

15

20

25

30

35

40

�1 2 3 4 5 6 7 8 9

0 10�

1 2 3 4 5 6 7 8 9 0 10�

1 2 3 4 5 6 7 8 9

0 10�

1 2 3 4 5 6 7 8 9

�0

�0

Fig. 7. Load capacity (W ) and attitude angle (f0) due to Ps ¼ 5 with respect to restriction parameter (G) for bearing with six entries (a) L=1.0, (b)L=5.0, – – – – – for

inherence and ——— for orifice. m, n:e0=0.1; ’, &:e0=0.3; K, J:e0=0.5.


ARTICLE IN PRESS


the film will whirl at this frequency and attempt to maintain theflow balance. Based on a small disturbance from the equilibriumposition of the journal, the non-dimensional perturbationmotion equations of journal center about the equilibrium axisof the rotor in the fixed reference coordinates (Ob, x, y) aredescribed by

Mg2 0

0 Mg2

" #€x€y

" #þ

gBxx gBxy

gByx gByy

" #_x_y

" #þ

K xx K xy

K yx K yy

" #x

y

" #¼

0

0

� �

ð12Þ

where M ¼mco2=ð2PaLDÞ is the non-dimensional massparameter, g¼O=o is the whirl ratio. Substituting

x ¼ Xest and y ¼ Y est ð13Þ

where X and Y are constant, into Eq. (12) gives a set ofsimultaneous and linear equations of X and Y amplitudedepending on the initial conditions, and s¼ ðcb=OÞ7 i representsthe characteristic frequency ratio of the journal motion.

Mg2s2þgBxxsþK xx gBxysþK xy

gByxsþK yx Mg2s2þgByysþK yy

" #X

Y

" #¼

0

0

� �ð14Þ

For the ideal flow of a non-viscous lubricant the half-frequencywhirl (g=0.5) can be induced by a flow balance; however, thewhirl frequency within a real rotor-bearing system can beinfluenced by both inertia and damping effects. The changes ofwhirl frequency will be determined from the eigensolutions of thecharacteristic equation. When a journal is subjected to a criticalwhirling condition, there will be at least one set of purelyimaginary roots as expressed by

s¼7 i ð15Þ

At this critical condition, cb is zero and gc ¼O=oc , where oc isthe critical journal speed.

0.0 0.90.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

W

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

W

�0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

0.0 0.9�0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

Fig. 8. Load capacity (W ) and attitude angle (f0) due tor P s ¼ 5 with respect to eccent

——— orifice with Co=2. K, J: six entries; ’, &: type 1 of three entries; ., ,: type

Substituting Eq. (13) into Eq. (15) gives

�IcþK xx�igcBxx K xy�igcBxy

Ic�igcByx �IcþK yy�igcByy

" #X

Y

" #¼

0

0

� �ð16Þ

where stability threshold Ic ¼ ðMg2Þc represents the criticalinertial force which is impedance to the self-excited vibrationcharacterizing by the critical whirl ratio gc, at the threshold ofinstability.

The non-trivial solutions of Eq. (16) for X and Y are derivedfrom the determinant of coefficients matrix being

IcðBxxþByyÞ�ðK xxByyþK yyBxxÞþK xyByxþK yxBxy ¼ 0 ð17Þ

ðIc�K xxÞðIc�K yyÞ�g2c ðBxxByy�BxyByxÞ�K xyK yx ¼ 0 ð18Þ

From Eq. (17) directly, Ic can be obtained by

Ic ¼K xxByyþK yyBxx�K xyByx�K yxBxy

BxxþByy

ð19Þ

Substituting Eq. (18) into Eq. (19) gives the critical whirl ratioas

g2c ¼ðIc�K xxÞðIc�K yyÞ�K xyK yx

BxxByy�BxyByx

ð20Þ

For aerostatic bearings the stiffness and damping coefficientsare implicit functions of whirl ratio g, Eqs. (19) and (20) must besolved iteratively due to modifying g by g=g+a(gc�g), (0oar1)until jgc�gj=g converges to an error allowance. Consequently,when a rotor operates at a specific rotating speed o self-excitedvibration with a frequency O=go occurs when grgc or Mg2 ofthe rigid rotor-aerostatic bearing system is larger than Ic.

0

5

10

15

20

25

30

0

5

10

15

20

25

0.0 0.9�0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

0.0 0.9

�0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

�0

�0

ricity ratio (e0) for (a) L=1.0, (b) L=5.0, – – – – – for inherence with Ch=2, and for

2 of three entries.

ARTICLE IN PRESS


4. Results and discussion

This study compares the static, dynamic characteristics andwhirl instability of a rigid rotor supported by two identicalaerostatic bearings with double-array entries, which are eitherorifices or inherences, as shown in Fig. 1. The ratio of bearing’slength to its diameter L/D is 1.0 and land width of axial flow a

equals to L/4. As shown in Fig. 4(b), each array comprises six orthree entries which are arranged symmetrically around circum-ference surface of such bearing. For the number of n=6, theentries are located equally at yr=01, 601, 1201, 1801, 2401 and 3001separately. Two types of arrangements for three entries are takeninto consideration, for type 1 of three entries locate at yr=01 (one

0.00.4

0.6

0.8

1.0

1.2

1.4

1.6

Kxx

0.4

0.6

0.8

1.0

1.2

1.4

1.6

Kyy

-0.40

-0.35

-0.30

-0.25

-0.20

-0.15

-0.10

Kxy

0.10

0.15

0.20

0.25

0.30

0.35

0.40

Kyx

�0.2 0.4 0.6 0.8 1.0

0.0�

0.2 0.4 0.6 0.8 1.0

0.0�

0.2 0.4 0.6 0.8 1.0

0.0�

0.2 0.4 0.6 0.8 1.0

Fig. 9. Stiffness (K ) and damping (B) coefficients with respect to whirl ratio (g) due to

with Co=2. K, J: six entries; ’, &: type 1 of three entries; ., ,: type 2 of three ent

at top), 1201 and 2401, and for type 2 of three entries locate atyr=601, 1801 (one at bottom) and 3001, respectively. The entriesare installed orifices or not in order to feed air with constantrestriction due to orifice and inherence compensations, respec-tively. All the air feed is assuming with an identical parameter ofrestriction.

The variations in load capacity W and its correspondingattitude angle fo with respect to restriction parameters G ofthose six entries and both types of three entries, are shown fromFigs. 5–7, respectively. For all cases, the load capacity of aerostaticbearings compensated by orifices is larger than that compensatedby inherences, and their attitude angles are smaller than thatcompensated by inherences. At low journal speed, the differences

0.30

0.35

0.40

0.45

0.50

0.55

0.60

0.65

Bxx

0.45

0.50

0.55

0.60

0.65

0.70

0.75

0.80

Byy

0.10

0.15

0.20

0.25

0.30

0.35

Bxy

-0.25

-0.20

-0.15

-0.10

-0.05

Byx

0.0�

0.2 0.4 0.6 0.8 1.0

0.0�

0.2 0.4 0.6 0.8 1.0

0.0�

0.2 0.4 0.6 0.8 1.0

0.0�

0.2 0.4 0.6 0.8 1.0

P s ¼ 5, and L=1 as e0=0.3, – – – – – for inherence with Ch=2, and ——— for orifice

ries.

ARTICLE IN PRESS


of the load capacity between both compensations are large;however, such difference is very small at high journal speed. For aconstant pressure supply, maximum load capacities will beobtained with various optimal restriction parameters used forboth compensations individually. Under a condition of individualoptimal restriction parameter, when it is at low journal speed, theload capacity of six entries is larger than that of both types ofthree entries; and when it is at high speed, the load capacity of sixentries is equal to that due to type 1 of three entries, however, it islarger than that due to type 2 of three entries.

The variations of load capacity W and attitude angle f0 forPs ¼ 5with respect to the eccentricity ratio e0 are shown in Fig. 8(a)

0.01.0

1.2

1.4

1.6

1.8

2.0

2.2

Kxx

1.2

1.4

1.6

1.8

2.0

2.2

2.4

Kyy

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

Kxy

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

1.2

Kyx

�0.2 0.4 0.6 0.8 1.0

0.0�

0.2 0.4 0.6 0.8 1.0

0.0�

0.2 0.4 0.6 0.8 1.0

0.0�

0.2 0.4 0.6 0.8 1.0

Fig. 10. Stiffness (K ) and damping (B) coefficients with respect to whirl ratio (g) due to

with Co=2. K, J: six entries; ’, &: type 1 of three entries; ., ,: type 2 of three ent

and (b) of L=1.0 and 5.0, respectively. The load capacity of orificerestriction can be larger than that of inherent restriction whene0o0.5, contrarily, in the case of type 2 of three entries, its loadcapacity is different when e0Z0.5. When individual optimalrestriction parameters of both compensations are used, their loadcapacities are very similar. Especially for type 2 of three entries withvarious compensations, their load capacities are almost equal undera condition of large eccentricity. Wherein, the load capacity ofinherent restriction is larger than that of orifice restriction, butthere are large differences between the attitude angles of these twocompensations, the attitude angle due to orifice compensation isabout 51 less in both rotation speeds cases.

0.0

0.2

0.4

0.6

0.8

1.0

1.2

Bxx

0.0

0.2

0.4

0.6

0.8

1.0

1.2

Byy

0.20.40.60.81.01.21.41.61.82.0

Bxy

-2.0-1.8-1.6-1.4-1.2-1.0-0.8-0.6-0.4-0.2

Byx

0.0�

0.2 0.4 0.6 0.8 1.0

0.0�

0.2 0.4 0.6 0.8 1.0

0.0�

0.2 0.4 0.6 0.8 1.0

0.0�

0.2 0.4 0.6 0.8 1.0

P s ¼ 5, and L=5 as e0=0.3, – – – – – for inherence with Ch=2, and ——— for orifice

ries.

ARTICLE IN PRESS


In Figs. 9 and 10, we can compare these dynamic coefficients ofsix entries and both types of three entries under conditions ofrestriction parameter Go=2 and Gh=2 respectively, eccentricity

0 110.0

0.2

0.4

0.6

0.8

1.0

1.2

Stability

InstabilityI c

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

Stability

Instability

�1 2 3 4 5 6 7 8 9 10

0 11�

1 2 3 4 5 6 7 8 9 10

2

I c

Fig. 11. Critical inertial force (Ic) and square of critical whirl ratio (gc2) due to P s ¼ 5

(a) L=1, (b) L=5, – – – – – for inherence and ——— for orifice. m, n:e0=0.1; ’, &:e0=

0 110.0

0.2

0.4

0.6

0.8

1.0

1.2

Stability

Instability

I c

0.0

0.4

0.8

1.2

1.6

Instability

Stability

I c

�1 2 3 4 5 6 7 8 9 10

0 11�

1 2 3 4 5 6 7 8 9 10

22

Fig. 12. Critical inertial force (Ic) and square of critical whirl ratio (gc2) due to P s ¼ 5

(a) L=1, (b) L=5, – – – – – for inherence and ——— for orifice. m, n:e0=0.1; ’, &:e0=

ratio e0=0.3, and at rotation speed L=1 and 5, respectively. Directstiffness coefficient of orifice restriction is larger than that of in-herent restriction. Direct damping coefficient of inherent restriction

-0.4

-0.3

-0.2

-0.1

0.0

0.1

0.2

0.3 Stability

Instability

0.0

0.1

0.2

0.3Stability

Instability

0 11�

1 2 3 4 5 6 7 8 9 10

0 11�

1 2 3 4 5 6 7 8 9 10

��2

cc

with respect to restriction parameter (G) for bearing with type 1 of three entries

0.3; K, J:e0=0.5.

-0.3

-0.2

-0.1

0.0

0.1

0.2

0.3

Instability

Stability

0.1

0.2

0.3

Instability

Stability

0 11�

1 2 3 4 5 6 7 8 9 10

0 11�

1 2 3 4 5 6 7 8 9 10

��

cc

with respect to restriction parameter (G) for bearing with type 2 of three entries

0.3; K, J:e0=0.5.

ARTICLE IN PRESS


is larger than that of orifice restriction. Whereas, the cross stiffnessand cross damping do not have the uniform relationships. Thedifferences of stiffness and damping between both restrictions atlow journal speed are larger than those at high journal speed.

0 110.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

Stability

InstabilityI c

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8Instability

Stability

I c

�

1 2 3 4 5 6 7 8 9 10

0 11

�

1 2 3 4 5 6 7 8 9 10

�22

Fig. 13. Critical inertial force (Ic) and square of critical whirl ratio (gc2) due to Ps ¼ 5 wit

– – – – – for inherence and ——— for orifice. m, n:e0=0.1; ’, &:e0=0.3; K, J:e0=0.5.

0.00.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

Stability

Instability

I c

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

Stability

Instability

I c

�0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

0.0�0

0.1 0.2 0.3 0.4 0.5 0.6 0.7

22

Fig. 14. Critical inertial force (Ic) and square of critical whirl ratio (gc2) due to P s ¼ 5 wit

Ch=4, and ——— for orifice with Co=6. K, J: six entries; ’, &: type 1 of three entri

Determinations of the stiffness and damping coefficients cannotgive any useful data for facilitating the bearing design. However,these coefficients are useful for determining the instabilitythreshold induced by film whirl of a rotor-bearing system.

0.05

0.10

0.15

0.20

0.25

0.30Stability

Instability

0.1

0.2

0.3Stability

Instability

0 11

�

1 2 3 4 5 6 7 8 9 10

0 11

�

1 2 3 4 5 6 7 8 9 10

�c

c

h respect to restriction parameter (G) for bearing with six entries (a) L=1, (b) L=5,

-0.2

-0.1

0.0

0.1

0.2

0.3Stability

Instability

-0.2

-0.1

0.0

0.1

0.2

0.3

Instability

Stability

0.0�0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

0.0�0

0.1 0.2 0.3 0.4 0.5 0.6 0.7

��

cc

h respect to eccentricity ratio (e0) for (a) L=1, (b) L=5, – – – – – for inherence with

es; ., ,:type 2 of three entries.

ARTICLE IN PRESS


The variations in critical inertial force Ic with respect torestriction parameters G for six entries and both types of threeentries are shown from Figs. 11–13, respectively. The thresholdsof critical inertial forces due to orifice restrictions are larger thanthose due to inherent restrictions at both rotation speeds.Inversely, the thresholds of critical whirl ratios due to theorifice restriction are smaller than those due to the inherentrestriction. The differences of critical inertial forces, under acondition of low journal speed, between both restrictions arelarger than those at high speed. The critical inertial force isgetting while the eccentricity is getting larger. The criticalinertial forces due to high speed are larger than those due tolow speed.

The variations in the square of critical whirl ratio gc2 with

respect to the restriction parameters G for three kinds of bearingsare shown at the right portions from Figs. 11–13. It reveals thatthe rotor motion is unstable as the frequency square g24gc

2, also,when gc decreases to zero as the restriction parameter increases,the air film will be no longer to endure dynamic load exerted bythe rotor. Hence, both regions of the dynamic instability arevaried due to the whirl of air film which is induced by thedynamic effects of high journal rotation speed.

The stability thresholds of three kind aerobearings at highspeed of journal rotation are similar. When restriction parameteris low, the instability of the rotor is induced by the hydrodynamicpressure. Also, at low journal speed, the stability is worse whilethe eccentricity is larger, the stability of orifice compensated isbetter than that of inherence compensated among three kinds ofeccentricities.

The critical inertial force Ic and its corresponding square ofcritical whirl ratio (gc

2) with respect to eccentricity ratios e0 atboth low and high speed of journal rotation are shown inFig. 14(a) and (b) respectively. In comparison of critical inertialforces in the selection of optimal restriction parameters, thosedue to three entries of both types are equal, however, for sixentries that is smallest among three cases. While selecting theoptimal restriction parameters for both restrictions, the criticalinertial force of six entries is larger than that due to both types ofthree entries. At low journal speed, the threshold becomes smalleras the eccentricity is increasing. Otherwise, at high journal speed,the threshold becomes larger while the eccentricity is increasing.

For the same condition, the square of critical whirl ratio due toinherence is better than that due to orifice compensation. At lowrotor speed, that due to six entries is larger than that due to type 2of three entries, and both are larger than that due to type 1 ofthree entries. At high rotor speed, that due to type 2 of threeentries is larger than that due to six entries and type 1 of threeentries. At high speed of journal rotation, the Ic increases as theeccentricity ratio is increasing, since the hydrodynamic effectincreases due to the journal speed. On the other hand, at lowspeed, the Ic decreases as the eccentricity ratio increases.

5. Conclusions

The simulation results are summarized below:

(1)
For all cases, the load capacity of aerostatic bearingscompensated by the orifices is larger than that compensatedby inherences, and wherein, the corresponding attitude angleis smaller than that compensated by inherences. In theconditions of individual optimal restriction parameter, forlow journal speed the load capacity due to six entries is largerthan that due to type 1 of three entries, and larger than thatdue to type 2 of three entries as well; moreover, for highjournal speed the load capacities due to both six entries and
type 1 of three entries are equal, which is larger than that dueto type 2 of three entries.

(2)
Direct stiffness coefficients due to orifice restriction are largerthan those due to inherence restriction. Inversely, directdamping coefficients due to inherence restriction are largerthan those due to orifice restriction.
(3)
The critical inertial forces due to orifice restriction are largerthan those due to the inherence restriction. Inversely, thethresholds of critical whirl ratio due to the orifice restrictionare smaller than those due to inherence restriction. Thedifference of stability threshold between both restrictions atlow journal speed is larger than that at high journal speed.
(4)
The critical inertial force is getting larger as the eccentricity isgetting smaller. The critical inertial force at high speed islarger than that at low speed. An individual case with thecondition of various numbers and locations of entries as wellas optimal restriction parameters is corresponding to themaximum critical inertial forces.
(5)
In comparison of critical inertial force, for optimal restrictionparameters that of three entries in both types are equal andare smaller than that of six entries in both conditions of lowand high journal speed. The critical inertial force of orificerestriction surpasses those of inherence restriction in allsituations.
(6)
In all situation cases and under the same condition, thecritical whirl ratio of the inherent restriction is superior to theorifice restriction. At low rotor speed, the threshold of sixentries is larger than that due to type 2 of three entries; bothare all larger than that due to type 1 of three entries. At highrotor speed, the threshold due to type 2 of three entries islarger than that that due to six entries which are also largerthan that due to type 1 of three entries as well.
Acknowledgement

This study was supported by the National Science Councilunder grant NSC 92-2623-7-033-003-ET.

References

[1] Lund JW. The hydrostatic gas journal bearing with journal rotation andvibration. J Basic Eng, Trans ASME J Lub Tech 1964;87(2):328–36.

[2] Lund JW. A theoretical analysis of whirl instability and pneumatic hammerfor a rigid rotor in pressurized gas journal bearing. Trans ASME J Lub Tech1967;89(2):154–65.

[3] Lund JW. Calculation of stiffness and damping properties of gas journalbearing. Trans ASME J Lub Tech 1968;90(4):793–803.

[4] Leonard R, Rowe WB. Dynamic force coefficient and the mechanism ofinstability in hydrostatic journal bearings. Wear 1973;23:277–82.

[5] Wadhwa SS, Sinhasant R, Singh DV. Analysis of orifice compensatedexternally pressurized gas bearing. Trib Int 1983;16(4):203–11.

[6] Han DC, Park SS, Kim WJ, Kim JW. A study on the characteristics of externallypressurized air bearings. Precis Eng 1994;16(3):164–73.

[7] Zhang RQ, Chang HS. A new type of hydrostatic/hydrodynamic gas journalbearing and its optimization for maximum stability. Trib Trans 1995;38(3):589–94.

[8] Talukder HM, Stowell TB. Pneumatic hammer in an externally pressurizedorifice compensated air journal bearing. Trib Int 2003;36(8):585–91.

[9] Su JCT, Lie KN. Rotor dynamic instability analysis on hybrid air journalbearings. Trib Int 2006;39(1):238–48.

[10] Pink EG, Stout KJ. Design procedures for orifice compensated gas journalbearings based on experimental data. Trib Int 1978;11(1):63–75.

[11] Kazimierski Z, Trojnarski J. Investigations of externally pressurized gasbearings with different entry systems. Trans ASME J Lub Tech 1980;102(1):59–64.

[12] Kassab SZ, Noureldeen EM, Shawky MA. Effects of operating conditions andsupply hole diameter on the performance of a rectangular aerostatic bearing.Trib Int 1997;30(7):533–45.

[13] Kwan YB, Post JB. A tolerancing procedure for inherently compensated,rectangular aerostatic thrust bearings. Trib Int 2000;33(8):581–5.

ARTICLE IN PRESS


[14] Renn IC, Hsiao CH. Experimental and CFD study on the mass flow ratecharacteristic of gas through orifice-type restrictor in aerostatic bearing. TribInt 2004;37(4):309–15.

[15] Lo CY, Wang CC, Lee YH. Performance analysis of high-speed spindleaerostatic bearings. Trib Int 2005;38(1):5–14.

[16] Belforte G, Raparelli T, Viktorov V, Trivella A, Colombo F. An experimentalstudy of high-speed rotor supported by air bearings: test RIG and firstexperimental results. Trib Int 2006;39(8):839–45.

[17] Belforte G, Raparelli T, Viktorov V, Trivella A. Discharge coefficients of orifice-type restrictor for aerostatic bearings. Trib Int 2007;40(8):512–21.

[18] Yang DW, Chen CH, Kang Y, Hwang RM, Shyr SS. Influence of orificeson stability of rotor-aerostatic bearing system. Trib Int 2009;42(8):1206–19.

[19] Laurene VFausett. Applied Numerical Analysis Using Matlabs. Prentice-Hall,Inc.; 1999.

Documents

The comparison in stability of rotor-aerostatic bearing system compensated by orifices and inherences