Tribology International 37 (2004) 227–234www.elsevier.com/locate/triboint

The influences of orifice restriction and journal eccentricity on thestability of the rigid rotor-hybrid bearing system

Cheng-Hsien Chena, Yuan Kangb,∗, Ching-Chu Huanga

a Department of Refrigeration and Air Conditioning, Chin Min College, No. 110, Syuefu Rd., Toufen Township, Miaoli County, 351, Taiwan,R.O.C.

b Department of Mechanical Engineering, Chung Yuan Christian University, Chung-Li 320, Taiwan, R.O.C.

Received 3 April 2003; received in revised form 19 June 2003; accepted 10 July 2003

Abstract

The influences of restriction parameter and eccentricity on the stability of the rigid rotor supported by single-row, six-recessedhybrid bearings with orifice compensation are studied. The load capacity, stability threshold, and the critical whirl ratio versus thechange of restriction parameter are each simulated for both small and large eccentricity ratio cases, also for both the shallow-recessed and the deep-recessed bearings in various land-width ratios.

Simulation results will reveal that the significant influence of design parameters. The appropriate selections of restriction parameterand land-width ratio for the design of a hybrid bearing, superior load capacity and stability threshold can be obtainable. Thesimulated results have been concluded for the design of hybrid bearing and its restrictors. 2003 Elsevier Ltd. All rights reserved.

Keywords: Hybrid bearing; Rigid rotor; Orifice compensation; Restriction parameter; Eccentricity; Stability

1. Introduction

The hybrid bearings are commonly used in precisionmachinery applications due to the combination of themerits of both hydrostatic and hydrodynamic bearings.However, since the hybrid bearings are characterized bytheir versatile geometric configurations and diversifiedfluid-feed flow control devices, the restriction parameterstogether with the optimization of values of bearing para-meters become more important in the dynamic character-istics of a rotor-bearing system.

Many researchers have extensively investigated vari-ous types of hybrid bearings and their efforts have beenfocused on the determinations of both the static anddynamic characteristics. Rowe et al.[1] first presentedplain hybrid bearings whose design relies mainly on thehydrodynamic effects besides the hydrostatic effects toachieve the necessary load support. Later, Rowe and

∗ Corresponding author. Tel.:+886-3-265-4315; fax:+886-3-265-4351

E-mail address: yk@cycu.edu.tw (Y. Kang).

0301-679X/$ - see front matter 2003 Elsevier Ltd. All rights reserved.doi:10.1016/S0301-679X(03)00139-7

Koshal [2] proposed an optimal strategy that is basedon the maximum hybrid load for minimum total powerdissipation. Both the theoretical and experimental inves-tigations of the optimal hybrid performances of the oil-lubrication plain hybrid bearings have been implementedby Koshal and Rowe[3,4]. They have concluded thatin optimal performances, the plain hybrid bearings aresuperior to recessed hydrostatic bearings at a low eccen-tricity ratio and low speed, while, at a high eccentricityand high speed, the plain hybrid bearings are comparableto axial groove hydrodynamic bearings with advantagesfor variable directions of loading. To predict the stabilityof plain hybrid bearings, Rowe and Chong[5] have com-pared two methods of computation of their dynamicforce coefficients.

In addition, Ghosh[6] presented a perturbation analy-sis and used the finite difference method to evaluate thedynamic coefficients of a hydrostatic bearing, as shaftdoes not rotate. Ghosh et al.[7] utilized the sameapproach for a capillary compensated hybrid bearing toconsider the effect of speed parameter on dynamic coef-ficients of hybrid bearings. Rowe[8] demonstrated thatthe properties of linearity in small displacement analysis

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Nomenclature

Ae effective recess areaa axial flow land widthBxx, Byy, Bxy, Byx direct and cross damping coefficients in x�y coordinateBij non-dimensional damping coefficients, Bij = Bijcw /PsLD, (i,j = x,y)c radial clearance of bearingCd orifice discharge coefficientD journal diameterdo diameter of the orifice restrictore eccentricityFx, Fy, Fx, Fy dimensional and non-dimensional fluid-film force components in the x and y

directions, (Fx,Fy) = (Fx,Fy) /PsLDh, h dimensional and non-dimensional film thickness, h = h /c = 1+ecosqhp recess depthhr, hr dimensional and non-dimensional film thickness at the center of rth recess, hr = hr /cKxx,KyyKxy,Kyx direct and cross stiffness coefficients in x�y coordinateKij non-dimensional stiffness coefficients, Kij = Kijc /PsLD, (i,j = x,y)L axial length of bearingM non-dimensional mass parameter, M = mcw2 /PsLDMc stability thresholdm rotor massOb,Oj bearing center and journal centerPr, Pr dimensional and non-dimensional recess pressure at the rth recess, Pr = Pr /Ps

Pr0 concentric recess pressurePs fluid supply pressureQr, Qr dimensional and non-dimensional flow rate at the rth recess, Qr = 12mQr /Psc3

Qr0, Qr0 dimensional and non-dimensional concentric flow rate at the rth recess, Qr0 = 12mQr0 /Psc3

R journal radiust timeV0, V0 dimension and dimensionless recess volume, V0 = Aehp, V0 = V0 /cAe

x, y, z Cartesian coordinatesx, y, z non-dimensional Cartesian coordinates, (x,y) = (x,y) /c, z = z /L /2

Greek symbols

b concentric pressure ratio, b = Pr0 /Ps

do orifice restriction parameter, do = 3√2pmCdd2o/√rPsc3

e eccentricity ratio, e = e /cg fluid compressibility parameter, g = V0�Ps

� speed parameter, � = 6mw /Ps(c /R)2

l whirl ratio, l = wp /wlc critical whirl ratio, lc = wc /wm absolute viscosity of lubricantq angular coordinate� inverse of fluid bulk modulusr density of lubricants squeeze number, s = 12mwp /Ps(c /R)2

t non-dimensional time, t = wptw journal spin speedwc critical whirl frequency of oil-filmwp whirl frequency of the journal center about the equilibrium pointy recess frequency parameter, y = sA /R2

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could be employed to obtain general equations forhydrostatic stiffness, hydrodynamic stiffness and sqeezedamping. Ghosh and Viswanath [9] have shown that therecess fluid volume compressibility drastically alters thedynamic characteristics of multi-recess hydrostatic bear-ings in the high frequency range. Ghosh et al. [10] andGuha et al. [11] presented theoretical analysis in additionto the influence of inertial effect to evaluate the dynamiccoefficients matrix for high frequency vibrations of adeep-recessed hybrid journal bearing with capillary com-pensation. Recently Chen et al. [12] have used the samemethod of Ghosh [10] to study the influences of capillaryrestriction parameters on the stability of a Jeffcott rotor-hybrid bearing system, however, the influence of journaleccentricity is significant yet has not been studied.

Various types of devices for flow control can beemployed in a hybrid bearing system. Both the capillaryand orifice restrictors are most common and have beenwidely adopted in many industrial applications due totheir advantages of large load capacity, high stiffnessand ease manufacturing. Raimondi and Boyd [13] havestudied the static characteristics of hydrostatic journalbearings by using both capillary and orifice with variousrestriction parameters. Their results indicated that themaximum load capacity could be obtained at a criticalvalue of restriction parameter for bearing with both typesrestrictors. They also indicated that orifice-compensatedbearing could provide more capacity than the capillary-compensated bearing. Cheng and Rowe [14] proposedthat the orifice restrictors are more compact than thecapillary restrictors and give fractionally greater stiff-ness, whilst stiffness corresponding to capillary is morelinear. In addition, if the stiffness or load variations areimportant for performance requirement, the design of therestrictors becomes critical and must be undertaken inparallel with the bearing configuration. However, rarestudies have not yet been reported in the literature toinvestigate the influence of orifice restriction parameterscoupled with parameters such as land-width ratios andeccentricity ratios on the stability of rotor-bearing sys-tems.

This study is to extensively evaluate the effects of ori-fice restriction parameters on the stability characteristicsof a rigid rotor with both the deep-recessed and shallow-recessed bearing for large and small eccentricity ratios.On the basis of the simulation drawn from this study,the numerical results will be useful for engineers in thedesign of hybrid bearing and its restrictors.

2. Governing equations of lubrication

An incompressible flow of viscous fluid situated ina thin film bearing is described by the following non-dimensional Reynolds equation as [10]

∂∂q�h3

∂P∂q� � �2R

L �2 ∂∂z�h3

∂P∂z� � ��∂h

∂q� 2l

∂h∂t� (1)

where non-dimensional parameters are defined by

� �6mw

Ps(c /R)2, h �hc, q �

XR

, z �z

L /2, P �

PPs

.

The generalized Reynolds equations governing thestatic and dynamic characteristics of thin film are achi-eved by using the perturbation method. This is solvedfor the determination of pressure distribution by usingthe finite difference method with successive over-relax-ation scheme to satisfy the boundary conditions [10] andthe flow continuity equation of orifice restrictors whichis expressed in a non-dimensional form as

do(1�Pr)0.5 � Qr � y∂∂t

(hr) � yg∂∂t

(Pr) (2)

where do = 3√2pmCdd2o�√rPsc3, being the restriction

parameter of orifice. For practical use with conditions ofzero shaft rotation speed and the design condition h =c, a relation between the orifice restriction parameter andconcentric pressure ratio can be obtained from Eq. (2) as

do(1�b)0.5 � Qr0 (3)

where b = Pr0 /Ps, being concentric pressure ratio.

3. The rigid rotor-hybrid bearing model

For a rigid rotor, supported horizontally on two ident-ical hybrid journal bearings, the perturbation equationsfor the journal motion are described as [12]:

Ml2d2xdt2

� �2Fx � �2(Kxxx � Kxyy � lBxxx (4)

� lBxyy )

Ml2d2ydt2

� �2Fy � �2(Kyxx � Kyyy � lByxx (5)

� lByyy )

Components of bearing force, Fx and Fy, along withthe stiffness and damping coefficients, are determined bythe integration of the fluid–film pressure distribution. Inthis study, the formulation and programming of the solv-ing process are carried out in accordance with Ghosh etal. [10].

4. Determination of stability threshold and whirlratio

The stability threshold and critical whirl ratio of arigid rotor-hybrid bearing system described by Eqs. (4)

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and (5) can be obtained by using the Routh–Hurwitzmethod and expressed by:

Mc �2A1A3A5

A21 � A2A2

5�A1A4A5(6)

lc � �(Keq�Kxx)(Keq�Kyy)�KxyKyx

BxxByy�BxyByx�1

2(7)

where A1 = KxxByy + KyyBxx�KxyByx�KyxBxy, A2 =KxxKyy�KxyKyx, A3 = BxxByy�BxyByx, A4 = Kxx + Kyy, A5

= Bxx + Byy.The detailed derivation of the above Eqs. (6) and (7)

are described in Chen et al. [12]. The self-excitedvibration occurs when value M of the rigid rotor hybridbearing system is larger than Mc. Conversely, for M �Mc the vibration induced by initial perturbations dies outexponentially with time. However, M = Mc relates to thecritical condition at which the rotor whirls to a limitcycle through transient vibration.

5. Results and discussion

According to the procedures and programs outlined inGhosh et al. [10], this study has evaluated the dynamiccharacteristics of a rigid rotor system which is supportedby single-row, six-recessed and orifice compensatedhybrid bearings as shown in Fig. 1.

For three different land-width ratio (a /L) cases thevariations of load capacity (W) with the eccentricity ratio(e) for three values of restriction parameter (do) areshown in Figs. 2 and 3 for both the shallow-recessedbearing (hp /c = 1.0) and the deep-recessed bearing(hp /c = 5.0). It is observed that for all cases the Wincrease with an increase of e but decrease when do

increases. For small land-width cases (a /L = 0.1) no sig-nificant difference of W are presented in do = 2 and10 when e � 0.5.

The same results for various values of eccentricityratio (e) and land-width ratio (a /L) cases are rearranged

Fig. 1. Rigid rotor-bearing model.

Fig. 2. The load capacity versus eccentricity for hp /c = 1.0: (a) a/L= 0.1; (b) a/L=0.25; (c) a/L=0.4.

to illustrate the variations of load capacity (W) versusthe restriction parameter (do) which are shown in Fig.4(a) for the shallow-recessed bearing (hp /c = 1.0), andin Fig. 4(b) for the deep-recessed bearing (hp /c = 5.0).Results obtained indicate that the W decreases when do

increases, since the hydrostatic load which is induced bythe restriction resistance will be reduced with an increaseof do. For do less than 10, W is decreasing violently sincethe hydrostatic effect decreases rapidly due to do chang-ing. While in the region of 10 � do � 40, the changeof W is alleviated with the weakening of hydrostatic

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Fig. 3. The load capacity versus eccentricity for hp /c = 5.0: (a) a/L= 0.1; (b) a/L=0.25; (c) a/L=0.4.

effects. When do � 40, the W appears to be unchange-able since the hydrostatic effects will not help matters.

When do � 10, for both small land-width ratio (a /L= 0.1) and moderate land-width ratio (a /L = 0.25) cases,both deep-recessed bearings with large eccentricity ratioand shallow-recessed bearings with small eccentricityratio maximum load capacity occur at a critical do andthat the latter has smaller value of critical do. Whilst, forlarge land-width ratio (a /L = 0.4) cases, no maximumW corresponding to a critical do exist in the region of

Fig. 4. The load capacity (W) versus restriction parameter (do) for (a)shallow recess (b) deep recess, both with � = 6.0.

do � 10 being due to the predominance of hydrodynamiceffect over hydrostatic effect.

In order to investigate the influences of recess depthratio on load capacity Fig. 2 is redrawn into Fig. 5(a)for a large eccentricity ratio (e = 0.5), and into Fig. 5(b)for a small eccentricity ratio (e = 0.1). For both casesof large and small eccentricity ratio , the load capacityof a shallow-recessed bearing is higher than that of adeep-recessed bearing for the same land-width ratiosince the shallow recesses can provide more hydrodyn-amic pressure than deep recesses. With the exception ofsmall eccentricity ratio case, when restriction parameteris less than 2 the W of a deep-recessed bearing is higherthan that of a shallow-recessed bearing.

For hybrid bearings the load capacity is dependent onnot only eccentricity but also restriction parameter. Thus,for a case of known load capacity corresponding to afixed eccentricity, to choose an optimal restriction para-meter for maximum stability is worthy to do.

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Fig. 5. The load capacity (W) versus restriction parameter (do) for (a)large eccentricity (b) small eccentricity, both with � = 6.0.

All the simulated results are presented in terms ofstability threshold (Mc) and critical whirl ratio (lc) as afunction of restriction parameter (do) with various non-dimensional parameters including land-width ratio(a /L), eccentricity ratio (e) and recess depth ratio(hp /c). In these cases, the journal speed parameter is keptconstant, i.e. � = 6.0.

The simulation results of stability threshold (Mc) areshown in Fig. 6(a) for the shallow-recessed bearings, andin Fig. 6(b) for the deep-recessed bearings. For bothcases of the shallow-recessed and deep-recessed bear-ings the stability threshold decreases as restriction para-meter increases, since the hydrodynamic effect becomesdominate, while the hydrostatic effect is saturated andgets losing its superiority on stability when δo is increas-ing. The influence of land-width ratio on stability is con-trary to load capacity; the stability threshold gets largeras land-width ratio decreasing.

Fig. 6. The stability threshold (Mc) versus restriction parameter (do)for (a) shallow recess (b) deep recess, both with � = 6.0.

The same results of Fig. 6 are redrawn into Fig. 7(a)for small eccentricity ratio, and into Fig. 7(b) for largeeccentricity ratio in order to illustrate the influence ofrecess depth on stability threshold (Mc). Hydrodynamiceffects are generated not only in lands but also in shal-low recesses, therefore, for large eccentricity ratio theMc of shallow-recessed bearings is superior to that ofdeep-recessed bearings. Contrarily, for small eccentricitythe stability threshold of deep-recessed bearings issuperior to that of shallow-recessed bearings. Since thecoupled hydrostatic and hydrodynamic effects are pro-vided by a shallow-recessed bearing in case of smalleccentricity, which may induce in a smaller Mc than thatof a deep-recessed bearing. Whilst, for both small andmoderate land-width ratio the stability threshold of adeep-recessed bearing is very close to that of a shallow-

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Fig. 7. The stability threshold (Mc) versus restriction parameter (do)for (a) small eccentricity (b) large eccentricity, both with � = 6.0.

recessed bearing with small land-width ratio, providesthe maximum stability threshold.

The variation of the critical whirl ratio (lc) withrestriction parameter (δo) are provided in Fig. 8(a) forsmall eccentricity ratio, and in Fig. 8(b) for large eccen-tricity ratio. In all cases, lc increases along with anincrease in δo or in a /L, since it is inevitable that thelower value of critical whirl ratio is induced by thehigher instability threshold.

6. Conclusions

For hybrid bearings with orifice compensation, thesimulated results indicate that both restriction parameterand eccentricity are the most essential factors of theinfluence on the load capacity and stability of rotor-

Fig. 8. The critical whirl ratio (lc) versus restriction parameter (do)for (a) small eccentricity (b) large eccentricity, both with � = 6.0.

hybrid bearing systems. The reasons are listed in the fol-lowing.

1. The recess depth do little influence on the loadcapacity, in most cases, the load capacity of shallow-recessed bearings is only slightly greater than that ofdeep-recessed bearing.

2. The influences of land-width ratio on the loadcapacity are affected by the value of restriction para-meter. For small restriction parameters the larger sillarea the larger load capacity, contrarily, for largerestriction parameters the smaller sill area the largerload capacity.

3. For stability of large eccentricity cases, the simulatedresults in most cases indicate that the shallow-recessed bearings are superior to the deep-recessed

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bearings. Contrarily, for stability of small eccentricitycases the deep-recessed bearings are better than theshallow-recessed bearings.

4. From the view for load capacity in bearing design,optimal design parameters which including restrictionparameter, land-width ratio and recess depth for theminimization of eccentricity ratio are searched in do

� 10.5. When stability threshold is concerned in bearing

design, simulation results indicate that the smaller therestriction parameter and the smaller the land-widthratio, the larger the stability threshold, and smalleccentricity deep-recessed cases are superior to shal-low-recessed cases, while, contrarily, large eccen-tricity shallow-recessed cases are superior to deep-recessed cases.

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