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1Fundamentals ofHydrodynamic Bearings

Hydrodynamic (fluid film) bearings are used extensively in different kinds ofrotating machinery in the industry. Their performance is of utmost importancein chemical, petrochemical, automotive, power generation, oil and gas, aerospaceturbo-machinery, and many other process industries around the globe.Hydrodynamic bearings are generally classified into two broad categories: jour-

nal bearings (also called sleeve bearings) and thrust bearings (also called sliderbearings). In this book, we exclusively focus our attention on journal bearings.Figure 1.1a shows a schematic illustration of a rotor bearing system, which con-

sists of a shaft with a central disk symmetrically supported by two identical journalbearings at both ends. Figure 1.1b shows the geometry and system coordinates ofthe journal rotating in one of the two identical journal bearings. To easily identifythe bearing’s physical wedge effect and annotate the multiple parameters of a rotorbearing system, the clearance between the journal and the bearing bushing is exag-gerated. θ is the circumferential coordinate starting from the line going through thecenters of the bearing bushing and the rotor journal. ϕ is defined as the system atti-tude angle. e is the rotor journal center eccentricity from the center of the bearingbushing.W represents the vertical load imposed on the shaft and supported by thebearing. p is the hydrodynamic pressure applied by the thin fluid film onto the jour-nal surface. f is the hydrodynamic force obtained by integrating the hydrodynamicpressure p generated around the journal circumference.

Thermohydrodynamic Instability in Fluid-Film Bearings, First Edition.J. K. Wang and M. M. Khonsari.© 2016 John Wiley & Sons, Ltd. Published 2016 by John Wiley & Sons, Ltd.

COPYRIG

HTED M

ATERIAL

In most cases, except in a floating ring configuration, the bearing bushing isfixed and the rotor rotates at the speed of ω inside the bearing bushing. InFigure 1.1, the journal center positionOj is described as (e, ϕ) relative to the centerOb of the fixed journal bearing bushing.Radial clearance C is defined as the clearance between the bearing and the rotor

journal (i.e.,C =Rb −Rj, where Rb is the inside radius of the bearing bushing and Rj

z

Ob

Journal bearing

(a)

(b)

Fluidinlet

pi

y

Journal bearing

Bearing

Journal

W

2Wx

y

θ

Θi

Θ

x270°

0°

90°Ob

p

ω

e

fOj

fε

eε

eϕ

fϕ

ϕ

ϕ

p.cosθ

p.sinθ

180°

Central disk

Figure 1.1 (a) Model of a rotor supported by two identical journal bearings; (b) geometryand system coordinates of a journal rotating in a fluid film journal bearing

2 Thermohydrodynamic Instability in Fluid-Film Bearings

is the radius of the rotor journal). In terms of this radial clearance, the journal centereccentricity from the bearing center can be normalized as ε = e C. The dimension-less parameter ε is called eccentricity ratio. Due to the physical constraint of thebearing bushing, the rotor journal must be designed to operate inside of the bearingbushing, that is, 0 ≤ ε ≤ 1. Therefore, the journal center position Oj within the fluidfilm journal bearing can be redefined as (Cε, ϕ). When ε= 0, the center of the shaftOj coincides with the center of the bearing bushing Ob and the fluid film bearing istheoretically incapable of generating hydrodynamic pressure by wedge effect andits corresponding load-carrying capacity is nil. When ε= 1, the shaft comes intointimate contact with the inner surface of the bushing, and depending on the oper-ating speed, bearing failure becomes imminent due to the physical rubbingbetween the shaft and the bushing.Based on the above physics, the important concept of rotor bearing clearance

circle is introduced to easily describe the rotor journal position within any hydro-dynamic journal bearing. Figure 1.2a shows the rotor bearing clearance circle inboth polar and Cartesian coordinate systems. The radius of the clearance circleis equal to the radial clearance C defined earlier and the center of the clearancecircle is the bearing center Ob. The journal center Oj is always either within oron the clearance circle. In other words, it will never go beyond the clearance circledue to the physical constraint of bearing bushing. Figure 1.2b shows the dimen-sionless rotor bearing clearance circle in both polar and Cartesian coordinatesystems.The fundamental equation that governs the pressure distribution in a hydrody-

namic bearing was first introduced by Osborne Reynolds in 1886. In this chapter,we begin by describing the Reynolds equation and provide closed-form analyticalsolutions for two simplified extreme cases commonly known as the short andlong bearing solutions. At the end, a brief discussion will be provided to addressthe numerical methods to solve the Reynolds equation for finite-length journalbearings.

1.1 Reynolds Equation

The Reynolds equation assuming that thin-film lubrication theory holds for a per-fectly aligned journal bearing system lubricated with an incompressible Newtonianfluid is given by Equation 1.1.

∂

R2∂θGθ

h3

μ

∂p

∂θ+

∂

∂zGz

h3

μ

∂p

∂z=ω

2∂h

∂θ+∂h

∂t1 1

where z is the axial coordinate with the origin at the mid-width of the journalbearing.

3Fundamentals of Hydrodynamic Bearings

Ob (0,0) Cε

y

y

180° Dimensional clearance circle

Dimensionless clearance circle

(0, C)

270°(–C, 0)

(0, –C)

ϕ

ϕ

(C, ϕ)

0°

180°

(a)

(b)

(0, 1.0)

(1.0, 0)

(0, 0.75)

(0.75, 0)

(0, 0.5)

(0.5, 0)

(0, 0.25)

(0.25, 0)

(1.0, ϕ)

(0, –1.0)0°

(C, 0) x

x

90°

90°270°(–1.0, 0)

ε

(0.25, ϕ)

Oj (Cε, ϕ)

Oj (ε, ϕ)

(0.75, ϕ)

Ob (0, 0)

Figure 1.2 (a) Dimensional and (b) dimensionless rotor bearing clearance circles


Detailed derivation of the Reynolds equation is available in tribology textbooks(see for example, Khonsari and Booser, 2008). In Equation 1.1, θ is the circumfer-ential coordinate and z is the axial coordinate perpendicular to the paper inFigure 1.1, R is the journal radius, μ is the fluid viscosity, and the fluid film thick-ness h is given by Equation 1.2. The parameters Gθ andGz are the turbulent coeffi-cients given by Equations 1.3 and 1.4 (See Hashimoto and Wada (1982) andHashimoto et al. (1987)).

h =C 1 + εcosθ 1 2

Gθ =1

12 aθ + bθεcosθ1 3

Gz =1

12 az + bzεcosθ1 4

where aθ = 1 + 0 00069Re0 95, az = 1+ 0 00069Re0 88, bθ = 0 00066Re0 95, bz =0 00061Re0 88, and Re = ρRωC μ is the Reynolds number. The turbulent coeffi-cients Gθ and Gz given by Equations 1.3 and 1.4 agree well with those given byNg and Pan (1965).

On the left-hand side of Reynolds Equation 1.1, the first term ∂R2∂θ Gθ

h3

μ∂p∂θ is

the pressure-induced flow in the circumferential direction while the second term∂∂z Gz

h3

μ∂p∂z is the pressure-induced flow in the axial direction. On the right-hand

side, the first term ω2∂h∂θ is the physical wedge effect in the circumferential direc-

tion between the bearing bushing and the rotor journal, and the second term∂h ∂t is the normal squeeze action of the fluid film in the radial direction.Under the simplified isothermal assumption and neglecting the pressure influ-

ence on the fluid viscosity (i.e., constant fluid viscosity throughout the fluid film),the Reynolds Equation 1.1 can be simplified to

∂

R2∂θGθh

3 ∂p

∂θ+

∂

∂zGzh

3 ∂p

∂z=ωμ

2∂h

∂θ+ μ

∂h

∂t1 5

For a steady-state fluid film, the fluid film thickness h is not a function of time,that is, ∂h ∂t = 0. Then, the Reynolds equation can be further reduced to

∂

R2∂θGθh

3 ∂p

∂θ+

∂

∂zGzh

3 ∂p

∂z=ωμ

2∂h

∂θ1 6


while for laminar flow (aθ = 1, az = 1, bθ = 0, bz = 0), the simplified Reynolds equa-tion (Eq. 1.1) can be rewritten as

∂

R2∂θ

h3

μ

∂p

∂θ+

∂

∂z

h3

μ

∂p

∂z= 6ω

∂h

∂θ+ 12

∂h

∂t1 7

Therefore, for a rotor bearing system with steady-state and laminar fluid film,Equation 1.8 presents the further reduced but commonly used Reynolds equation.

∂

R2∂θ

h3

μ

∂p

∂θ+

∂

∂z

h3

μ

∂p

∂z= 6ω

∂h

∂θ1 8

Reynolds Equation 1.1 is a time-dependent second-order partial differentialequation. To predict the pressure distribution through solving the Reynolds equa-tion, in addition to the initial condition, four boundary conditions are needed interms of the geometrical parameters θ and z. For steady-state Reynolds equationssuch as Equations 1.6 and 1.8, only the four boundary conditions are needed todefine the pressure distribution.

1.1.1 Boundary Conditions for Reynolds Equation

In most hydrodynamic bearing applications, the fluid lubricant flows out of thebearing at ambient pressure. In other words, the gauge pressure at the geometricalboundary is equal to 0. Inside the bearings, since a conventional fluid lubricantcannot withstand negative pressure, it cavitates if the liquid pressure falls belowthe atmospheric pressure.Depending on how to define and handle the cavitation region, there are three

classical types of boundary conditions: full-Sommerfeld boundary conditions (cav-itation is fully neglected and p= 0 when θ = 2π), half-Sommerfeld boundary con-ditions (also called Gümbel boundary conditions, i.e., p = 0 when 180 ≤ θ ≤ 360 ),and Reynolds boundary conditions (also called Swift–Stieber boundary condi-tions, i.e., both pressure and pressure gradient approach 0 where cavitation begins).All three classical types of boundary conditions assume that the fluid film starts atθ = 0. The detailed definitions of these boundary conditions will be introduced inthe related chapters that follow. For further reference, Khonsari and Booser (2008)have given a complete summary of these boundary conditions on both their impli-cations and limitations. In recent years, by combining the Reynolds boundary con-dition with some new experimental findings on when and how the fluid film starts,a more complete type of boundary conditions (Reynolds–Floberg–Jakobsson orRFJ boundary conditions) has been derived and applied successfully into differentapplications (Wang and Khonsari, 2008). The RFJ boundary conditions will be dis-cussed in Section 1.3.


As Equations 1.1, 1.5–1.8 read, even for the most simplified two-dimensionalReynolds equation—which is still a nonlinear partial differential equation—aclosed-form analytical solution is practically impossible.On the other hand, based on the physical implications of different applications,

two kinds of extreme-condition approximation of the Reynolds equation have beenwell developed and applied widely to predict the bearing performance analytically.One of them is called the infinitely short bearing theory (often called the short bear-ing theory) for the application of bearing length-over-diameter ratio far less than 1;the other approximation is the infinitely long bearing theory (often called the longbearing theory) for the application of bearing length-over-diameter ratio far morethan 1.Generally speaking, to obtain sufficiently accurate results, the short bearing the-

ory is often applied to bearings with length-over-diameter ratio up to 0.5 and theinfinitely long bearing theory is recommended for bearings with length-over-diameter ratio of 2.0 or greater. The bearing having length-over-diameter ratiomore than 0.5 while not exceeding 2.0 is called finite bearing. For finite bearingswith length-over-diameter ratio more than 0.5 while not exceeding 1.0, the shortbearing theory still could render a reasonable approximation of the bearing perfor-mance. If the finite bearing length-over-diameter ratio is more than 1.0, the longbearing theory might be a reasonable approximation, particularly if one is inter-ested in trends. However, if an accurate prediction of the bearing performanceis desired, then the full Reynolds equation should be treated with an appropriatenumerical solution.

1.1.2 Short Bearing Approximation

For short journal bearings, the second term (side leakage in the axial direction)in the Reynolds Equations 1.1, 1.5–1.8 is so dominant that the first term (the pres-sure-induced flow in the circumferential direction) can be neglected to obtain ananalytical solution to the Reynolds equations. The side leakage controls the fluidfilm pressure distribution and then the bearing loading capacity. Section 1.2 willshow the simplified Reynolds equation for short bearings under certain commonlyused boundary conditions.

1.1.3 Long Bearing Approximation

For long journal bearings, side leakage and fluid pressure gradient in the axialdirection are negligible (i.e., ∂p ∂z≈0). This implies that the second term (sideleakage in the axial direction) in the Reynolds Equations 1.1, 1.5–1.8 can bedropped and it becomes possible to obtain an analytical solution to the Reynoldsequations. Section 1.3 will show the detailed derivation of an analytical solution forlong bearings under certain boundary conditions.


1.2 Short Bearing Theory

1.2.1 Analytical Pressure Distribution

In infinitely short journal bearings, side leakage controls the fluid film pressuredistribution and the bearing load-carrying capacity. Because of the dominantpressure-induced side leakage in the axial direction, the pressure-induced flowin the circumferential direction (i.e., the partial differentials of the pressure p interms of θ on the left-hand side of Reynolds Equation 1.1) is neglected to obtainan analytical solution to the Reynolds equations. Assuming constant fluid viscositythroughout the fluid film, the Reynolds equation for infinitely short bearingsincluding the turbulent effects can be further reduced from Equations 1.5 to 1.9.

Gz∂2p

∂z2=μω

2h3∂h

∂θ+

μ

h3∂h

∂t1 9

The boundary conditions of the fluid film in the axial directions are given byEquations 1.10 and 1.11.

∂p

∂z z= 0

= 0 1 10

p z =± L 2 = 0 1 11

where L is the bearing length.Integrating the Reynolds Equation 1.9 twice and substituting the above bound-

ary conditions yields the following expression for the pressure distribution of thefluid film formed around the journal surface:

p =3μ az + bzε cos θ

h3ω∂h

∂θ+ 2

∂h

∂tz2−

L2

41 12

where az and bz are defined in Section 1.1.Substituting Equation 1.2 into Equation 1.12, the expression for hydrodynamic

pressure distribution becomes (Wang and Khonsari, 2006)

p =3μC2

z2−L2

4

2azεcosθ−az ω+ 2θ εsinθ + 2bzεεcos2θ−bz ω+ 2θ ε2 sinθcosθ

1 + εcosθ 3

1 13

where “.” represents d/dt.


Referring to Figure 1.1, the position of the journal center can be representedas Oj (Cε, ϕ), where ϕ denotes the attitude angle. Note that as ϕ changes, sodoes the reference line for θ. Increasing ϕ will result in decreasing θ (Lund,1966); so,

dϕ

dt= −

dθ

dti e θ = −ϕ 1 14

With the aid of Equation 1.14, the expression for the pressure distribution,Equation 1.13, can be rewritten as (Wang and Khonsari, 2006)

p =3μC2

z2−L2

4

2azεcosθ−az ω−2ϕ εsinθ + 2bzεεcos2θ−bz ω−2ϕ ε2 sinθcosθ

1 + εcosθ 3

1 15

If the fluid film has a laminar flow (az = 1, bz = 0), the pressure distribution fur-ther simplifies to the following expression (Wang and Khonsari, 2006):

p=3μC2

z2−L2

4

2εcosθ− ω−2ϕ εsinθ

1 + εcosθ 3 1 16

1.2.2 Hydrodynamic Fluid Force

Figure 1.3 illustrates the radial and tangential components of the fluid force exertedon the rotor journal surface. By definition, the fluid force components in the radial(subscript ε) and tangential (subscript ϕ) directions can be obtained by integratingthe pressure distribution around the rotor journal surface:

fε =RL 2

−L 2

2π

0pcosθdθdz 1 17

fϕ =RL 2

−L 2

2π

0psinθdθdz 1 18

Substituting Equation 1.15 into Equations 1.17 and 1.18, applying the halfSommerfeld boundary condition where the negative hydrodynamic fluid pressureat the geometrically diverging region is assumed to be negligible (i.e., p = 0 when


180 ≤ θ ≤ 360 ), and utilizing the integrals given in the appendix A by Wang andKhonsari (2006), the following expressions for the hydrodynamic fluid forcecomponents yield (Wang and Khonsari, 2006)

fε = −μRL3

2C2ω−2ϕ

2 az−2bz ε2 + 2bz1−ε2 2 −

bzεln

1 + ε1−ε

+ πεaz + 5bz + 2 az−3bz ε2− 2bz ε2

1−ε2 5 2+2bzε2

1 19

fϕ =μRL3

2C2επ ω−2ϕ

2bz + az−3bz ε2

2 1−ε2 3 2−bz

+ 4εbz + az−2bz ε2

1−ε2 2 −bz2ε

ln1 + ε1−ε

1 20

where “.” represents d/dt.Normalizing the time t parameter by substituting t =ωt, the dimensionless radial

and tangential components of the hydrodynamic fluid force in the journal bearingbecome

Ob

Oj

y

W

p

x

fε

eε

f

eϕ

fϕ

ϕp ·cosθ

p · sinθ

θ

Cε

Figure 1.3 Radial and tangential components of the fluid force exerted on the journal(Wang and Khonsari, 2006)


f ε =fε

mCω2= −

μRL3

2mC3ω1−2ϕ

2 az−2bz ε2 + 2bz1−ε2 2 −

bzεln

1 + ε1−ε

+ πεaz + 5bz + 2 az−3bz ε2− 2bz ε2

1−ε2 5 2+2bzε2

1 21

f ϕ =fϕ

mCω2=

μRL3

2mC3ωεπ 1−2ϕ

2bz + az−3bz ε2

2 1−ε2 3 2−bz

+ 4εbz + az−2bz ε2

1−ε2 2 −bz2ε

ln1 + ε1−ε

1 22

where “.” represents d/ωdt.Under laminar flow condition (az = 1 and bz = 0), the hydrodynamic fluid force

components (Eqs. 1.19 and 1.20) are simplified as

fε = −μRL3

2C2

2 ω−2ϕ ε2

1−ε2 2 +πε 1 + 2ε2

1−ε2 5 21 23

fϕ =μRL3

2C2

π ω−2ϕ ε

2 1−ε2 3 2+

4εε

1−ε2 2 1 24

where “.” represents d/dt.Equations 1.25 and 1.26 show the dimensionless hydrodynamic fluid force com-

ponents for laminar flow applications.

f ε =fε

mCω2= −

μRL3

2mC3ω

2 1−2ϕ ε2

1−ε2 2 +πε 1 + 2ε2

1−ε2 5 21 25

f ϕ =fϕ

mCω2=

μRL3

2mC3ω

π 1−2ϕ ε

2 1−ε2 3 2+

4εε

1−ε2 2 1 26

where “.” represents d/ωdt.

1.2.3 Static Performance of Short Journal Bearings

Assuming that the rate of speed change is zero (ε =ϕ= 0), the fluid pressure, inte-grated fluid force components, attitude angle, and the relation between eccentricity


ratio ε and Sommerfeld number S (S = μωLR3 πWC2 ) can be derived usingEquations 1.16 and 1.23–1.24 for laminar flow applications and the force equilib-rium between the hydrodynamic fluid force f and the externally applied loadW on

the journal (W = f 2ε + f 2ϕ and tanϕ = − fϕ fε). Table 1.1 shows the comparison

between the results derived by applying half-Sommerfeld boundary conditionand those based on full-Sommerfeld boundary condition.The difference between full-Sommerfeld and half-Sommerfeld boundary condi-

tions is in the treatment of the hydrodynamic pressure in the geometrically diver-gent region of flow. With the half-Sommerfeld boundary condition, the negativefluid pressure p is set to p= 0 when 180 < θ < 360 while with Full-Sommerfeldboundary condition the negative fluid pressure is assumed to remain intact. Underthe full-Sommerfeld boundary condition, the skew-symmetric hydrodynamic

Table 1.1 Comparison of static performance based on different types of boundarycondition

Half-Sommerfeld (Gümbel) boundarycondition Full-Sommerfeld boundary condition

Pressure: Pressure:

p=3μωεsinθ

C2 1 + εcosθ 3

L2

4−z2

when 0 ≤ θ ≤ 180 ;

p=3μωεsinθ

C2 1 + εcosθ 3

L2

4−z2 when

0 ≤ θ < 360 and −L 2 ≤ z ≤ L 2.p= 0 when 180 < θ < 360 .

For both cases, −L 2 ≤ z ≤ L 2

Fluid force components: Fluid force components:

fε = −μRL3ωε2

C2 1−ε2 2

fε = 0

fϕ =πμRL3ωε

4C2 1−ε2 3 2fϕ =

πμRL3ωε

2C2 1−ε2 3 2

Relation between Sommerfeld numberand eccentricity ratio:

Relation between Sommerfeld number andeccentricity ratio:

S=4R2 1−ε2

2

πL2ε 16ε2 + π2 1−ε2S=

2R2 1−ε23 2

π2L2ε

Attitude angle: Constant attitude angle:

ϕ = tan−1 π 1−ε2

4εϕ=

π

2


pressure distribution in the whole circumferential coordinate (0 ≤ θ < 360 ) yieldsa constant attitude angle of ϕ = π 2. Numerous studies have proven that cavitationoften exists at the geometrically diverging region and the half-Sommerfeld bound-ary condition setting the negative pressure to zero actually renders a more accurateanalytical solution to Reynolds equation, especially for short journal bearings.

1.3 Long Bearing Theory

With identical bearing material of the same PV limit (contact pressure × surfacevelocity), long jounral bearings normally have a higher load-bearing capacity thanshort journal bearings. Proper distribution of lubricant requires special considera-tion, particularly for long journal bearings to function. Often an axial groove isdesigned to distribute the lubricant over the entire length of the journal to improvethe lubrication condition and control the temperature field. The location of the fluidinlet and the associated fluid supply pressure can have a pronounced influence onthe bearing performance. For journal bearings with length-over-diameter greaterthan 1, the infinitely long bearing theory should be used to obtain an analyticalsolution to the Reynolds Equations 1.1, 1.5–1.8. In addtion, due to its physicalimplications and the oversimplication in the circumferential direction of the fullReynolds equation, the short bearing theory described in Section 1.2 cannot beused to evaluate the axial groove effects. To obtain an analytical solution of thegoverning Reynolds equation including the axial groove effects, the long bearingtheory is the only option.

1.3.1 Analytical Pressure Distribution of Long Journal Bearings

Figure 1.4 shows the geometry and system coordinates used in long journal bear-ings. An absolute circumferential coordinateΘ is introduced to facilitate a compar-ison of the bearing performance under different fluid inlet positions and inletpressures. It is measured from the upper load line, that is, Θ = θ +ϕ, where θ isthe circumferential coordinate starting from the line of the centers of bearing bush-ing and rotor journal and ϕ is the attitude angle.Theoretically in infinitely long journal bearings, there is no side leakage and the

fluid pressure gradient in the axial direction is zero (i.e. ∂p ∂z = 0). Assuming con-stant fluid viscosity throughout the fluid film, the Reynolds Equation 1.5 for aninfinitely long journal bearing is simplified to

∂

∂θGθh

3 ∂p

∂θ=μωR2

2∂h

∂θ+ μR2 ∂h

∂t1 27

where h =C 1 + ε cos θ , μ is constant throughout the fluid film, and the turbulentcoefficient Gθ is given by Equation 1.3.


If the flow is laminar (az = 1, bz = 0), the Reynolds equation for a long bearingcan be further simplified to (Wang and Khonsari, 2008)

∂

∂θ

h3∂p

∂θ= 6μωR2 ∂h

∂θ+ 12μR2 ∂h

∂t1 28

The boundary conditions are

p = 0 at θ = θs 1 29

p = 0 at θ = θc 1 30

∂p

∂θ= 0 at θ = θc 1 31

where θs is the fluid pressure starting position and θc is the circumferential locationwhere the fluid film ruptures; that is, cavitation begins. According to the Reynoldsboundary condition, both the pressure and its gradient are equal to zero at θc.

Fluid inletgroove

pi

y

Bushing

Journal

Wθ

Θi

Θ

x

0°

90°

45°

Ob

pf

fε

eε

eϕ

fϕ

ϕ

p.cosθ

p.sinθ

180°

CεOj

Figure 1.4 Geometry and system coordinates used in long journal bearing. From Wangand Khonsari (2008) © Elsevier Limited.


Equations 1.30 and 1.31 reflect the Reynolds boundary conditions. According tothe Floberg–Jakobsson boundary condition, the pressure at the fluid pressure start-ing position θs is zero while, to satisfy the fluid flow continuity in journal bearing,the pressure gradient ∂p ∂θ at the fluid pressure starting position θs must be non-negative (Jakobsson and Floberg, 1957), that is,

∂p

∂θ≥ 0 at θ = θs 1 32

Since the fluid pressure at the fluid inlet position is always maintained at a spe-cified supply pressure pi (Zhang, 1989), the following equation introduces an extraboundary condition at the fluid inlet position θi.

p = pi at θ = θi 1 33

where θi =Θi−ϕ.Since h=C 1 + εcosθ as given in Equation 1.2, the Reynolds Equation 1.28 for

long journal bearing can be rewritten as

∂

∂θ

C3 1 + ε cos θ 3∂p

∂θ= −6μωCR2εsinθ + 12μCR2 εcosθ−θεsinθ 1 34

where “.” represents d/dt.Further, substituting Equation 1.14 into Equation 1.34 and rearranging the equa-

tion yields

d

dθ

C3 1 + εcosθ 3dp

dθ= 6μCR2ε 2ϕ−ω sinθ + 12μCR2εcosθ 1 35

Integrating Reynolds Equation 1.35 twice with respect to θ and applying theboundary conditions given by Equations 1.31 and 1.33 (see Appendix A fordetailed derivation) yields the following solution for the hydrodynamic pressuredistribution around the journal circumference (Wang and Khonsari, 2008):

p = pi +3μR2

C2 1−ε2 2

ω−2ϕ ε 1−ε20 5

1−εcosαc2 sinα− sinαi 1 + εcosαc

+ 2cosαc + ε αi−α −ε sinαcosα− sinαi cosαi

+ ε2sinαc

1−εcosαc

2−εcosα−εcosαi cosαi− cosα 1−εcosαcsinαc

− 2 + ε2 α−αi + 4ε sinα− sinαi −ε2 sinαcosα− sinαi cosαi

1 36


The dynamic pressure given by Equation 1.36 is valid for αs ≤ α ≤ αc since thefluid film pressure only exists in the range of θs ≤ θ ≤ θc per definition. Beyond thisrange, the fluid film pressure is zero. The relations between αs and θs, αc and θc, andαi and θi are defined by Equations A.5–A.7, respectively.By applying the boundary conditions described by Equations 1.29 and 1.30

to determine the parameters in Equation 1.36, the fluid pressure starting position θsand the starting position of cavitation θc are given by Equations 1.37 and 1.38(Wang and Khonsari, 2008).

ω−2ϕ ε 1−ε20 5

2 + εcosαc sinαc− sinαi −εsinαi cosαc− cosαi

+ 2cosαc + ε αi−αc + 2ε 2cosαi−2cosαc + εsin2αi 1−εcosαc

−εsinαi sinαc 4−εcosαi − 2 + ε2 sinαc αc−αi + 3εsin2αc

+C2pi3μR2

1−ε221−εcosαc = 0

1 37

ω−2ϕ ε 1−ε20 5

2 sinαs− sinαi 1 + εcosαc + 2cosαc + ε αi−αs

−ε sinαs cosαs− sinαi cosαi + 2ε 4εsinαc sinαs− sinαi

+ 2−εcosαs−εcosαi cosαi− cosαs 1−εcosαc − 2 + ε2 sinαc αs−αi

−ε2 sinαc sinαs cosαs− sinαi cosαi +C2pi3μR2

1−ε221−εcosαc = 0

1 38

Equations 1.37 and 1.38 show that both αc and αs are a function of ε, ε, ϕ, and αi.The parameter αi is a function of ε and ϕ with a specified fluid inlet position Θi.Upon determining αc by solving Equation 1.37 while π < αc ≤ 2π + αi with αi givenby Equation A.7, αs can be predicted using Equation 1.38 while αc−2π ≤ αs ≤ αi.If eccentricity ε is small—that is, when the bearing is lightly loaded—or when

the supply pressure pi is high, it is possible that cavitation would cease to exist(Dowson et al., 1985). In this case, there exists one set of αc and αs such that αsequals αc−2π and Equation 1.31 holds. Since the boundary conditions describedby Equations 1.31 and 1.33 still hold, the expression for the hydrodynamic fluidpressure given by Equation 1.36 remains valid. However, Equations 1.37 and 1.38cannot be used any more to determine αc and αs since neither of the Equations1.29 and 1.30 holds. Under this situation, the cyclic boundary conditionp αc = p αc−2π applies. Equation 1.39 is derived by applying boundary condi-tion p αc = p αc−2π on Equation A.8 (Wang and Khonsari, 2008).


ω−2ϕ ε 1−ε20 5

2cosαc + ε + 2ε 2 + ε2 sinαc = 0 1 39

When cavitation does not exist, Equation 1.39 is used to determine αc whileπ < αc ≤ 2π + αi. After determining αc, αs = αc−2π .Thus, the hydrodynamic pressure profile in a long journal bearing can be pre-

dicted using Equation 1.36 with αc, which can be determined using eitherEquation 1.37 or Equation 1.39 depending on whether the cavitation exists.The hydrodynamic pressure given by Equation 1.36 can be normalized using

Equation 1.40.

p =C2

μωR2 p1 40

1.3.2 Hydrodynamic Fluid Force of Long Journal Bearings

The hydrodynamic fluid force components in the radial (subscript ε) and tangential(subscript ϕ) directions of long journal bearings are given by Equation 1.41.

fε =RLθc

θs

pcosθdθ

fϕ =RLθc

θs

psinθdθ

1 41

Integrating Equation 1.41 by parts and applying the boundary conditions (whencavitation exists, p θ = θc = 0 and p θ = θs = 0; when cavitation does not exist,p θ = θc = p θ = θs = θc−2π) yields

fε =RL p sin θ θcθs−

θc

θs

dp

dθsinθdθ = −RL

θc

θs

dp

dθsinθdθ

fϕ =RL −pcosθ θcθs+

θc

θs

dp

dθcosθdθ =RL

θc

θs

dp

dθcosθdθ

1 42

Substituting Equation A.1 with C1 given by Equation A.2 into Equation 1.42,solving the integrals (see Appendix B), and then substituting θc with αc usingEquation A.6, the hydrodynamic fluid force components in the radial and tangen-tial directions are given by (Wang and Khonsari, 2008).


fε = −3μLR3

C2

ω−2ϕ ε cosαc− cosαs2

1−ε2 1−εcosαc

+ 2εαc−αs + sinαs cosαs 1−εcosαc + εcos2αs−2cosαs + cosαc sinαc

1−ε2 1 5 1−εcosαc

1 43

fϕ =3μLR3

C2

ω−2ϕ ε

1−ε2 1 5 1−εcosαc1 + 2εcosαc αc−αs

−2 ε + cosαc sinαc− sinαs + sinαc cosαc− sinαs cosαs

+2ε

1−ε2 2 1−εcosαc2ε cosαc−ε − 2εcosαs + sin2αs 1−εcosαc

− sin2αc + 3εsinαc αc−αs + 2ε2 + 2−εcosαs sinαs sinαc

1 44

where “.” represents d/dt.In Equations 1.43 and 1.44, when cavitation exists, αc is determined by

Equation 1.37 and then αs is determined by Equation 1.38; when cavitation doesnot exist, αc is determined by Equation 1.39 and then αs = αc−2π .Normalizing the time t by substituting t =ωt, the dimensionless radial and tan-

gential components of the hydrodynamic fluid force in the journal bearing aredefined as follows:

f ε =fε

mCω2= −

3μLR3

mωC3

1−2ϕ ε cosαc− cosαs2


+ 2εαc−αs + sinαs cosαs 1−εcosαc + εcos2αs−2cosαs + cosαc sinαc


1 45

f ϕ =fϕ

mCω2=3μLR3

mωC3

1−2ϕ ε

1−ε2 3 2 1−εcosαc1 + 2εcosαc αc−αs

−2 ε + cosαc sinαc− sinαs + sinαc cosαc− sinαs cosαs

+2ε

1−ε2 2 1−εcosαc2ε cosαc−ε − 2εcosαs + sin

2αs 1−εcosαc

− sin2αc + 3εsinαc αc−αs + 2ε2 + 2−εcosαs sinαs sinαc

1 46

where “.” represents d/ωdt.


1.3.3 Static Performance of Long Journal Bearings

Similar to the short bearing analysis given in Section 1.2.3, assuming ε =ϕ= 0, thefluid pressure, fluid force components, attitude angle, and the relation betweeneccentricity ratio ε and Sommerfeld number S are derived from Equations1.36–1.39 and 1.43–1.44 and the force equilibrium between the fluid force f

and the externally applied load W on the journal (W = f 2ε + f 2ϕ and tanϕ =

− fϕ fε). Table 1.2 compares the results derived under RFJ boundary conditionwith those based on the Reynolds boundary condition. The Reynolds boundarycondition—sometimes referred to as the Swift-Stieber boundary condition—actually is one special case of RFJ boundary condition with some simplificationsas (αs = αi = 0 and pi = 0).Based on the analysis in Section 1.3.2 and Table 1.2, the fluid film configura-

tion, pressure distribution, and steady-state journal position in axially groovedjournal bearing with different dimensionless fluid inlet pressures (in the rangeof 0 ≤ pi ≤ 1) and different fluid inlet positions (in the range of 0 ≤Θi ≤ 90 ) are cal-culated and summarized below.

1.3.3.1 Comparison of Different Types of Boundary Conditions

Assuming an inlet condition of (Θi = 0, pi = 0), Figure 1.5 compares how the eccen-tricity ratio (ε) varies as a function of the Sommerfeld number (S) based on the RFJboundary condition and the half-Sommerfeld boundary condition. Figure 1.6shows how the attitude angle (ϕ) varies as a function of eccentricity ratio (ε) basedon different boundary conditions. Since the journal position can be fully defined as(ε, ϕ) in a polar coordinate, Figure 1.6 also compares the journal position as a func-tion of eccentricity ratio (ε) based on different boundary conditions.Figure 1.5 shows that the eccentricity ratio ε predicted based on the assumed

half-Sommerfeld boundary condition is always overestimated compared to thatpredicted based on the RFJ boundary condition. However, Figure 1.6 shows thatthe attitude angle (ϕ) predicted based on the half-Sommerfeld boundary conditionis underestimated when ε is less than about 0.85. These inaccuracies are caused byneglecting the cavitation developed in the geometrically diverging region under theassumption of half-Sommerfeld boundary condition. The detailed cavitation willbe shown together with the fluid-film configuration and pressure distribution pre-sented in the next subsection.

1.3.3.2 Influence of Fluid Inlet Pressure

Assuming the fluid inlet position held at Θi = 0 , the influence of different fluidinlet pressures (pi = 0, 0.5, or 1.0) on the fluid film configuration, pressure


Table 1.2 Static performances based on different types of boundary condition

RFJ boundary condition Reynolds boundary condition

Pressure solution domain: αs ≤ α ≤ αc, Pressure solution domain: 0 ≤ α ≤ αc,

p= pi + 3μωR2ε C2 1−ε21 5

1−εcosαc

2 sinα−sinαi 1 + εcosαc + 2cosαc + ε

αi−α −ε sinαcosα−sinαi cosαi

;

elsewhere, p= 0.

p=6μωR2

C2 1−ε2 1 5 α−ε sinα

−

α+ε2

2α−2εsinα +

ε2 sin2α4

1−εcosαc

;

elsewhere, p= 0.

If cavitation exits, αc and αs are obtained from αc is obtained from

ε 2 + εcosαc sinαc−sinαi −εsinαi

cosαc−cosαi + 2 cos αc + ε αi−αc

+C2pi 1−ε21 5

1−ε cos αc 3μωR2 = 0

ε αc−sinαc cosαc

= 2 sinαc−αc cosαc

ε 2 sinαs−sinαi 1 + εcosαc + 2cosαc + ε

αi−αs −ε sinαs cosαs−sinαi cosαi

+C2pi 1−ε21 5

1−εcosαc 3μωR2 = 0

If no cavitation exits, αc and αs are obtainedfromαc = cos−1 −ε 2 ,αs = αc−2π

Fluid force components: Fluid force components:

fε = −3μωLR3

C2

ε cosαc−cosαs2

1−ε2 1−εcosαcfε = −

3μωLR3

C2

ε 1−cosαc2


fϕ =3μωLR3

C2

ε

1−ε2 1 5 1−εcosαc1 + 2εcosαc αc−αs −2 ε+ cosαcsinαc−sinαs + sinαc cosαc−sinαs cosαs

fϕ =6μωLR3

C2

sinαc−αc cosαc


Relation between Sommerfeld number andeccentricity ratio:

Relation between Sommerfeld numberand eccentricity ratio:

S=1−ε2

1 51−εcosαc

3πε 1−ε2 cosαc−cosαs4 +A2

1

S=1−ε2 1−εcosαc

3πε 1−cosαc4 + 1−ε2 A2

1

Attitude angle: Attitude angle:

ϕ = tan−1 A1 1−ε2−0 5

cosαc−cosαs−2 ϕ = tan−1 A1 1−ε2

0 51−cosαc

−2

where A1 = 1 + 2εcosαc αc−αs−2 ε+ cosαc

where A1 = αc−sinαccosαc.

sinαc−sinαs + sinαc cosαc−sinαs cosαs.

From Wang and Khonsari (2008) © Elsevier Limited.


distribution, and steady state equilibrium position (Cε, ϕ) are summarized and dis-cussed in this subsection.Figure 1.7 shows how the fluid film configuration and the pressure distribution

vary as a function of the eccentricity ratio ε in terms of the absolute circumferential

1.0

RFJ boundary condition

Half-Sommerfeld boundary condition

90°

180°

270°

0°

00..00 00..550.0 0.5

Figure 1.6 Comparison of the journal position based on different boundary conditions(Θi = 0, pi = 0). From Wang and Khonsari (2008) © Elsevier Limited.

0.001 0.01 0.1 1 10

Sommerfeld No., S

0.0

0.2

0.4

0.6

0.8

1.0

Ecc

entr

icity

rat

io, ε

RFJ boundary condition

Half-Sommerfeld boundary condition

Figure 1.5 Comparison of the curve S vs. ε based on different boundary conditions (Θi = 0,pi = 0). From Wang and Khonsari (2008) © Elsevier Limited.


coordinate Θ, which starts from the fixed position at the upper load line. The posi-tion of θ = 0 is defined at Θ=ϕ since Θ = θ +ϕ.

With the fluid inlet condition of (Θi = 0, pi = 0), Figure 1.7a shows that the fluidpressure starting position θs coincides with the fluid inlet position θi. The fluid

0.80.6Eccentricity ratio, ε

0.40.2

0 360270

180

Absolute circumferential coordinate, Θ90

0

Cavitation

Cavitation

Cavitation

1 0.80.6Eccentricity ratio, ε

0.40.2

0 360270

180


0


0.40.2

0 360270

180


Dim

ensi

onle

ss p

ress

ure,

pD

imen

sion

less

pre

ssur

e, p

Dim

ensi

onle

ss p

ress

ure,

p

0.0 0.2 0.4 0.6 0.8 1.00

45

90

135

180

225

270

315

360

Abs

olut

e ci

rcum

fere

ntia

l coo

rdin

ate,

Θ

Eccentricity ratio, ε

θ = 0°

θc

θi

0.0 0.2 0.4 0.6 0.8 1.00

45

90

135

180

225

270

315

360

Abs

olut

e ci

rcum

fere

ntia

l coo

rdin

ate,

Θ


θ = 0°

θc

θi

0.0 0.2 0.4 0.6 0.8

40

(a) pi = 0

(b) pi = 0.5

(c) pi = 1.0

35

30

25

20

15

10

5

0

1.00

45

90

135

180

225

270

315

360

θs

θs+ 360°

θs+ 360°

θs+ 360°

θ = 0°

θc

Abs

olut

e ci

rcum

fere

ntia

l coo

rdin

ate,

Θ


θi

40

35

30

25

20

15

10

40

35

30

25

20

15

10

5

0

5

1

0

1

Figure 1.7 Fluid film configuration and hydrodynamic pressure distribution change withchanging the steady-state eccentricity ratio ε in terms of the absolute circumferentialcoordinate Θ (Θi = 0) at different supply pressures pi = 0, 0.5, and 1.0. From Wang andKhonsari (2008) © Elsevier Limited.


pressure always starts from the fluid inlet position. It also shows that cavitationalways exists with the fluid inlet condition of (Θi = 0, pi = 0). The cavitation regionshrinks as the steady state eccentricity ratio ε decreases.Figure 1.7b shows that cavitation exists only if the steady state eccentricity ratio ε

is greater than about 0.4 with the fluid inlet condition of (Θi = 0, pi = 0 5). Whenε > 0 4, the cavitation region shrinks to a smaller portion of the journal surface witha smaller ε down to 0.4; otherwise, a full 2π fluid film exists in the fluid-film jour-nal bearings.Figure 1.7c shows that cavitation exists only if the steady-state eccentricity ratio

ε is greater than about 0.49 with the fluid inlet condition of (Θi = 0, pi = 1 0). Whenε > 0 49, the cavitation region shrinks to a smaller portion of the journal surfacewith a smaller ε down to 0.49; otherwise, a full 2π fluid film exists in the fluid-film journal bearings.Figure 1.8 compares the pressure distributions corresponding to different fluid

inlet pressures pi = 0, pi = 0 5, and pi = 1 0, respectively with ε= 0 5. Cavitation isshown to exist for all of the three fluid inlet conditions. However, the cavitationregion shrinks as the inlet pressure pi increases from 0 to 1.0. The peak pressureincreases with increasing the fluid inlet pressure pi from 0 to 1.0.Figure 1.9 shows a comparison of the curve S vs. ε corresponding to different

fluid inlet pressure pi = 0, pi = 0 5, and pi = 1 0, respectively. The effect of fluidinlet pressure on the curve S vs. ε is subtle. For a given Sommerfeld number S,the eccentricity ratio ε decreases slightly with increasing the fluid inlet pressurepi from 0 to 1.0.A comparison of the steady-state equilibrium positions corresponding to differ-

ent fluid inlet pressures pi = 0, pi = 0 5, and pi = 1 0 is shown in Figure 1.10. It is

0

1

2

3

4

5

6

7

8

Dim

ensi

onle

ss p

ress

ure,

p

0 45 90 135 180 225 270 315 360

Absolute circumferential coordinate, Θ

pi = 1.0

pi = 0.5

pi = 0

Figure 1.8 Comparison of the pressure distributions with ε= 0 5 and Θi = 0. From Wangand Khonsari (2008) © Elsevier Limited.


0.001 0.01 0.1 1 10

Sommerfeld No., S

0.0

0.2

0.4

0.6

0.8

1.0E

ccen

tric

ity r

atio

, ε

0.4

0.6

0.8

1.0

Ecc

entr

icity

rat

io, ε

0.01

Sommerfeld No., S

pi = 1.0pi = 0.5

pi = 0

Figure 1.9 Comparison of the curve S vs. ε corresponding to different fluid inlet pressures(Θi = 0). From Wang and Khonsari (2008) © Elsevier Limited.

90°

180°

1.0

0°

270°

pi = 1.0

pi = 0.5

pi = 0

0.50.0

Figure 1.10 Comparison of the steady-state equilibrium position corresponding todifferent fluid inlet pressures (Θi = 0). FromWang and Khonsari (2008) © Elsevier Limited.


shown that the effect of fluid inlet pressure on the steady-state equilibrium positionis substantial. For a given eccentricity ratio ε, the attitude angle ϕ increases as thefluid inlet pressure pi increases from 0 to 1.0. In addition, under the fluid inlet con-dition of (Θi = 0, pi = 0 5), the attitude angle ϕ stays at 90 when the eccentricityratio ε is in the range of 0–0.4. In other words, irrespective of how the eccentricitychanges within the range of 0–0.4, since the radial hydrodynamic fluid force com-ponent fε does not exist, the journal is always in a force equilibrium and is free to bemoved to and then stay at any position with a new eccentricity and the same attitudeangle. This feature makes the rotor bearing system very sensitive to external per-turbation in the horizontal direction. Any external perturbation in the horizontaldirection will move the journal to a new position and no force is available to bring itback to its original position. This is sometimes referred as inherent instability to anyexternalperturbations in thehorizontaldirection,whichoftenexists in realapplications.With a higher inlet pressure (Θi = 0, pi = 1 0), the range where the attitude angle ϕstays at 90 regardless of any change in eccentricity ratio ε expands to 0–0.49.

1.3.3.3 Influence of Fluid Inlet Position

We will examine the influence of different fluid inlet positions (Θi = 0 , 45 , or90 ) on the fluid film configuration, fluid pressure distribution, and steady-statejournal position (Cε, ϕ) in this subsection. The fluid inlet pressure is held at pi = 0.Figure 1.11 shows how the fluid film configuration and pressure distribution

vary as a function of the eccentricity ratio ε corresponding to different fluid inletconditions (Θi = 45 , pi = 0) and (Θi = 90 , pi = 0), respectively. Figures 1.7a and1.11a and b show that cavitation always exists with fluid inlet pressure pi = 0and the cavitation region extends to a larger portion of the journal surface asthe fluid inlet position angle Θi increases from 0 to 90 .Assuming the eccentricity ratio ε = 0 5, Figure 1.12 shows a comparison of the

fluid pressure distributions corresponding to different fluid inlet position Θi = 0,Θi = 45 , and Θi = 90 , respectively. Cavitation region extends to a larger portionof the journal surface as the fluid inlet position Θi changes from 0 to 90 . Withincreasing the fluid inlet position Θi from 0 to 90 , the peak pressure decreasesand moves to a higher angle position.With the fluid inlet pressure pi = 0, Figures 1.13 and 1.14 show a comparison of

the curve S vs. ε and a comparison of the steady-state equilibrium position corre-sponding to three different fluid inlet positions Θi = 0, Θi = 45 , and Θi = 90 ,respectively.Figure 1.13 shows that for a given Sommerfeld number S, the eccentricity ratio ε

increaseswith increasing the fluid inlet positionΘi from0 to 90 . Figure 1.14 showsthat, for a given eccentricity ratio ε, the attitude angleϕ decreaseswith increasing thefluid inlet positionΘi from0 to 90 . For lightly loaded journal bearings (that is smallsteady-state eccentricity ratio ε), the fluid inlet position effect is more pronounced.


Based on the results presented in Figures 1.7, 1.8, 1.9, 1.10, 1.11, 1.12, 1.13, and1.14, a set of curve-fitting functions are obtained and presented inAppendixC.Underdifferent fluid supply conditions, Equations C.1–C.20 define how θc (circumferentiallocation where the fluid film ruptures), θs (fluid pressure starting position), and ϕ(attitude angle) change with changing the steady-state eccentricity ratio ε and howthe steady-state eccentricity ratio ε changes as the Sommerfeld number (S) changes.

1.4 Finite Bearing Solution

As discussed in the previous sections, for finite bearings with length-over-diameterratio more than 0.5 while not exceeding 2, either the infinitely short bearing theoryor the infinitely long bearing theory can only provide a rough estimation of thebearing performance. If more accurate prediction of the bearing performance isexpected, numerical solutions to the Reynolds equations would be necessary.

Cavitation

Cavitation

Cavitation

Cavitation

0.0 0.2 0.4 0.6 0.8 1.00

45

90

135

180

225

270

315

360

Abs

olut

e ci

rcum

fere

ntia

l coo

rdin

ate,

Θ


0.0 0.2 0.4 0.6 0.8

(a) Θi = 45°

(b) Θi = 90°

1.00

45

90

135

180

225

270

315

360

Abs

olut

e ci

rcum

fere

ntia

l coo

rdin

ate,

Θ


40

35

30

25

20

15

10

5

01


0.40.2

0 360270

180


0

40

35

30

25

20

15

10

5

01

0.80.6


0.40.2

0 360270

180


θs

θc

θc

θi

θsθi

Dim

ensi

onle

ss p

ress

ure,

pD

imen

sion

less

pre

ssur

e, p

θ= 0°

θ= 0°

Figure 1.11 Fluid film configuration and hydrodynamic pressure distribution change withchanging the steady-state eccentricity ratio ε in terms of the absolute circumferentialcoordinate Θ (pi = 0) for different inlet positions Θi = 45 and 90 . From Wang andKhonsari (2008) © Elsevier Limited.


Two methods commonly used to obtain the numerical solutions to the Reynoldsequations are finite element methods (FEA) and finite difference methods (FDM).Application of FEA and/or FDM to numerically solve the full Reynolds equationsis out of the scope of this book. Detailed applications of numerical methods havebeen presented by Khonsari and Booser (2008), Khonsari and Wang (1991).

Ecc

entr

icity

rat

io, ε

Sommerfeld No., S

0.0

0.2

0.4

0.6

0.8

1.0

0.001 0.01 0.1 1 10

Θi = 0°

Θi = 45°

Θi = 90°

Figure 1.13 Comparison of the curve S vs. ε corresponding to different fluid inletpositions (Θi = 0). From Wang and Khonsari (2008) © Elsevier Limited.

0

1

2

3

4

5

6

7

Dim

ensi

onle

ss p

ress

ure,

p

0 45 90 135 180 225 270 315 360

Absolute circumferential coordinate, Θ

Θi = 0°

Θi = 45°

Θi = 90°

Figure 1.12 Comparison of the pressure distributions with ε= 0 5 and pi = 0. From Wangand Khonsari (2008) © Elsevier Limited.


ReferencesDowson, D., Ruddy, A.V., Sharp, R.S., Taylor, C.M., 1985, “Analysis of the Circumferentially Grooved

Journal Bearing with Consideration of Lubricant Film Reformation,” Proceedings of the Institution ofMechanical Engineers, Part C: Journal of Mechanical Engineering Science, 199, pp. 27–34.

Floberg, L. 1957, “The Infinite Journal Bearing, Considering Vaporization,” Transactions of ChalmersUniversity of Technology, 189, Goteborg, Sweden, pp. 1–82.

Hashimoto, H.,Wada, S., 1982, “Influence of Inertia Forces on Stability of Turbulent Journal Bearings,”Bulletin of the JSME, 25 (202), pp. 653–662.

Hashimoto, H., Wada, S., Ito, J.I., 1987, “An Application of Short Bearing Theory to Dynamic Char-acteristic Problems of Turbulent Journal Bearings,” ASME Journal of Tribology, 109, pp. 307–314.

Jakobsson, B., Floberg, L., 1957, “The Finite Journal Bearing, Considering Vaporization,” Transac-tions of Chalmers University of Technology, 190, Goteborg, Sweden, pp. 1–116.

Khonsari, M.M., Wang, S.H., 1991, “On the Fluid-Solid Interaction in Reference to Thermohydrody-namic Analysis of Journal Bearings,” ASME Journal of Tribology, 113, pp. 398–404.

Khonsari, M.M., Booser, E.R., 2008, Applied Tribology: Bearing Design and Lubrication, 2nd edition,John Wiley & Sons, Inc., New York.

Lund, J.W., 1966, “Self-excited, Stationary Whirl Orbits of a Journal in a Sleeve Bearings,” PhD thesis,Department of Mechanics, Rensselaer Polytechnic Institute, Troy.

Ng, C.W., Pan, C.H.T., 1965, “A Linearized Turbulent Lubrication Theory,” ASME Journal of BasicEngineering, 87, pp. 675–688.

Wang, J.K., Khonsari, M.M., 2006, “Application of Hopf Bifurcation Theory to the Rotor-bearingSystem with Turbulent Effects,” Tribology International, 39(7), pp. 701–714.

Wang, J.K., Khonsari, M.M., 2008, “Effects of Oil Inlet Pressure and Inlet Position of Axially GroovedInfinitely Long Journal Bearings, Part I: Analytical Solutions and Static Performance,” TribologyInternational, 41, pp. 119–131.

Zhang, Y., 1989, “Dynamic Properties of Flexible Journal Bearings of Infinite Width Considering OilSupply Position and Pressure,” Wear, 130, pp. 53–68.

1.090°

180°

270°

0°

Θi = 0°Θi = 45°

Θi = 90°

0.50.0

Figure 1.14 Comparison of the steady-state equilibrium position corresponding todifferent fluid inlet positions (pi = 0). FromWang and Khonsari (2008) © Elsevier Limited.


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