11Hydrodynamic and

ElastohydrodynamicLubrication

11.1 Basic Equations Lubrication Approximation • The Reynolds Equation • The Energy Equation

11.2 Externally Pressurized Bearings Annular Thrust Pad • Optimization • Operation with Capillary Restrictor

11.3 Hydrodynamic LubricationJournal Bearings • Thrust Bearings

11.4 Dynamic Properties of Lubricant FilmsLinearized Spring and Damping Coefficients • Stability of a Flexible Rotor

11.5 Elastohydrodynamic LubricationContact Mechanics • Dimensional Analysis • Film-Thickness Design Formulas • Minimum Film Thickness Calculations

The friction force generated when two bodies rub against each other can be drastically decreased, andthe subsequent wear almost entirely eliminated, if a lubricant is interposed between the surfaces in contact.Lubricants are generally considered to be oils, particularly petroleum oils because we have so muchexperience with them. However, almost any material can be a lubricant. As da Vinci observed 400 yearsago, “All things and anything whatsoever, however thin it be, which is interposed in the middle betweenobjects that rub together lighten the difficulty of this friction.” Nevertheless, in this chapter lubricationwith fluids, in particular incompressible fluids, is the only type of lubrication that will be discussed.

The equations that describe lubrication with continuous fluid films are derived from the basic equa-tions of fluid dynamics through specialization to the particular geometry of the typical lubricant film;lubricant films are distinguished by their small thickness relative to their lateral extent. If Ly and Lxz denotethe characteristic dimensions across the film thickness and the ‘plane’ of the film, respectively, as indicatedin Figure 11.1, then typical industrial bearings are characterized by (Ly /Lxz) = O(10–3). This fact alone,and the assumption of laminar flow, allows us to combine the equations of motion and continuity intoa single equation in lubricant pressure, the so called the Reynolds equation. The theory of lubrication isconcerned with the solution of the Reynolds equation, sometimes in combination with the equation ofenergy, under various lubricating conditions. When the surfaces are deformable, as in rolling contactbearings or human and animal joints, the equations of elasticity and the pressure dependence of lubricantviscosity must also be included in the solution of the problem.

Andras Z. SzeriUniversity of Delaware

11.1 Basic Equations

The flow of an incompressible, constant viscosity fluid is governed by the equations of motion, the so-called Navier–Stokes equation (Szeri, 1998),

(11.1)

and the equation of continuity

(11.2)

We render these equations dimensionless by normalizing the orthogonal Cartesian coordinates (x,y,z)with the corresponding length scales and the velocities with the velocity scales, U* in the in the x-z planeand V* perpendicular to it

(11.3a)

(11.3b)

We note that the characteristic velocity in the y direction, V*, cannot be chosen independent of U*, asthe terms in the equation of continuity must balance. Substitution of Equation 11.3 into Equation 11.2yields

(11.4)

indicating that the terms of the continuity equations will balance provided that

(11.5)

FIGURE 11.1 Schematic of lubricant film.

ρ ρ∂∂

+ ⋅∇( )

= −∇ + µ∇ +vv v v f

tp 2

div v = 0

x L x y L y z L zxz y xz= = =, ,

u U u v V v w U w= = =∗ ∗ ∗, ,

∂∂

+ ∂∂

+ ∂∂

=∗

∗

u

x

V L

U L

v

y

w

zxz

y

0

V L

U LO V

L

LUxz

y

y

xz

∗

∗∗ ∗= ( ) =

1 or

This establishes the velocity scale in the y direction. To nondimensionalize pressure and time we choose

(11.6)

The reduced Reynolds number, re, and the nondimensional frequency,–Ω, have the definition

where Ω is the characteristic frequency of the system. For journal bearings, the characteristic frequencyis related to the rotational velocity of the journal. Taking the shaft surface speed as the characteristicvelocity, U* = Rω, and the journal radius as the characteristic length in the “plane” of the bearing, Lxz =R, we have re ≈

–Ω. Making use of this approximation, we write the equations of motion and continuityin terms of the normalized quantities as

(11.7a)

(11.7b)

(11.7c)

(11.8)

11.1.1 Lubrication Approximation

Resulting from the normalization in Equations 11.3 and 11.6, we anticipate that each of the variableterms in Equations 11.7 and 11.8 are of the same order, say O(1). But then the relative importance ofthe variable terms is decided only by the magnitude of the dimensionless parameters (Ly /Lxz) and re thatmultiply them.

Under normal conditions lubricant films are thin relative to their lateral extent. For liquid lubricatedbearings (Ly /Lxz) = O(10–3). For gas bearings a film thickness of h ≈ 2.5 µm is not uncommon. The so-called lubrication approximation of Reynolds is developed under the assumptions that (Ly /Lxz) → 0 andre → 0. Under these conditions Equations 11.7 indicates that

(11.9)

and Equations 11.7 and 11.8 reduce to

p rp

Ut te= =

∗ρ 2, Ω

rL

L

L U L

L

L Le

y

xz

y y

xz

y xz=

=

( )∗

ν ν and Ω

Ω

ru

tu

u

xv

u

yw

u

z

p

x

u

y

L

L

u

x

u

zey

xz

∂∂

+ ∂∂

+ ∂∂

+ ∂∂

= − ∂∂

+ ∂∂

+

∂∂

+ ∂∂

2

2

22

2

2

2

L

Lr

v

tu

v

xv

v

yw

v

z

v

y

L

L

v

x

v

z

p

yy

xze

y

xz

∂∂

+ ∂∂

+ ∂∂

+ ∂∂

− ∂∂

−

∂∂

+ ∂∂

= − ∂∂

22

2

22

2

2

2

rw

tu

w

xv

w

yw

w

z

p

z

w

y

L

L

w

x

w

zey

xz

∂∂

+ ∂∂

+ ∂∂

+ ∂∂

= − ∂∂

+ ∂∂

+

∂∂

+ ∂∂

2

2

22

2

2

2

∂∂

+ ∂∂

+ ∂∂

=u

x

v

y

w

z0

∂∂

= ( ) ∂∂

≈ ∂∂

= ( )−p

yO

p

x

p

zO10 16 while

(11.10)

(11.11)

written now in terms of the primitive variables.

11.1.2 The Reynolds Equation

The second part of Equation 11.10 asserts that to the order of approximation employed here, the pressureis invariant in the y direction, i.e., across the film. But then the first and last parts of Equation 11.10 canbe integrated to obtain

(11.12a)

(11.12b)

The integration constants A, B, C, D are evaluated with the aid of the boundary conditions on velocity(Figure 11.1),

(11.13)

to obtain the velocity components in the plane of the bearing

(11.14)

The pressure distribution appearing in Equation 11.14 is not arbitrary but must be such that the equationof continuity is satisfied. For this, we substitute Equation 11.14 into the averaged (across the film)equation of continuity, which, upon interchanging the indicated differentiation and integration, becomes

(11.15)

Recognizing that

(11.16)

∂∂

= µ ∂∂

∂∂

= ∂∂

= µ ∂∂

p

x

u

y

p

y

p

z

w

y

2

2

2

20, ,

∂∂

+ ∂∂

+ ∂∂

=u

x

v

y

w

z0

up

xy Ay B=

µ∂∂

+ +1

22

wp

zy Cy D=

µ∂∂

+ +1

22

u U w y

u U w y h

= = =

= = =

1

2

0 0

0

,

,

at

at

up

xy yh

y

hU

y

hU

wp

zy yh

=µ

∂∂

−( ) + −

+

=µ

∂∂

−( )

1

21

1

2

21 2

2

vx

p

xy yh dy

z

p

zy yh dy

x

y

hU

y

hU dy

h x t h x t h x t

h x t

[ ] = − ∂∂ µ

∂∂

−( )

− ∂

∂ µ∂∂

−( )

− ∂∂

−

+

( ) ( ) ( )

( )

∫ ∫0

2

0

2

0

1 20

1

2

1

2

1

, , ,

,

∫∫ + ∂∂

Uh

x2

v V Vdh

dty

h x t[ ] = − −( ) ==

( )0

1 2

,

where V1, V2 are the normal, i.e., approach, velocities of the bearing surfaces, we obtain the Reynoldsequation that governs the pressure distribution in a lubricant film (Szeri, 1998)

(11.17)

If the relative motion between the surfaces in contact is pure translation, as is in conventional thrustbearings, Equation 11.17 can be cast in the form

(11.18)

where

and we assume that ∂(U1 + U2)/∂x = 0.If, however, relative rotation of the surfaces is encountered, as in journal bearings, then a component

of the rotational velocity will augment the relative motion in the tangential direction (see Figure 11.10)as the surfaces are not parallel. Equation 11.17 now takes the form

(11.19)

Here U0 = U1 + U2 and V0 = V2 – V1. Thus while in thrust bearings it is the difference in tangentialvelocities that creates pressure, in journal bearings it is their sum; if journal and bearing rotate in oppositedirections but with the same speed, there is no pressure generated. However, for both journal and thrustbearings positive pressures are generated only when the film is convergent, both in space and time.

(11.20)

11.1.2.1 Turbulent Flow

For turbulent flow of the lubricant, most theories currently in use yield a formally similar Reynoldsequation, but now the film thickness is weighted by the functions kx

1/3 and kz1/3, respectively, in the

direction of relative motion and perpendicular to it (Szeri, 1998). For a journal bearing, the turbulentReynolds equation takes the form

(11.21a)

where–P is the average value of p.

The weighting functions are related (in Constantinescu’s model) to the local Reynolds number as

(11.21b)

∂∂ µ

∂∂

+ ∂

∂ µ∂∂

= −( ) ∂

∂+

∂ +( )∂

+ −( )x

h p

x z

h p

zU U

h

xh

U U

xV V

3 3

1 2

1 2

2 16 6 12

∂∂ µ

∂∂

+ ∂

∂ µ∂∂

= ∂

∂+

x

h p

x z

h p

zU

h

xV

3 3

0 06 12

U U U V V V0 1 2 0 2 1= − = −,

∂∂ µ

∂∂

+ ∂

∂ µ∂∂

= ∂

∂+

x

h p

x z

h p

zU

h

xV

3 3

0 06 12

Uh

x

h

t0 0 0

∂∂

< ∂∂

<,

∂∂ µ

∂∂

+ ∂

∂ µ∂∂

= ∂

∂x

h

k

P

x z

h

k

P

zU

h

xx z

3 3

0

1

2

k kx h= + ( )12 0 53 20 725

..

Re

(11.21c)

and the local Reynolds number, Reh = Rωh/v, is calculated from the reduced Reynolds number throughReh = re(h/C)(R/C).

11.1.2.2 Surface Roughness

Surface roughness should be taken into account if it is excessive. This can be done by applying thePatir–Cheng flow factors (Patir and Cheng, 1978), obtained by numerically solving the Reynolds equationfor microbearings possessing real, rough, surfaces (Figure 11.2). To calculate the pressure flow factors

–φx

and–φz, the microbearing is subjected to mean pressure gradients (p1 – p2)/(x2 – x1) and (p1 – p2)/(z2 –

z1), respectively, where (x2 – x1)(z2 – z1) is the projected area of the microbearing. The pressure flow factorφx, for example, is calculated from

where the overscore bar indicates statistical averaging. In the next step, the flow factors themselves areaveraged, having been calculated for a large number of statistically identical microbearings. The Reynoldsequation for rough surfaces now takes the form

(11.22)

where σ is the standard deviation of the surface roughness.

11.1.3 The Energy Equation

It is well to remember that bearings rarely operate in the isothermal mode. In almost all cases of practicalbearing operations, sufficient heat is generated by dissipation to invalidate the isothermal assumption.Some of the generated heat is conducted away, predominantly by the bearing pad, but the remainder isused to increase the temperature of the lubricant. Increasing the temperature of the lubricant changes itsviscosity, and as the rate of heat generation in the film is nonuniform, so will be the viscosity distribution.

Under conditions of light load, low speed, and large bearing-thermal capacity, bearing performancecan be adequately predicted when using an average or operating viscosity, obtained by iteration betweenthe bearing performance charts and the viscosity–temperature chart (Szeri, 1998).

In other, more general cases, the actual point by point variation of the lubricant’s viscosity must betaken into account; thus the fluid equations and the energy equations must be solved simultaneously(Suganami and Szeri, 1979).

The equation of energy for a heat-conducting Newtonian fluid is

(11.23)

If the fluid is incompressible cv = cp = c and –pdivv = 0 by Equation 11.2. Assuming further that theheat conductivity is constant, Equation 11.23 becomes

(11.24a)

k kz h= + ( )12 0 296 20 65

..

Re

φ x hp

xh

p p

x x= ∂

∂

−( )−( )

3 2 1

2 1

∂∂ µ

∂∂

+ ∂

∂ µ∂∂

= + ∂

∂+ − ∂

∂+ ∂

∂x

h p

x z

h p

z

U U h

x

U U

x

h

tx zsφ φ σ φ3 3

1 2 1 2

12 12 2 2

ρd c T

dtpdiv k T

v( )= − + + ( )v µΦ div grad

ρcT

tu

T

xv

T

yw

T

zk

T

x

T

y

T

z

∂∂

+ ∂∂

+ ∂∂

+ ∂∂

= ∂∂

+ ∂∂

+ ∂∂

+

2

2

2

2

2

2µΦ

In orthogonal Cartesian coordinates the dissipation function is

(11.24b)

FIGURE 11.2 Flow factors for rough surfaces, (a) pressure flow factor, (b) shear flow factor. (From Patir, N. andCheng, H.S. (1979), Application of average flow model to lubrication between rough sliding surfaces, ASME J. Lub.Tech., 101, 220-30. With permission.)

Φ = ∂∂

+ ∂∂

+ ∂∂

+ ∂∂

+ ∂∂

+ ∂∂

+ ∂∂

+ ∂∂

+ ∂∂

2

2 2 2 2 2 2u

x

v

y

w

z

u

y

v

x

v

z

w

y

w

x

u

z

Applying the lubrication approximation, Equations 11.3 and 11.6, we obtain

(11.25)

where the Peclet number and the dissipation number have the definition, respectively,

the starred quantities are evaluate in the reference state, and–T = UT/T*,

–µ = µ/µ*.The thermohydrodynamic (THD) theory of fluid film lubrication accounts for pointwise variation of

lubricant viscosity and relies on the simultaneous solution of Equation 11.25, the equation of heatconduction in the bearing, and the extended form of the Reynolds equation that is written for variableviscosity (Suganami and Szeri, 1979)

(11.26)

for journal bearings and

(11.27)

for slider bearings. Here we employed the notation

11.1.3.1 Effective Viscosity

In numerous applications, the temperature rise in the lubricant remains relatively small and uniformthroughout the film. In these cases the isothermal theory will provide useful approximations to actualbearing performance, provided that an average or effective viscosity value that is compatible with thebearing temperature rise is used in the computations (Kaufman et al., 1978). This calculation might bebased on the assumptions that:

1. All heat, H, generated in the film by viscous dissipation is removed from the bearing by thelubricant.

2. The lubricant that exits the bearing by its sides has an average temperature Ts = (Ti + ∆T/2), where∆T = To – Ti is the average temperature rise across the bearing.

uT

xv

T

yw

T

z Pe

T

y

u

y

w

y

∂∂

+ ∂∂

+ ∂∂

= ∂∂

+ µ ∂∂

+ ∂∂

1 2

2

2 2

Λ

Pec

kr

cT

L

Lexz

y

= µ = µ

∗ ∗

∗

, Λ ωρ

2

∂∂ µ

∂∂

+ ∂

∂ µ∂∂

= ∂

∂−

( )( )

+

∗ ∗x

h P

x z

h P

zU

xh

x h z

x h zV

Γ Γ3 3

0

2

1

0

ξ

ξ

, ,

, ,

∂∂ µ

∂∂

+ ∂

∂ µ∂∂

= ∂

∂( )( ) +

∗ ∗x

h P

x z

h P

zU

x

x h z

x h zV

Γ Γ3 3

0

2

1

0

ξ

ξ

, ,

, ,

Γ x z x y zx h z

x h zx y z dy

x zdy

x y zx z

ydy

x y z

, , ,, ,

, ,, ,

, ,, ,

, , ,, ,

( ) = − ( ) −( )( ) ( )

( ) =µ( ) ( ) =

µ( )

∫

∫ ∫

ξξ

ξξ

ξ η ξ ηη η

2

2

1

1

0

1

1

0

2

0

This average temperature rise, ∆T, can be calculated from a simple energy balance

(11.28)

For typical petroleum oils ρc ≈ 1.39 MPa/C and ρc = 4.06 MPa/C for water.

11.2 Externally Pressurized Bearings

The operation of externally pressurized bearings is simple in principle. Figure 11.3a shows a hydrostaticpad with the runner covering the recess and resting on the land, sealing the pressurized lubricant in therecess. As there is no flow, the recess pressure equals the supply pressure. Increasing the supply pressure tothe point that the pressure force on the runner equals the external load, the runner lifts off the land andflow out of the recess commences, as indicated in Figure 11.3b. As now there is flow, the pressure force overthe recess becomes somewhat smaller than before, but this decrease is compensated by the pressure forcedeveloped over the land, and the total pressure force still equals the external load. Increasing (decreasing)the supply pressure ps, the recess pressure pr will simply adjust itself through increasing (decreasing) thefilm thickness so that the generated pressure force always balances the external load.

To deal with unsymmetrical load distribution, two or more hydrostatic pads are applied in conjunctionwith flow restrictors (compensated hydrostatic bearing). Flow restrictors are simple devices, such as cap-illaries, orifices, or flow-control valves, for which a small change in flow rate results in a large pressuredrop across the device. Figure 11.4 is a schematic of a two-pad hydrostatic bearing, equipped with flowrestrictors. The two recesses are fed from a common pressurized manifold. When the load is distributedsymmetrically, the film thickness over the two pads has the same value, leading to equal recess pressures(Position I). If now the external load vector is disturbed so that the film thickness decreases over pad A(Position II), the flow rate across the flow-control device in that same pad is decreased. As there is nowless pressure drop across the restrictor due to the decreased flow rate, the recess pressure in pad A willincrease and yield a righting moment.

Ordinarily, hydrostatic bearings are characterized by constant film thickness h = 0. Then the Reynoldsequation valid for simple relative translation of the bearing surfaces, Equation 11.18, reduces to

(11.29)

FIGURE 11.3 Hydrostatic bearing schematics (a) before and (b) after lift-off. (From Szeri, A.Z. (1998). Fluid FilmLubrication, Theory and Design, Cambridge University Press. With permission.)

∆t H c Q Qs= −( )[ ]ρ 2

∂∂

+ ∂∂

=2

2

2

20

p

x

p

z

The boundary conditions that accompany Equation 11.29 are

(11.30)

where Γ1 and Γ2 represent inside (recess) and outside contours, respectively, of the pad. Equation 11.29,subject to boundary conditions Equation 11.30, is easily solved in any of several orthogonal curvilinearcoordinate systems where Laplace’s equation separates, provided that Γ1, Γ2 are composed of coordinatelines. Otherwise, though the solution is still elementary, it must be performed on the computer.

Irrespective of the method of solution, i.e., analytical or numerical, and the geometry and size of thebearing pad, the performance of hydrostatic bearings can be written in terms of a set of dimensionlessparameters: the load factor af , the flow factor qf , and the loss factor hf . If W, Q, Hp , Hf represent the externalload, the lubricant flow, the pressure power loss, and the frictional power loss, respectively, then we canwrite

(11.31a)

(11.31b)

(11.31c)

(11.31d)

Here A is the combined area of pad and recess, ps is the supply pressure, pr is the recess pressure, andUM is the maximum relative velocity between pad and runner.

11.2.1 Annular Thrust Pad

We illustrate the evaluation of the load, flow, and friction factors on the analytically simplest of hydrostaticpads, the annular thrust pad. The geometry is characterized by the inner (recess) radius R1 and the outer(pad) radius R2 > R1. We use the pad radius and the recess pressure to normalize Equations 11.29 and 11.30

FIGURE 11.4 Schematic of a two-pad compensated hydrostatic bearing, (I) symmetrical, (II) unsymmetrical loaddistribution. (From Szeri, A.Z. (1998), Fluid Film Lubrication, Theory and Design, Cambridge University Press. Withpermission.)

p p

p p

r

a

=

=

on

on

Γ

Γ

1

2

W a Apf r=

Q qh

pf r=µ

3

H qh

p pp f r s=µ

3

H hU A

hf fM= µ 2

(11.32)

and obtain

(11.33a)

in place of Equation 11.29, while the boundary conditions take the form

(11.33b)

Integrating of Equation 11.33a and enforcing the boundary conditions, Equation 11.33b, lead to

(11.34)

Having found the pressure distribution, we are ready to evaluate the performance of the pad.

Load capacity:

(11.35a)

Flow rate:

(11.35b)

where

is the (radial) flow velocity, obtained as in Equation 11.14.

Pumping power:

(11.35c)

p p p r R rr= =, 2

d

drr

dp

dr

= 0

p rR

Rp r= = = =1 0 11

2

at at ,

pr

R R= ( )

ln

ln 1 2

W R p rpdr R pR

R R Rr rdr

R R

R RAp

r

R

R

r

R R

r

= π + π = π

+ ( )

=− ( )

( )

∫ ∫12

22 1

2

2

1 2

1

1 2

2

2 1

22

1

2

1

2

1 2

lnln

ln

Q ru dyR R

h pr

h

r= π = π( ) µ∫2

60 2 1

3

ln

udp

dry y hr =

µ−( )1

2

H p Q

R R

h p p

p s

s r

=

= π( ) µ6 2 1

3

ln

Frictional power loss:

(11.35d)

where τ = µdur/dr is the shear stress on the runner.A comparison of Equations 11.31 to 11.35 yields

(11.36a)

(11.36b)

(11.36c)

The total power loss is the sum of frictional loss and pumping power

(11.37)

11.2.2 Optimization

The curves of Figure 11.5 indicate the existence of optimum values hopt, µopt of the film thickness and theviscosity, respectively, defined by the conditions

(11.38)

Substitution of HT from Equation 11.37 into Equation 11.38 yields the optimum film thickness atconstant viscosity and the optimum viscosity at constant film thickness, respectively (Szeri, 1998)

(11.39a,b)

Substituting Equation 11.39a into Equations 11.31c and 11.31d, we find that at constant viscosity thetotal power loss is minimum when Hf /Hp = 3 and the total power consumption of the bearing is

(11.40a)

H r dAR R U A

hfR

RM= =

− ( ) µ∫ ωτ1

2 1

21 2

42

aR R

R Rf =− ( )

( )1

2

1 2

2

2 1ln

qR Rf = π

( )6 2 1ln

hR R

f =− ( )1

21 2

4

H H H qh

p p hU A

hT p f f r s fM= + =

µ+ µ3 2

∂∂

= ∂∂µ

=H

h

HT T0 0,

hh U A

q p p

q h p p

h U Aoptf M

f r sopt

f r s

f M

=µ

µ =

2 21 4

4

2

1 2

3,

H q h p p U AT h f f r s Mopt, = µ( )4

33 4

3 2 6 31 4

When Equation 11.39b is used, we find that Hf /Hp = 1 and the minimum power at constant film thicknessis

(11.40b)

FIGURE 11.5 Power loss for an annular hydrostatic bearing as function of (a) the dimensionless film thickness and(b) the dimensionless velocity. (From Szeri, A.Z. (1998), Fluid Film Lubrication, Theory and Design, CambridgeUniversity Press. With permission.)

H q h Ap p hUT f f r s Mopt,µ = 2

Obviously, the conditions Hf /Hp = 1 and Hf /Hp = 3 cannot be satisfied simultaneously. However, bycalculating the variation of HT when the ratio Hf /Hp is in the interval 1 ≤ Hf /Hp ≤ 3 we find

(11.41)

Equation 11.41 shows that as long as we keep 1 ≤ Hf /Hp ≤ 3, the total power will not exceed its optimalvalue by more than 16%. But this is not the full story, as we should also keep in mind the temperaturerise as the result of dissipation. Assuming that all heat generated is used for increasing the temperatureof the lubricant and that none is conducted away, a simple energy balance yields

(11.42)

Assuming, for the sake of simplicity, that ps /pr = const., the condition ∂∆T/∂ps = 0 assures us that thetemperature rise is a minimum when Hf /Hp = 1, thus we do well to design for this condition.

11.2.3 Operation with Capillary Restrictor

The recess pressure is related to the supply pressure through pr(RC + RB) = psRB where RB is the resistanceto flow over the land and RC is the resistance in the capillary. The capillary flow Qc, on the other hand,is determined by the pressure drop ∆p and can be calculated from the Hagen–Poisseuille law Qc = ∆p/RC.We can also relate the external load to the supply pressure

(11.43)

where lC is the length of the capillary and dC is its diameter.Let W0 be the load at the reference film thickness h0, then the ratio of load to reference load is

(11.44)

where

and the ratio of supply pressure to recess pressure is given by ps/pr = 1+ ξ. The dimensionless bearingstiffness is obtained by differentiating W/W0 with respect to h/h0

(11.45)

1 1 1398

1 1547 1

3 1

≤ ≤

≥ ≥

≥ ≥µ

µ

H

H

H

H

H

H

T h

T h

T

T

f

p

opt

opt

,

,

,

,

.

.

for

∆Tc

h

q

U A

h ppf

f

M

r

s= µ +

1 2 2

4ρ

W a ApR

R RR q h R

df sB

C BB f C

C

C

=+

= µ = µπ

, , 32

4

128 l

W

W X03

1

1= +

+ξ

ξ

ξ = =R

RX

h

hC

Bo

and 0

λξ ξ

ξ≡ −

∂( )∂( ) =

+( )+( )

W W

h h

X

X

0

0

2

32

3 1

1

Figure 11.6 shows the variation of λ/(1 + ξ) with X.

(i) Operation without regulator.Single bearing: If there are no flow regulators installed so that the recess is fed directly by the

pump, then the flow in the recess is defined by the pump flow-pressure characteristics. Theapplied load and the geometry specifies the recess pressure pr

(1) = W(1)/af A. The pump willsupply flow Q(1) = qf h(1)3/µpr, and we may find h(1) for a given µ, or µ at given h(1). If the loadwere to be increased to W (2) > W(1), we would obtain pr

(2) > pr(1), h(2) < h(1), and Q(2) < Q(1).

Two bearings: Let us assume, for simplicity, that the two bearings, which operate from a singlemanifold without flow regulators, are geometrically identical but carry loads W(1) and W(2) >W(1), respectively. The required recess pressures are

At the start of operation the pump delivery pressure is increased from zero and reaches the lower ofthe lift-off pressures pL

(1) = W(1)/Ar, where Ar is the area of the recess. Now that the passage is open forthe lubricant, the delivery pressure will drop back to pr

(1) and there it will stay. There is no way for thepump to reach pL

(2) = W(2)/Ar > pL(1) > pr

1, the lift required by pad no. 2, i.e., pad no. 2 will not comeinto operation at all.

(ii) Operation with capillary restrictor. The size of the restrictor to be installed in bearing no. 1 of theprevious example must be such that ∆p(1) + pr

(1) > pL(2) at the required flowrate Q(1).

FIGURE 11.6 Operation with capillary restrictor: dimensionless bearing stiffness vs. film thickness. (From Szeri,A.Z. (1998), Fluid Film Lubrication, Theory and Design, Cambridge University Press. With permission.)

pW

a A

W

a Apr

f fr

11 2

2( ) ( ) ( ) ( )= < =

Example 1. Hydrostatic Pad

Design an annular hydrostatic pad with the following requirements:

For this geometry Equation 11.36 yields af = 0.54 and qf = 0.755, so that

We select a pump that delivers 6.309 cm3/sec. At this flowrate the film thickness is an acceptable

We thus need a pump that delivers pL = 44,500/(0.0635)2 π = 3.5129 MPa at zero flow and 1.626 MPa atQ = 6.31 cm3/s.

Assume a supply pressure ps = 2.0684 MPa at Q = 6.309 cm3/s, then from the Hagen–Poisseuille law

Using this last equation, for standard capillary inside diameters we construct Table 11.1.Only capillaries with 0.084 < dC < 0.18 are satisfactory, and we choose dC = 0.12 cm and lC = 12.0 cm:

as lC/dC > 20, the required length is practical, and dC > 0.0635 cm (clogging).The choice of flow restrictors will influence bearing performance under dynamic conditions. Table 11.2

lists advantages and disadvantages of flow restrictors (rating 1 is best or most desirable).

11.3 Hydrodynamic Lubrication

Hydrodynamic lubrication relies on the relative motion of nonparallel bearing surfaces. To generatepositive, i.e., load-carrying, pressure, the film must be convergent in the direction of relative motion

TABLE 11.1 Capillary Restrictors for Hydrostatic Pad

dC (cm) lC (cm) lC/dC

0.38 1184 31200.27 302 11000.18 60 3330.12 12 960.084 2.8 180.069 1.3 170.051 0.38 7.4

W N R cm R cm

h m Pa s

= = =

≥ µ µ = × ⋅−

44 500 6 35 12 7

50 8 3 03 10

1 2

2

, , . , . ,

. , .

p MPa

Q m s

r =× π

=

=×( )×

× = ×−

−−

44500

0 54 0 1271 626

0 7555 08 10

3 03 101 627 10 5 3115 10

2

63

2

6 6 3

. ..

..

.. .

h m m= × × ×× ×

= µ > µ

− −6 309 10 3 03 10

0 755 1 626 1053 80 50 8

6 2

6

1 3

. .

. .. .

lC C Ccm d d cm( ) =π −( ) ×

× × × ×= × × ( )− −

2 0684 1 626 10

128 3 03 10 6 309 105 68 10

6

2 6

4 4 4. .

. ..

(Figure 11.1). No outside agency is required to create and maintain a load-carrying film, provided thatadequate lubricant is made available. Hydrodynamic films are easy to obtain; in fact they often occureven when their presence is deemed undesirable, e.g., in hydroplaning of automobile tires on wetpavement.

Prototypes of conformal hydrodynamic bearings are journal bearings and thrust bearings; these bear-ings might also be called “thick film” bearings. A journal bearing at load per projected bearing area of1.36 MPa, speed 60 rps, and lubricated with an ASTM Grade 315 oil at 52°C would have a minimumfilm thickness of the order of 88.4 µm. This film is thick in comparison to film found in counterformalbearings, such as ball and roller bearings.

Journal bearings are designed to support radial loads on rotating shafts, while thrust bearings, as theirname implies, support axial or thrust loads. Although the mode of lubrication is identical in these twobearing types, their geometry is sufficiently distinct for us to discuss them under separate headings; injournal bearings, in general, the clearance geometry is convergent–divergent, and film rupture occurs inthe divergent part. Conventional thrust bearings, on the other hand, are purely convergent and their filmremains continuous throughout the clearance.

The third main heading of this section introduces the idea of lubricant film instability in the dynamicsense; in most cases of application, the thermomechanical load on the bearing varies with time and, asa result, the bearing surfaces undergo cyclic oscillation that can lead to catastrophic film failure.

11.3.1 Journal Bearings

In its most elementary form, a journal bearing is a short, rigid, metal cylinder that surrounds and supportsthe rotating shaft, as in Figure 11.7. The clearance space between bearing and shaft is filled with alubricant, usually a petroleum oil.

Under zero load and negligible body weight, the rotating journal is concentric with its bearing, but asthe external load on the journal is increased, the shaft center sinks below the center of the bearing. For

TABLE 11.2 Advantages/Disadvantages of Flow Restrictors

Compensating Elements Capillary Orifice Valve

Initial cost 2 1 3Cost to fabricate, install 2 3 1Space requirement 2 1 3Reliability 1 2 3Useful life 1 2 3Availability 2 3 1Tendency to clog 1 2 3Serviceability 2 1 3Adjustability 3 2 1

FIGURE 11.7 Schematic of a full journal bearing. (From Szeri, A.Z. (1998), Fluid Film Lubrication, Theory andDesign, Cambridge University Press. With permission.)

isothermal operations, somewhat of a rarity in practice, the loading conditions on the shaft can becharacterized by a single dimensionless group, the Sommerfeld number S, defined by

where N is the rotation and R is the radius of the shaft, C is the radial clearance between bearing andshaft, µ is the viscosity of the fluid, and P is the specific load. The specific load has the definition P =W/LD, where L is the length and D is the diameter of the bearing; note that this definition of P is usedeven for partial arc bearings, for which the projected bearing area might be less than LD.

For an unloaded bearing P → 0 and S → ∞, a weightless shaft runs concentric with its bearing. Onincreasing the external load or decreasing the speed, i.e., on decreasing S, the journal will move awayfrom its concentric position, the journal trajectory approximating a semicircular arc (Figure 11.8). Underextreme load or vanishing speed, S → 0, metal-to-metal contact occurs at the point where the load linecuts the bearing.

In most applications there is considerable heat generation, and the viscosity of the lubricant does notremain uniform throughout the film. In such cases we need more than a single dimensionless group tocharacterize journal bearing operations. In fact, under nonisothermal conditions the number of charac-terizing parameters is so large that tabulation of bearing performance becomes impractical.

Denote the radial clearance, i.e., the difference in radii between cylinder and shaft, by C as before, theradius of the shaft by Rs and the radius of the cylinder by Rb, so that C = Rb – Rs, then in normal designpractice (C/R) = O(10–3). This signifies that the lubrication approximation to Equations 11.1 and 11.2is valid, and we can employ the Reynolds equation for evaluating bearing performance. The small valueof C/R further signifies that the curvature of the film can be neglected to the same order, thus the analysismay be performed in an orthogonal Cartesian coordinate system. The (x,z) plane of this coordinatesystem lies in the surface of the bearing, so that x is the “circumferential” coordinate and z is parallel tothe axis of the shaft. The y coordinate is normal to the “plane” of the bearing, i.e., it points toward thecenter of the bearing arc, yet to the approximation involved here the y arrays appear parallel to one

FIGURE 11.8 Shaft trajectory during static loading. (From Szeri, A.Z. (1998), Fluid Film Lubrication, Theory andDesign, Cambridge University Press. With permission.)

SN

P

R

C= µ

2

another. Because of the smallness of the clearance, it makes no difference whether we put R = Rs or R =Rb in the definition of the Sommerfeld number.

Using plane trigonometry and the binomial expansion (Szeri, 1998), it can be shown that the filmthickness between eccentric cylinders is well approximated by the formula

(11.46)

where e is the eccentricity, h the film thickness, and θ the angle measured from the line of centers in thedirection of shaft rotation. When the load-line, and therefore the line of centers, is not fixed but oscillating,as when there is a rotating out of balance force on the shaft, the film thickness relates to the fixed position through the formula

(11.47)

Here φ is the angle between load-line and line of centers, the so-called attitude angle, and ψ defines theload-line relative to the fixed position = 0. Equation 11.47 can be used to evaluate the right-hand sideof the Reynolds equation, Equation 11.19, to obtain (Szeri, 1998)

(11.48)

Here ω and ωW = dψ/dt are the angular frequencies of the shaft and the load vector, respectively.Equation 11.48 is only an approximation to the governing equations of lubricant flow, good to order C/Rprovided that ·e,

·φ, and ωW are of the same order of magnitude as ω or smaller.Though Equation 11.48 was arrived at through simplification of the full nonlinear equations,

Equations 11.1 and 11.2, its solution is still difficult to obtain except in numerical form. For this reason,before the advent of high-speed computing Equation 11.48 was further simplified to make it amenableto analytical solutions. These simplifications, known as the short-bearing and the long-bearing approxi-mations to the Reynolds equation, must be used with great caution, however, as they may yield incorrectperformance parameter values. Before discussing these approximations to Equation 11.49 we shall makethe equation nondimensional.

The Reynolds equation is nondimensionalized via the substitutions

(11.49)

Assuming that ·e =·φ = ωW = 0 in Equation 11.48, the nondimensional form of the Reynolds equation,

valid for journal bearings under static loading, is

(11.50)

The individual terms on the left-hand side of Equation 11.50

h C e C= + = +( )cos cosθ ε θ1

h C e= + − +( )[ ]cos Ξ φ ψ

∂∂ µ

∂∂

+ ∂

∂ µ∂∂

= ∂

∂+ + +( )[ ]x

h p

x z

h p

zR

h

xe e W

3 3

6 12ω θ φ ω θ˙cos ˙ sin

x R zL

z h CH C p NR

Cp= = = = +( ) = µ

θ ε θ, , cos , 2

1

2

∂∂

∂∂

+

∂∂

∂∂

= π ∂∂θ θ θ

Hp D

L zH

p

z

H3

2

3 12

∂∂

∂∂

∂∂

∂∂

θ θ

Hp D

L zH

p

z3

2

3 and

represent the rate of average (across the film thickness) pressure flow in the circumferential and axialdirections respectively, the sum of which is balanced by the shear flow

By the normalizing transformation, Equation 11.49, each of the variable terms of Equation 11.50 are ofthe same order of magnitude; thus for long-bearings for which (D/L)2 → 0 the following approximationis acceptable

(11.51)

whereas for short-bearings (L/D)2 → 0, and we may approximate Equation 11.50 in the form

(11.52)

When solved with zero pressure boundary conditions, the pressure distributions specified byEquation 11.50 or its approximations, Equations 11.51 and 11.52, are 2π-periodic functions, antisym-metric with respect to the position of minimum film thickness, θ = π. They yield negative pressures ofthe same magnitude as positive pressures. However, unless special care is taken to remove all impurities,liquids cannot withstand large negative pressures and the lubricant film will rupture within a shortdistance downstream from the position of the minimum film thickness. Though the cavitation zonemight be preceded by a short range of subambient pressures, in most performance calculations this regionof subambient pressures is disregarded.

The boundary condition at film-cavity interface is complicated, particularly when dynamic loadingconditions prevail, and is still under investigation. Most computer calculations are based on the so-calledSwift–Stieber boundary conditions

(11.53)

that preserve flow continuity at the film-cavity interface. Here θcav = θcav (–z) denotes the angular positionof the film-cavity interface. Equation 11.50 also implies that –p ≥ 0 everywhere in the film.

The Swift–Stieber boundary condition cannot be implemented in the short-bearing approximation,as the latter is governed by a differential equation in –z, and only with some difficulty in the long-bearingapproximation. The closest we can come to Equation 11.53 when using the short-bearing approximationis to disregard negative (below ambient) pressures in calculating bearing performance, i.e., assume thefilm to cavitate at the position of minimum film thickness. The conditions

(11.54)

are used extensively for performance calculations in both short-bearing and long-bearing approximationsand are known as the Gümbel boundary conditions. We will evaluate bearing performance, for both short-bearing and long-bearing approximations, under the conditions specified in Equation 11.54, but for finitebearings we use the Swift–Stieber conditions, Equation 11.35. Figure 11.9 displays pressure distributionunder Sommerfeld, Gümbel, and Swift–Stieber boundary conditions.

12π ∂∂H

θ

d

dH

d p

d

d H

θ θ θ3 12

= π∂

∂∂

∂∂

= π

∂∂z

Hp

z

L

D

H3

2

12θ

pp= ∂

∂= =

θθ θ0, at cav

p

p

θ θ

θ θ

( ) ≥ ≤ ≤ π

( ) = π ≤ ≤ π

0 0

0 2

,

,

Bearing performance is calculated from the pressure distribution by substitution into the followingformulas

Load capacity: The force balance (Figure 11.10)

(11.55)

where FR and FT are the components of the pressure force along the line of centers and normal to it,respectively, and W is the external load on the shaft, yield

(11.56a)

(11.56b)

Equations 11.56 define the he nondimensional force components fR, fT. The Sommerfeld numberreemerges here as the inverse of the nondimensional pressure force

FIGURE 11.9 Journal bearing pressure distribution under (a) Sommerfeld, Gümbel, and (b) Swift–Stieber bound-ary conditions. (From Szeri, A.Z. (1998), Fluid Film Lubrication, Theory and Design, Cambridge University Press.With permission.)

W F

W F

R

T

cos

sin

φ

φ

+ =

− + =

0

0

F f LD N R C p dxdzR R

R

L

L cav

≡ µ ( ) = ∫∫−

2

02

2

cosθθ

F f LD N R C p dxdzT T

R

L

L cav

≡ µ ( ) = ∫∫−

2

02

2

sinθθ

(11.57)

Attitude angle: The attitude angle, i.e., the angle between the load vector and the line of centers is given by

(11.58)

Friction variable: The friction force exerted on the shaft is found from

(11.59)

where τxyh(x) is the shear stress on the shaft and cµ is the friction variable, which is the conventionalfriction coefficient scaled with (C/R) to numerically convenient values.

Lubricant flow: The rate of inflow can be calculated from

(11.60)

where u is given by Equation 11.14 and qi is the dimensionless inflow variable.

FIGURE 11.10 Journal bearing nomenclature. (From Szeri, A.Z. (1998), Fluid Film Lubrication, Theory and Design,Cambridge University Press. With permission.)

SN

P

R

Cf fR T≡ µ

= +( )−2

2 21 2

φ = arctanf

fT

R

F cC

RW dxdzxy h x

RL

µ µ ( )π

≡

= ∫∫ τ0

2

0

Q q NRLC u y z dydzi i

h xR

≡ = ( )( )π

∫∫2 000

2

, ,

Table 11.3 lists the performance characteristics for both short- and long-bearings. These calculationsare based on the Gümbel conditions, Equation 11.54.

As the cavitated film does not contribute to load capacity but only to unwanted friction, it serves nouseful purpose. This leads to the idea of employing partial arc pads instead of 360°, or full, bearings tosupport the shaft. Different types of fixed pad partial journal bearings, each of arc β, are shown sche-matically in Figure 11.11.

TABLE 11.3 Performance of Short- and Long-Bearings

Short-Bearing Approximation Long-Bearing Approximation(Gümbel condition) (Gümbel condition)

Equation for pressure, –p

Boundary condition

Pressure, –p

Radial force, fR

Tangential force, fT

Attitude angle, φ

Sommerfeld number, S

Friction variable, cµ

Flow variable, qi 2πε —

FIGURE 11.11 Fixed type journal bearings: (a) full 360° bearing, (b) centrally loaded partial bearing, (c) offsetloaded partial bearing (offset parameter α/β). (From Raimondi, A.A. and Szeri, A.Z. (1984), Journal and thrustbearings, in CRC Handbook of Lubrication, Vol. 2, Booser, E.R., (Ed.), CRC Press, Boca Raton, FL. With permission.)

∂∂

= π

∂∂

p

zH

dp

dz

L

D

H3

2

12θ

∂∂

= π ∂∂θ θ θ

Hdp

d

H3 12

p p za= = ± at 1 p p pi0 2( ) = π( ) =

61

12

32π

∂∂

−( ) +L

D H

Hz paθ

12 2

2 12 2

π +( )+( ) +( )

+ε θ ε θ

ε ε θ

sin cos

cospi

−

π

−( )L

D

22

2

2

4

1

ε

ε− π

+( ) −( )12

2 1

2

2 2

εε ε

L

D

π−( )

2 2

2

3 21

εε

6

2 1

2

2 21 2

π

+( ) −( )ε

ε ε

arctanπ −

1

4

2εε

arctanπ −

1

2

2εε

D

L

−( )π π −( ) +

2 22

2 2 2

1

1 16

ε

ε ε ε

2 1

6 4 1

2 2

2 2 2

+( ) −( )π + π −( )

ε ε

ε ε ε

2

1

2

2

π

−( )S

εε φ

εsin + π

−

4

1

2

2

S

11.3.1.1 Finite Journal Bearings

The pressure distribution in a finite journal bearing of length L, diameter D and angular extent β, isgoverned by the Reynolds equation

(11.61)

and the boundary conditions (considering –p as gauge pressure)

(11.62a)

(11.62b)

If (θ1 + β) > π, Equation 11.61 yields both positive and negative pressures, leading to film rupture aspreviously discussed. At the film-cavity interface the Swift–Stieber conditions, Equation 11.35, are usuallyapplied.

Equation 11.61 and its boundary conditions, Equation 11.62, contain three parameters, (L/D), θ1, andβ. The only additional parameter of the problem, ε, appears in the definition of the lubricant filmgeometry

(11.63)

Thus the journal bearing problem is uniquely characterized by the parameter set

(11.64)

The first two parameters of this set define bearing geometry, while the last two characterize the geometryof the film. Having selected parameter set (11.64), we can find the pressure by solving the system consistingof Equations 11.61, 11.62, and 11.63. The nondimensional lubricant force components are obtained from(note the limits of integration on z)

(11.65a)

(11.65b)

and the attitude angle from

(11.66)

Knowledge of the force components fR, fT thus enables us to determine both the magnitude, f =, and the direction, φ, of the load the lubricant film will support under the specified conditions.

Instead of characterizing the oil-film force this way, however, it has been customary to employ an alternaterepresentation of the Sommerfeld number

∂∂

∂∂

+

∂∂

∂∂

= π ∂∂θ θ θ

Hp D

L zH

p

z

H3

2

3 12

p z= = ±0 1 at

p = = +0 1 1 at θ θ θ β,

Hx

R= + = + +

≤ ≤1 1 1 1ε θ ε θ θ θ βcos cos ,

L D , , ,β ε θ1

f p d dzR

cav

= ∫∫1

2 10

1

cosθ θθ

θ

f p d dzT

cav

= ∫∫1

2 10

1

sinθ θθ

θ

φ = arctanf

fT

R

fR2 fT

2+

(11.67)

and the offset parameter (α/β)

(11.68)

where α is the position of the load-line relative to the leading edge of the pad (Figure 11.11).As shown above, under isothermal conditions the computational problem is defined by the “design

parameters” L/D, β, ε, θ1, while the computations yield the “performance, parameters” L/D, β, S, α/β.The designer, however, must proceed in an inverse manner, so to speak. (In the following, we drop theparameters L/D and β from the list.)

What is known at the design stage are the pad geometry and the performance requirements, i.e., themagnitude and direction of the external load, the shaft speed and the viscosity of the lubricant, and itis easy for the designer to define the couple S, α/β; but the designer has no way of determining thecouple ε, θ1 that is required in order to compute the minimum film thickness, often the controllingparameter.

Let Ω(ε, θ1) and Ψ(ε, θ1) represent the Sommerfeld number and the offset parameter, respectively,obtained by the analyst at some given ε, θ1, and let S and α/β be the values that are requested by thedesigner. The task for the analyst is then to find that particular ε, θ1 that yields

(11.69a)

(11.69b)

We can solve this pair of nonlinear equations for the unknowns ε, θ1 by iteration, e.g., using Newton’smethod

(11.70)

The performance curves in Figure 11.12, taken from Raimondi and Szeri (1984), are for centrallyloaded fixed-pad partial bearings of L/D = 1, β = 160° and various values of the Reynolds number. Theturbulent data were obtained from Equation 11.21.

Example 2. Journal Bearing

Calculate the performance of a centrally loaded partial arc journal bearing given the following data

To start the design process, we assume an effective viscosity

S f=1

αβ β

θ φ= π − +( )[ ]11

S − ( ) =Ω ε θ, 1 0

αβ

ε θ− ( ) =Ψ , 1 0

∂∂

∂∂

∂∂

∂∂

−−

=−

−

= …( ) −( )( ) −( )

Ω Ω

Ψ Ψ

Ω

Ψε θ

ε θ

ε εθ θ

αβ

1

1

1

1 1

11 2 3

n n

n n

S, , , ,n

β ωρ

= °( ) = =( )= × =

= = =

−

2 79 160 40 251

2 10 831

0 508 355 84

3 3

. sec, sec

,

. .

rad N rad

C R kg m

D L m W kN

ISO VG 32

µ = × ⋅ =−3 447 10 0 41482 2. ; .Pa s cm sν

For Reynolds and Sommerfeld numbers, respectively, we obtain

FIGURE 11.12 Performance of an (L/D) = 1, β = 160° partial journal bearing: (a) minimum film thickness, (b)position of minimum film thickness, (c) power loss, (d) lubricant inflow, (e) lubricant side flow. (From Raimondi,A.A. and Szeri, A.Z. (1984), Journal and thrust bearings, in CRC Handbook of Lubrication, Vol. 2, Booser, E.R., (Ed.),CRC Press, Boca Raton, FL. With permission.)

Re. .

.,

.

..

= × × = <

= × ××

( ) =−

25 4 251 0 0508

0 4148781 1000

3 447 10 40

1 3789 10500 0 25

2

6

2

the flow is laminar

S

FIGURE 11.12 (continued)

Entering Figure 11.12c, d, and e in succession we find

The temperature rise across the bearing is calculated from a simple heat balance, Equation 11.28,

and, assuming an inlet temperature of Ti = 45 C, yields the operating temperature

On Figure 11.13 plot the point A(52.5, 34.47). Note that 1 Pa · s = 1000 cP.Assume another effective viscosity: µ = 6.8948 × 10–3 Pa · s, v = 0.083 cm2/s. The corresponding

performance parameters are

FIGURE 11.12 (continued)

H MW

Q m s

Q m ss

= × π × × × × × =

= × × × × × = ×

= × × = ×

−

− −

− −

3 05 2 355 84 10 40 5 08 10 0 1386

3 2 0 254 5 08 10 40 0 508 8 39 10

1 32 2 6219 10 3 46 10

3 4

4 3 3

3 3 3

. . . .

. . . . .

. . .

∆T C= ×× −( ) ×

=−

1 386 10

1 39 10 8 39 3 46 2 1014 97

5

6 3

.

. . ..

T T T Cs i= + × =0 5 52 5. .∆

Re ,

.

.

= >

=

= × )−

3904 1000

0 05

8 39 10 3 3

turbulent flow

S

Q m s

The operating temperature is Ts = 45 + 14.94/2 = 52.47 C, and we plot point B(52.5, 6.9) in Figure 11.13.The line drawn from A to B intersects the ISO VG 32 line at

giving the effective temperature and effective viscosity that is consistent with the operating conditionsof the bearing under the assumption that a single viscosity can portray bearing performance. The finalReynolds number, Sommerfeld number, and minimum film thickness are

11.3.1.2 Pivoted-Pad Journal Bearings

In many applications the bearings are constructed of several identical pads. Multipad bearings, in general,dissipate less energy than full bearings. However, they too are unable to damp out unwanted rotorvibrations. This becomes a problem especially when attempting to operate the rotor in the neighborhoodof a system critical speed. For this reason the pads are often pivoted in one point or along a line. Pivoted

FIGURE 11.13 ISO viscosity grade for lubricating oils. Points A, B, and the line drawn through them, refer toExample 2. (From Raimondi, A.A. and Szeri, A.Z. (1984), Journal and thrust bearings, in CRC Handbook of Lubri-cation, Vol. 2, Booser, E.R. (Ed.), CRC Press, Boca Raton, FL. With permission.)

.

.

.

= ×

=

=

−4 93 10

0 123

14 94

3 3Qs m s

H MW

T C∆

T C Pa ss s= µ = × ⋅−52 5 1 7 10 2. . and

Re , . , . . .= = = × =1583 0 123 0 4 0 0508 0 0203S h cmn

pads are free to swivel and follow the motion of the rotor, maintaining, at all times of normal operation,a load-carrying lubricant film between rotor and pad. Other advantages of pivoted-pad bearings are thatthe clearance can be closely controlled by making the pivots adjust radially, thus enabling operation withsmaller clearances than considered appropriate for a plain journal bearing, and the pads can be preloadedto achieve relatively high stiffness (important with a vertical rotor).

Figure 11.14 shows a pad machined to radius R + C (position 1). Assuming the pad does not tilt, thefilm thickness is uniform and equal to C, and the pad develops no hydrodynamic force. If the pad is nowmoved inward the distance (C – C ′) into position 2, the film thickness will no longer be uniform; theresulting hydrodynamic force preloads the pad. The degree of preload is indicated by the preload coefficientm = (C – C ′)/C, the value of which varies between m = 0 for no preload, to m = 1, for metal-to-metalcontact between pad and shaft.

Figure 11.15 is a schematic of the geometrical relationships in a pivoted-pad journal bearing. Thesymbols OB and OJ mark the positions of the bearing center and the instantaneous position of the shaftcenter. The center of the pad is at Ono when the pad is unloaded and moves to On when it is loaded. Thepivot point P is located at angle ψ relative to the vertical load line WB. The eccentricity of the journalrelative to the bearing center, OB, is e0 = C ′ε0 and its attitude angle φ0. Relative to the instantaneous padcenter, OJ, the journal eccentricity is e = Cε and the attitude angle φ, the latter measured from the loadline that, by necessity, intersects the pivot P. The radii of journal, pad, and pivot circle are R, , and

, respectively. From geometric consideration we have (Lund, 1964)

(11.71)

where m is the preload coefficient defined earlier, and the index n refers to the nth pad.Once the position of the journal relative to the bearing is specified, i.e., (ε0,φ0) is given, Equation 11.71

together with the constraint that the load must pass through the pivot are sufficient to determine (εn,φn),the position of the nth pad, n = 1,…,N, relative to the journal. Knowing (εn,φn) makes it possible tocalculate individual pad performance that can then be summed to yield the performance characteristicsof the bearing. Unfortunately (ε0,φ0) is not known at the design stage, and the best the designer can dois assume ε0 and use the condition that WB is purely vertical to calculate the corresponding φ0. Thisprocedure is, at least, tedious. If, however, the pivots are arranged symmetrically with respect to the load-line, the pads are centrally pivoted and are identical, the journal will move along the load line WB and

FIGURE 11.14 Preloading of a pad: (1) as machined, (2) preloaded (Ob, Oj, Op, bearing, journal, pad center; rb,rp, R, bearing, pad, journal radius). (From Raimondi, A.A. and Szeri, A.Z. (1984), Journal and thrust bearings, inCRC Handbook of Lubrication, Vol. 2, Booser, E.R. (Ed.), CRC Press, Boca Raton, FL. With permission.)

OnP

OBP

ε φ ε ψ φ ε ψ φn n n n

C

Cmcos cos cos= − ′ − −( ) = − −( )1 0 0 0 0

φ0 ≡ 0. Figure 11.16 plots performance data for a five-pad pivoted-pad journal bearing (ε′0 = ε0/max(ε0),and for the five-pad bearing max(ε0) from geometry).

Example 3

Find the performance for a five-pad tilting-pad bearing for a horizontal rotor, given the following data

Bearing performance is calculated as follows:

The normalized bearing eccentricity ratio ε′0 = ε0/1.236 will be used to obtain stiffness and dampingcoefficients.

FIGURE 11.15 Geometry of a pivoted-pad journal bearing. (From Lund, J.W. (1964), Spring and damping coeffi-cients for the tilting pad journal bearing, ASLE Trans., 7, 342. With permission.)

β = °( ) == µ = × ⋅= == ′ = =

−

1 05 60 11 12

12 7 1 379 10

6 35 0 1658

0 0127 60

2

2

. .

. .

. .

. sec

rad W kN

D cm Pa s

L cm v cm s

C C cm N r

Re. .

.

.

.

.

..

. . . ,

. . . . ,

= × π × × = < ( )

= × ××

=

= × =

= × π × × × × × =

′ =

−

−

6 35 2 60 0 0127

0 1658183 1000

1 379 10 60

1 3789 10

6 35

0 01270 15

0 26 0 127 0 0033

3 9 2 11 12 10 60 1 27 10 2 08

0

2

6

2

3 4

0

laminar

from Figure 11.16a

from Figure 11.16b

S

h cm

H kW

n

ε .. ,67 from Figure 11.16c

FIGURE 11.16 Performance of a five-pad pivoted-pad journal bearing (L/D) = 1, β = 160°: (a) minimum filmthickness, (b) power loss, (c) normalized bearing eccentricity ratio. (From Raimondi, A.A. and Szeri, A.Z. (1984),Journal and thrust bearings, in CRC Handbook of Lubrication, Vol. 2, Booser, E.R. (Ed.), CRC Press, Boca Raton, FL.With permission.)

11.3.2 Thrust Bearings

Thrust bearings in their most elementary form consist of two inclined plane surfaces in relative motionto one another. The geometry of the bearing surfaces is commonly rectangular, to accommodate a linearmotion, or sector shaped, to support a rotation, but other geometries are possible.

11.3.2.1 The Plane Slider

A schematic of a fixed plane slider is shown in Figure 11.17. The gradient of the pad surface m = (h2 –h1)/B is used to scale Equation 11.18 and its boundary conditions

(11.72a)

FIGURE 11.16 (continued)

FIGURE 11.17 Schematic of a plain slider. (From Szeri, A.Z. (1998), Fluid Film Lubrication, Theory and Design,Cambridge University Press. With permission.)

∂∂

∂∂

+

∂∂

∂∂

= −x

xp

x

B

L zx

p

z3

2

34 1

(11.72b)

(11.72c)

Here

(11.72d)

The nondimensional force f is calculated from

(11.73)

Its values at various values of –x1 and aspect ratio (L/B) are displayed in Table 11.4a.The center of pressure, xp, is calculated from

(11.74)

Let–xp =

–x1 + δ represent the dimensionless coordinate of the center of pressure, i.e., the location of the

pivot for a pivoted pad, then δ, the dimensionless distance between pad leading edge and pivot, is given by

(11.75)

Table 11.4b lists pivot position δ at various values of –x1 and aspect ratio (L/B). The nondimensional flowvariable at inlet, x2, is listed in Table 11.4c.

TABLE 11.4a Nondimensional Oil-Film Force: f = Fh12/µULB 2

–x1/(L/B) 8/3 2 4/3 1 2/3 1/2 1/3

0.125 0.0458 0.0424 0.0360 0.0303 0.0216 0.0161 0.00990.20 0.0700 — — 0.0441 0.0303 0.0218 0.01270.25 0.0825 0.0755 0.0623 0.0509 0.0344 0.0243 0.01390.50 0.1155 0.1047 0.0845 0.0675 0.0337 0.0298 —0.60 0.1205 — — 0.0697 0.0447 0.0303 0.01620.80 0.1238 — — — 0.0448 0.0301 0.01591.00 0.1225 0.1110 0.0879 0.0692 0.0437 0.0291 0.01521.25 0.1179 — — 0.0662 0.0415 0.0276 0.01441.67 0.1081 — — 0.0604 0.0377 0.0249 0.01292.00 0.1005 0.0903 0.0715 0.0559 0.0348 0.0230 0.01192.50 0.0901 — — 0.0500 0.0311 0.0205 0.01064.00 0.0676 0.0607 0.0479 0.0374 0.0232 0.0153 0.00785.00 0.0577 0.0518 0.0409 0.0319 0.0198 0.0130 0.006710.0 0.0330 0.0296 0.0234 0.0182 0.0113 0.074 0.0038

Source: Szeri, A.Z. and Powers, D. (1970), Pivoted plane pad bearings: a varia-tional solution, ASME Trans., Ser. F., 92, 466-72.

p x ,±( ) =1 0

p x z p x z1 2 0, ,( ) = ( ) =

x Bx y mBy zL

z pU

Bmp= = = = µ

, , , 2

62

fFh

ULBx p x z dxdz

x

x

≡µ

= ( )+

∫∫12

2 12

1

0

1

61

1

,

Fx xp x z dxdzpx

x

L

L

= ( )∫∫−,

1

2

2

2

δ = − + ( )+

∫∫xx

fxp x z dx dz

x

x

112 1

0

16

1

1

,

Example 4. Plane Slider

Calculate the performance of a fixed-pad slider bearing if the following data are specified

From Equation 11.73 we obtain the relationship between the external load and the dimensionless lubri-cant force

TABLE 11.4b Optimum Pivot Position: δ = (–xp ––x1)

–x1/(L/B) 8/3 2 4/3 1 2/3 1/2 1/3

0.125 0.2900 0.2858 0.2766 0.2663 0.2452 0.2268 0.20000.20 0.3236 — — 0.3021 0.2825 0.2649 0.23880.25 0.3397 0.3363 0.3286 0.3196 0.3009 0.2841 0.25880.50 0.3876 0.3851 0.3793 0.3725 0.3579 0.3444 —0.60 0.3992 — — 0.3854 0.3721 0.3597 0.34080.80 0.4161 — — — 0.3931 0.3825 0.36621.00 0.4281 0.4264 0.4226 0.4180 0.4081 0.3988 0.38441.25 0.4388 — — 0.4302 0.4217 0.4137 0.40131.67 0.4510 — — 0.4440 0.4372 0.4307 0.42072.00 0.4576 0.4567 0.4544 0.4516 0.4456 0.4399 0.43112.50 0.4649 — — 0.4599 0.4549 0.4503 0.44314.00 0.4568 0.4763 0.4750 0.4736 0.4704 0.4674 0.46315.00 0.4811 0.4806 0.4796 0.4784 0.4758 0.4735 0.470110.0 0.4883 0.4878 0.4868 0.4854 0.4820 0.4780 —

Source: Szeri, A.Z. and Powers, D. (1970), Pivoted plane pad bearings: a varia-tional solution. ASME Trans., Ser. F, 92, 466-72.

TABLE 11.4c Nondimensional Flow at Inlet: qi = Qx=β/ULh1

–x1/(L/B) 8/3 2 4/3 1 2/3 1/2 1/3

0.125 1.6990 1.9662 2.4632 2.8866 3.4626 3.7744 4.07100.20 1.3390 — — 2.0402 2.3825 2.5670 2.73620.25 1.2059 1.3318 1.5588 1.7544 2.0156 2.1584 2.29040.50 0.9168 0.9717 1.0735 1.1584 1.2728 1.3360 —0.60 0.8621 — — 1.0558 1.1485 1.1997 1.27890.80 0.7879 — — — 0.9905 1.0270 1.06251.00 0.7396 0.7638 0.8078 0.8449 0.8948 0.9229 0.95051.25 0.6990 — — 0.7791 0.8176 0.8393 0.86081.67 0.6550 — — 0.7123 0.7393 0.7549 0.77042.00 0.6322 0.6427 0.6621 0.6783 0.7004 0.7130 0.72602.50 0.6083 — — 0.6439 0.6611 0.6709 0.68094.00 0.5702 0.5750 0.5838 0.5912 0.6014 0.6072 0.61325.00 0.5569 0.5607 0.5676 0.5733 0.5813 0.5859 0.590610.0 0.5293 0.5310 0.5343 0.5370 0.5409 0.5431 0.5454

Source: Szeri, A.Z. and Powers, D. (1970), Pivoted plane pad bearings: a variationalsolution, ASME Trans., Ser. F, 92, 466-72.

W kN L cm B cm

U m s Pa s

= = =

= µ = ⋅

16 013 20 32 7 62

30 5 0 04137

. , . , . ,

. , .

fh

h m= × ×× × ×

= × ( )16 013 10

0 04137 30 5 0 2032 0 07621 0756 10

312

2

712.

. . . ..

For (L/B) = 8/3, the dimensionless force has its maximum value of f = 0.1238 at –x1 = 0.8, yielding thelargest value of the exit film thickness at which this bearing can carry the assigned load.

The corresponding surface slope is calculated from

The film thickness at inlet is h2 = 0.01073 + 0.00176 × 7.62 = 0.024 cm. The lubricant flow-rate at inletis Q = 0.7879 × 30.5 × 0.2032 × 1.073 × 10–4 = 524 × 10–6 m3/s and the optimum pivot position is xp –x1 = 0.4161 × 7.62 = 3.12 cm from the trailing edge.

11.3.2.2 Annular Thrust Bearing

The fixed-pad slider bearing is the most basic configuration. If the pads are arranged in an annularconfiguration with radial oil distribution grooves, a complete thrust bearing (Figure 11.18a) is achieved.

FIGURE 11.18 Fixed-pad thrust bearing: (a) arrangement of pads, (b) pad geometry. (From Raimondi, A.A. andSzeri, A.Z. (1984) Journal and thrust bearings, in CRC Handbook of Lubrication, Vol. 2, Booser, E.R. (Ed.), CRC Press,Boca Raton, FL. With permission.)

h m1 7

1 2

40 1238

1 0756 101 073 10=

×

= × −.

..

m = ×× ×

=−

−1 073 10

0 8 7 62 100 00176

4

2

.

. ..

Approximate performance calculations of this bearing can be made by relating the rectangular sliderbearing (width B, length L) to the sector configuration (Figure 11.18b).

The pads of pivoted-pad thrust bearings are supported on pivots (Figure 11.19). As the location of thepivot fixes the location of the center of pressure, on loading a pad will swivel until it occupies a positionthat places the center of pressure over the pivot. While performance of a pivoted pad is identical to thatof a fixed pad designed with the same surface slope, the pivoted-type bearing has the advantages of (a)being self-aligning, (b) automatically adjusting pad inclination to optimally match the needs of varyingspeed and load, and (c) having the capability of operating in either direction of rotation. Theoretically, thepivoted pad can be optimized for all speeds and loads by judicious pivot positioning, whereas the fixed-pad bearing can have optimum performance only for one operating condition. Although pivoted-padbearings involve somewhat greater complexity, standard designs are readily available for large machines.

The Reynolds equation, Equation 11.19, in cylindrical coordinates (r,θ) takes the form

(11.76)

This equation can be solved numerically. The resulting performance charts (Figures 11.20a to 11.20f)are conveniently entered on a trial basis with an assumed tangential slope parameter mθ and radial slopeparameter mr. Load capacity, minimum film thickness, power loss, flow and pivot location (if pivoted-pad type) are then determined and the procedure, if necessary, repeated to find an optimum design.Figures 11.20e and 11.20f provide pivot locations for tilting pad sectors.

The thrust bearing charts were prepared for a ratio of outside radius to inside radius of 2 and a sectorangular length of 40°. While this angle corresponds to seven sectors to form a full thrust bearing, theresults should generally give a preliminary indication of performance of other geometries with the samesurface area and mean radius. For other pad geometries see Pinkus and Sternlicht (1961).

Example 5. Sector Thrust Pad

Calculate thrust pad sector performance when given the following:

FIGURE 11.19 Pivoted-pad thrust bearing. Source: Raimondi, A. A. and Szeri, A. Z. 1984. Journal and thrustbearings. (From Raimondi, A.A. and Szeri, A.Z. (1984) Journal and thrust bearings, in CRC Handbook of Lubrication,Vol. 2, Booser, E.R. (Ed.), CRC Press, Boca Raton, FL. With permission.)

∂∂

∂∂

+ ∂∂

∂∂

= µ ∂∂r

rhp

r rh

pr

h3 316

θ θω

θ

β γ

γ θ

= ° = ( )= = × −

40 0

13 97 5 82 1024

r

R cm rad

no radial tilt

. .

Calculating first

Enter Figures 11.20a to 11.20d with mθ = 0.8 and mr = 0.0

FIGURE 11.20 Performance charts for fixed or tilting pad sector: (a) load capacity, (b) minimum film thickness,(c) power loss, (d) flow, (e) center of pressure (tangential location), and (f) center of pressure (radial location).(From Raimondi, A.A. and Szeri, A.Z. (1984) Journal and thrust bearings, in CRC Handbook of Lubrication, Vol. 2,Booser, E.R. (Ed.), CRC Press, Boca Raton, FL. With permission.)

ω

= µ = × ⋅

= µ = =( )−6 985 1 379 10

50 8 50 314

12

c

R cm Pa s

h m N r rad pad center

. .

. , sec sec

m R hcθ θγ= ( ) = × ×( ) × × =− − −1

2 6 46 985 10 50 8 10 5 82 10 0 80. . . .

W kN

h m

= × × × × × −( ) ×( ) =

= × × = µ

− −

−

0 058 6 1 379 10 314 0 1397 0 06985 50 8 10 13 9

0 45 50 8 10 22 86

2 4 62

6

. . . . . .

. . .min