Chapter 11: Hydrodynamic and Elastohydrodynamic Lubrication

11
Hydrodynamic and
Elastohydrodynamic
Lubrication
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 Lubrication
Journal Bearings • Thrust Bearings
11.4 Dynamic Properties of Lubricant Films
Linearized Spring and Damping Coefficients • Stability of a
Flexible Rotor
11.5 Elastohydrodynamic Lubrication
Andras Z. Szeri
University of Delaware
Contact Mechanics • Dimensional Analysis • FilmThickness Design Formulas • Minimum Film Thickness
Calculations
The friction force generated when two bodies rub against each other can be drastically decreased, and
the 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 much
experience with them. However, almost any material can be a lubricant. As da Vinci observed 400 years
ago, “All things and anything whatsoever, however thin it be, which is interposed in the middle between
objects that rub together lighten the difficulty of this friction.” Nevertheless, in this chapter lubrication
with 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 equations 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 denote
the characteristic dimensions across the film thickness and the ‘plane’ of the film, respectively, as indicated
in 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 into
a single equation in lubricant pressure, the so called the Reynolds equation. The theory of lubrication is
concerned with the solution of the Reynolds equation, sometimes in combination with the equation of
energy, under various lubricating conditions. When the surfaces are deformable, as in rolling contact
bearings or human and animal joints, the equations of elasticity and the pressure dependence of lubricant
viscosity must also be included in the solution of the problem.
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FIGURE 11.1
11.1
Schematic of lubricant film.
Basic Equations
The flow of an incompressible, constant viscosity fluid is governed by the equations of motion, the socalled Navier–Stokes equation (Szeri, 1998),
 ∂v

ρ + v ⋅∇v  = −∇p + µ∇ 2 v + ρf
 ∂t

(11.1)
div v = 0
(11.2)
(
)
and the equation of continuity
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 plane
and V* perpendicular to it
x = L xz x , y = L y y , z = L xz z
(11.3a)
u = U ∗u , v = V∗v , w = U∗w
(11.3b)
We note that the characteristic velocity in the y direction, V*, cannot be chosen independent of U*, as
the terms in the equation of continuity must balance. Substitution of Equation 11.3 into Equation 11.2
yields
∂u V∗L xz ∂v ∂w
+
+
=0
∂x U∗L y ∂y ∂z
(11.4)
indicating that the terms of the continuity equations will balance provided that
()
V∗L xz
=O 1
U∗L y
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or
L 
V∗ =  y  U∗
 L xz 
(11.5)
This establishes the velocity scale in the y direction. To nondimensionalize pressure and time we choose
p = re
p
,
ρU∗2
t = Ωt
(11.6)
–
The reduced Reynolds number, re, and the nondimensional frequency, Ω, have the definition
 L LU
re =  y  y ∗
 L xz  ν
(
 L  L y L xz Ω
and Ω =  y 
ν
 L xz 
)
where Ω is the characteristic frequency of the system. For journal bearings, the characteristic frequency
is related to the rotational velocity of the journal. Taking the shaft surface speed as the characteristic
velocity, 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 continuity
in terms of the normalized quantities as
2
 ∂u
∂u
∂u
∂u 
∂p ∂2u  L y   ∂2u ∂2u 
+u
+v
+w  = −
+
+
re 
+
∂x
∂y
∂z 
∂x ∂y 2  L xz   ∂x 2 ∂z 2 
 ∂t
 Ly 
L 
 xz 
2
2


2
2
2


r  ∂v + u ∂v + v ∂v + w ∂v  − ∂ v − L y  ∂ v + ∂ v   = − ∂p

 e  ∂t
∂y
∂x
∂y
∂z  ∂y 2  L xz   ∂x 2 ∂z 2  


(11.7a)
(11.7b)
2
 ∂w
∂w
∂w
∂w 
∂p ∂2w  L y   ∂2w ∂2w 
+
+u
+v
+w
=
−
+
+
re 

∂x
∂y
∂z 
∂z ∂y 2  L xz   ∂x 2 ∂z 2 
 ∂t
(11.7c)
∂u ∂v ∂w
+
+
=0
∂x ∂y ∂z
(11.8)
11.1.1 Lubrication Approximation
Resulting from the normalization in Equations 11.3 and 11.6, we anticipate that each of the variable
terms in Equations 11.7 and 11.8 are of the same order, say O(1). But then the relative importance of
the variable terms is decided only by the magnitude of the dimensionless parameters (Ly /Lxz) and re that
multiply them.
Under normal conditions lubricant films are thin relative to their lateral extent. For liquid lubricated
bearings (Ly /Lxz) = O(10–3). For gas bearings a film thickness of h ≈ 2.5 µm is not uncommon. The socalled lubrication approximation of Reynolds is developed under the assumptions that (Ly /Lxz) → 0 and
re → 0. Under these conditions Equations 11.7 indicates that
( )
∂p
= O 10 −6
∂y
and Equations 11.7 and 11.8 reduce to
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while
∂p ∂p
≈
=O 1
∂x ∂z
()
(11.9)
∂p
∂ 2u
=µ 2,
∂x
∂y
∂p
= 0,
∂y
∂p
∂ 2w
=µ 2
∂z
∂y
(11.10)
∂u ∂v ∂w
+ +
=0
∂x ∂y ∂z
(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 pressure
is invariant in the y direction, i.e., across the film. But then the first and last parts of Equation 11.10 can
be integrated to obtain
u=
1 ∂p 2
y + Ay + B
2µ ∂x
(11.12a)
w=
1 ∂p 2
y + Cy + D
2µ ∂z
(11.12b)
The integration constants A, B, C, D are evaluated with the aid of the boundary conditions on velocity
(Figure 11.1),
u = U1, w = 0 at y = 0
(11.13)
u = U 2 , w = 0 at y = h
to obtain the velocity components in the plane of the bearing
u=
 y
1 ∂p 2
y
y − yh + 1 − U1 + U 2
2µ ∂x
h
 h
(
)
(
1 ∂p 2
w=
y − yh
2µ ∂z
(11.14)
)
The pressure distribution appearing in Equation 11.14 is not arbitrary but must be such that the equation
of 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
[v] (
h x,t
0
)
=−
−
∂  1 ∂p

∂x  2µ ∂x
∂
∂x
∫


( )

∂ 1 ∂p
∫ ( y − yh)dy  − ∂z  2µ ∂z ∫ ( y − yh)dy 
2
0
h x,t
2
0
( ) 
h x,t
0
( )
h x,t
∂h
y
y 
1 −  U1 + U 2  dy + U 2
∂x
h
h




(11.15)
Recognizing that
[v] (
h x,t
y =0
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)
(
)
= − V1 − V2 =
dh
dt
(11.16)
where V1, V2 are the normal, i.e., approach, velocities of the bearing surfaces, we obtain the Reynolds
equation that governs the pressure distribution in a lubricant film (Szeri, 1998)
(
)
∂ U1 + U 2
∂  h 3 ∂p  ∂  h 3 ∂p 
∂h
+ 12 V2 − V1
+ 
+ 6h
= 6 U1 − U 2



∂x
∂x  µ ∂x  ∂z  µ ∂z 
∂x
(
)
(
)
(11.17)
If the relative motion between the surfaces in contact is pure translation, as is in conventional thrust
bearings, Equation 11.17 can be cast in the form
∂  h 3 ∂p  ∂  h 3 ∂p 
∂h
+ 
= 6U 0
+ 12V0



∂x  µ ∂x  ∂z  µ ∂z 
∂x
(11.18)
where
U 0 = U1 − U 2 ,
V0 = V2 − V1
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
∂  h 3 ∂p  ∂  h 3 ∂p 
∂h
+
= 6U 0
+ 12V0
∂x  µ ∂x  ∂z  µ ∂z 
∂x
(11.19)
Here U0 = U1 + U2 and V0 = V2 – V1. Thus while in thrust bearings it is the difference in tangential
velocities that creates pressure, in journal bearings it is their sum; if journal and bearing rotate in opposite
directions but with the same speed, there is no pressure generated. However, for both journal and thrust
bearings positive pressures are generated only when the film is convergent, both in space and time.
U0
∂h
∂h
< 0,
<0
∂x
∂t
(11.20)
11.1.2.1 Turbulent Flow
For turbulent flow of the lubricant, most theories currently in use yield a formally similar Reynolds
equation, but now the film thickness is weighted by the functions kx1/3 and kz1/3, respectively, in the
direction of relative motion and perpendicular to it (Szeri, 1998). For a journal bearing, the turbulent
Reynolds equation takes the form
∂  h 3 ∂P  ∂  h 3 ∂P  1 ∂h
+
= U
∂x  µk x ∂x  ∂z  µkz ∂z  2 0 ∂x
(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
(
k x = 12 + 0.53 k 2 Re h
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)
0.725
(11.21b)
(
kz = 12 + 0.296 k 2 Re h
)
0.65
(11.21c)
and the local Reynolds number, Reh = Rωh/v, is calculated from the reduced Reynolds number through
Reh = 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 the
Patir–Cheng flow factors (Patir and Cheng, 1978), obtained by numerically solving the Reynolds equation
–
for 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 factor
φx, for example, is calculated from
 ∂p 
φx = h 
 ∂x 
(
(

p −p
h 3 2 1
x2 − x1

) 
) 
where the overscore bar indicates statistical averaging. In the next step, the flow factors themselves are
averaged, having been calculated for a large number of statistically identical microbearings. The Reynolds
equation for rough surfaces now takes the form
h 3 ∂p  ∂  h 3 ∂p  U1 + U 2 ∂h U1 − U 2 ∂φs ∂h
∂ 
+
=
+
+
φx
φ
σ

∂x  12µ ∂x  ∂z  z 12µ ∂z 
∂x
∂x ∂t
2
2
(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 practical
bearing 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 is
used to increase the temperature of the lubricant. Increasing the temperature of the lubricant changes its
viscosity, 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 performance
can be adequately predicted when using an average or operating viscosity, obtained by iteration between
the 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 be
taken 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
ρ
( ) = − pdiv v + µΦ + div (k gradT )
d c vT
dt
(11.23)
If the fluid is incompressible cv = cp = c and –pdivv = 0 by Equation 11.2. Assuming further that the
heat conductivity is constant, Equation 11.23 becomes
 ∂2T ∂2T ∂2T 
 ∂T
∂T
∂T
∂T 
ρc 
+u
+v
+w
=
k

 ∂x 2 + ∂y 2 + ∂z 2  + µΦ
∂x
∂y
∂z 
 ∂t


© 2001 by CRC Press LLC
(11.24a)
FIGURE 11.2 Flow factors for rough surfaces, (a) pressure flow factor, (b) shear flow factor. (From Patir, N. and
Cheng, H.S. (1979), Application of average flow model to lubrication between rough sliding surfaces, ASME J. Lub.
Tech., 101, 220-30. With permission.)
In orthogonal Cartesian coordinates the dissipation function is
 ∂u  2  ∂v  2  ∂w  2   ∂u ∂v  2  ∂v ∂w  2  ∂w ∂u  2
+ 
Φ = 2  +   +    +  +  +  +
 +
 ∂x   ∂y   ∂z    ∂y ∂x   ∂z ∂y   ∂x ∂z 


© 2001 by CRC Press LLC
(11.24b)
Applying the lubrication approximation, Equations 11.3 and 11.6, we obtain
u
 ∂u  2  ∂w  2 
∂T
∂T
1 ∂2T
∂T
  + 
Λ
+v
+w
+
µ
=
 
∂x
∂y
∂z Pe ∂y 2
 ∂y   ∂y  


(11.25)
where the Peclet number and the dissipation number have the definition, respectively,
µ ωL 
Λ = ∗  xz 
ρcT∗  L y 
cµ
Pe = ∗ re ,
k
2
–
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 heat
conduction in the bearing, and the extended form of the Reynolds equation that is written for variable
viscosity (Suganami and Szeri, 1979)
(
(
)
)
ξ2 x , h, z 
∂  Γh 3 ∂P  ∂  Γh 3 ∂P 
∂ 
+
=
−
U
h

 + V0
0
∂x  µ∗ ∂x  ∂z  µ∗ ∂z 
∂x 
ξ1 x , h, z 
(11.26)
for journal bearings and
(
(
)
)
∂  Γh 3 ∂P  ∂  Γh 3 ∂P 
∂ ξ2 x , h, z
+ V0
+ 
= U0



∂x  µ∗ ∂x  ∂z  µ∗ ∂z 
∂x ξ1 x , h, z
(11.27)
for slider bearings. Here we employed the notation
) ((
) (
)

ξ2 x , h, z
Γ x , z = − ξ 2 x , y , z −
ξ1 x , y , z

ξ1 x , h, z
0 
1
( ) ∫ (
(
η
) ∫ µ( x , y , z ) ,
ξ1 x , η, z =
0
dy
(

) dy

η
) ∫ µ(xydy, y, z )
ξ2 x , η, z =
0
11.1.3.1 Effective Viscosity
In numerous applications, the temperature rise in the lubricant remains relatively small and uniform
throughout the film. In these cases the isothermal theory will provide useful approximations to actual
bearing performance, provided that an average or effective viscosity value that is compatible with the
bearing temperature rise is used in the computations (Kaufman et al., 1978). This calculation might be
based on the assumptions that:
1. All heat, H, generated in the film by viscous dissipation is removed from the bearing by the
lubricant.
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.
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This average temperature rise, ∆T, can be calculated from a simple energy balance
[ (
)]
∆t = H ρc Q − Qs 2
(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 hydrostatic
pad with the runner covering the recess and resting on the land, sealing the pressurized lubricant in the
recess. As there is no flow, the recess pressure equals the supply pressure. Increasing the supply pressure to
the point that the pressure force on the runner equals the external load, the runner lifts off the land and
flow out of the recess commences, as indicated in Figure 11.3b. As now there is flow, the pressure force over
the recess becomes somewhat smaller than before, but this decrease is compensated by the pressure force
developed 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) the
film 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 conjunction
with flow restrictors (compensated hydrostatic bearing). Flow restrictors are simple devices, such as capillaries, orifices, or flow-control valves, for which a small change in flow rate results in a large pressure
drop across the device. Figure 11.4 is a schematic of a two-pad hydrostatic bearing, equipped with flow
restrictors. The two recesses are fed from a common pressurized manifold. When the load is distributed
symmetrically, 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 now
less pressure drop across the restrictor due to the decreased flow rate, the recess pressure in pad A will
increase and yield a righting moment.
Ordinarily, hydrostatic bearings are characterized by constant film thickness h = 0. Then the Reynolds
equation valid for simple relative translation of the bearing surfaces, Equation 11.18, reduces to
∂2 p ∂2 p
+
=0
∂x 2 ∂z 2
(11.29)
FIGURE 11.3 Hydrostatic bearing schematics (a) before and (b) after lift-off. (From Szeri, A.Z. (1998). Fluid Film
Lubrication, Theory and Design, Cambridge University Press. With permission.)
© 2001 by CRC Press LLC
FIGURE 11.4 Schematic of a two-pad compensated hydrostatic bearing, (I) symmetrical, (II) unsymmetrical load
distribution. (From Szeri, A.Z. (1998), Fluid Film Lubrication, Theory and Design, Cambridge University Press. With
permission.)
The boundary conditions that accompany Equation 11.29 are
p = pr on Γ1
p = pa on Γ2
(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 curvilinear
coordinate systems where Laplace’s equation separates, provided that Γ1, Γ2 are composed of coordinate
lines. 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 the
bearing pad, the performance of hydrostatic bearings can be written in terms of a set of dimensionless
parameters: the load factor af , the flow factor qf , and the loss factor hf . If W, Q, Hp , Hf represent the external
load, the lubricant flow, the pressure power loss, and the frictional power loss, respectively, then we can
write
W = a f Apr
(11.31a)
h3
p
µ r
(11.31b)
H p = qf
h3
p p
µ r s
(11.31c)
H f = hf
µU M2 A
h
(11.31d)
Q = qf
Here A is the combined area of pad and recess, ps is the supply pressure, pr is the recess pressure, and
UM 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 hydrostatic
pads, 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
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p = pr p,
r = R2r
(11.32)
and obtain
d  dp 
r  = 0
dr  dr 
(11.33a)
in place of Equation 11.29, while the boundary conditions take the form
R1
,
R2
p = 1 at r =
p = 0 at r = 1
(11.33b)
Integrating of Equation 11.33a and enforcing the boundary conditions, Equation 11.33b, lead to
ln r
ln R1 R2
p=
(
(11.34)
)
Having found the pressure distribution, we are ready to evaluate the performance of the pad.
Load capacity:
2

 R1 
2

W = πR pr + 2π rpdr = πR pr   +
 R2  ln R1 R2
R1

R2
∫
2
1
=
2
2
(
( ) Ap
2 ln( R R )
1

r ln rdr 

R2

) ∫
R1
(11.35a)
2
1 − R1 R2
r
2
1
Flow rate:
h
∫
Q = 2πrur dy =
h 3 pr
π
6 ln R2 R1 µ
)
(11.35b)
h 3 ps pr
π
µ
6 ln R2 R1
(11.35c)
0
(
where
ur =
(
1 dp
y y −h
2µ dr
)
is the (radial) flow velocity, obtained as in Equation 11.14.
Pumping power:
H p = psQ
=
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(
)
Frictional power loss:
Hf =
∫
R2
rωτdA =
(
1 − R1 R2
)
4
2
R1
µU M2 A
h
(11.35d)
where τ = µdur /dr is the shear stress on the runner.
A comparison of Equations 11.31 to 11.35 yields
af
( )
=
2 ln( R R )
(11.36a)
π
6 ln R2 R1
(11.36b)
2
qf =
hf =
2
1 − R1 R2
1
(
)
(
)
1 − R1 R2
4
(11.36c)
2
The total power loss is the sum of frictional loss and pumping power
HT = H p + H f = q f
µU M2 A
h3
pr ps + h f
µ
h
(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 the
viscosity, respectively, defined by the conditions
∂H T
= 0,
∂h
∂H T
=0
∂µ
(11.38)
Substitution of HT from Equation 11.37 into Equation 11.38 yields the optimum film thickness at
constant viscosity and the optimum viscosity at constant film thickness, respectively (Szeri, 1998)
14
hopt
 h µ 2U M2 A 
= f
 ,
 3q f pr ps 
µ opt
 q h4 p p 
=  f 2r s 
 hf U M A 
12
(11.39a,b)
Substituting Equation 11.39a into Equations 11.31c and 11.31d, we find that at constant viscosity the
total power loss is minimum when Hf /Hp = 3 and the total power consumption of the bearing is
H T ,hopt =
© 2001 by CRC Press LLC
4
3
34
(q h p p µ U A )
f
3
f r
2
s
6
M
3
14
(11.40a)
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, Cambridge
University Press. With permission.)
When Equation 11.39b is used, we find that Hf /Hp = 1 and the minimum power at constant film thickness
is
H T ,µopt = 2 q f h f Apr ps hU M
© 2001 by CRC Press LLC
(11.40b)
Obviously, the conditions Hf /Hp = 1 and Hf /Hp = 3 cannot be satisfied simultaneously. However, by
calculating the variation of HT when the ratio Hf /Hp is in the interval 1 ≤ Hf /Hp ≤ 3 we find

≤ 1.1398 
H T , hopt
Hf

≥1
 for 3 ≥
HT , µ
Hp

1.1547 ≥
≥1

H T , µopt

1≤
HT ,h
(11.41)
Equation 11.41 shows that as long as we keep 1 ≤ Hf /Hp ≤ 3, the total power will not exceed its optimal
value by more than 16%. But this is not the full story, as we should also keep in mind the temperature
rise as the result of dissipation. Assuming that all heat generated is used for increasing the temperature
of the lubricant and that none is conducted away, a simple energy balance yields
∆T =

1  h f µ 2U M2 A
+ ps 

4
ρc  q f h pr

(11.42)
Assuming, for the sake of simplicity, that ps /pr = const., the condition ∂∆T/∂ps = 0 assures us that the
temperature 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 resistance
to 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
W = a f Aps
RB
,
RC + RB
RB = µ q f h 3 , RC =
128µlC2
πdC4
(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
W
1+ ξ
=
W0 1 + ξX 3
(11.44)
where
ξ=
RC
RBo
and X =
h
h0
and the ratio of supply pressure to recess pressure is given by ps /pr = 1+ ξ. The dimensionless bearing
stiffness is obtained by differentiating W/W0 with respect to h/h0
λ≡−
© 2001 by CRC Press LLC
(
∂ W W0
) = 3ξ(1 + ξ)X
( ) (1 + ξX )
∂ h h0
3
2
2
(11.45)
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.)
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. The
applied load and the geometry specifies the recess pressure pr(1) = W(1)/af A. The pump will
supply 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 load
were 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 single
manifold without flow regulators, are geometrically identical but carry loads W (1) and W (2) >
W (1), respectively. The required recess pressures are
(1)
pr =
(1)
(2 )
W
W
(2 )
<
= pr
af A af A
At the start of operation the pump delivery pressure is increased from zero and reaches the lower of
the lift-off pressures pL(1) = W(1)/Ar, where Ar is the area of the recess. Now that the passage is open for
the lubricant, the delivery pressure will drop back to pr(1) and there it will stay. There is no way for the
pump to reach pL(2) = W(2)/Ar > pL(1) > pr1, the lift required by pad no. 2, i.e., pad no. 2 will not come
into operation at all.
(ii) Operation with capillary restrictor. The size of the restrictor to be installed in bearing no. 1 of the
previous example must be such that ∆p (1) + pr(1) > pL(2) at the required flowrate Q(1).
© 2001 by CRC Press LLC
TABLE 11.1 Capillary Restrictors
for Hydrostatic Pad
dC (cm)
lC (cm)
lC/dC
0.38
0.27
0.18
0.12
0.084
0.069
0.051
1184
302
60
12
2.8
1.3
0.38
3120
1100
333
96
18
17
7.4
Example 1. Hydrostatic Pad
Design an annular hydrostatic pad with the following requirements:
W = 44, 500 N , R1 = 6.35 cm, R2 = 12.7 cm,
h ≥ 50.8 µm, µ = 3.03 × 10 −2 Pa ⋅ s
For this geometry Equation 11.36 yields af = 0.54 and qf = 0.755, so that
pr =
44500
= 1.626 MPa
0.54 × 0.1272 π
(5.08 × 10 ) 1.627 × 10 = 5.3115 × 10
Q = 0.755
−6
3
6
3.03 × 10−2
−6
m3 s
We select a pump that delivers 6.309 cm3/sec. At this flowrate the film thickness is an acceptable
 6.309 × 10−6 × 3.03 × 10−2 
h=

6
 0.755 × 1.626 × 10

13
= 53.80 µm > 50.8 µm
We thus need a pump that delivers pL = 44,500/(0.0635)2 π = 3.5129 MPa at zero flow and 1.626 MPa at
Q = 6.31 cm3/s.
Assume a supply pressure ps = 2.0684 MPa at Q = 6.309 cm3/s, then from the Hagen–Poisseuille law
( )
lC cm =
(
)
π 2.0684 − 1.626 × 106
−2
128 × 3.03 × 10 × 6.309 × 10
−6
( )
dC4 = 5.68 × 104 × dC4 cm
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 generate
positive, i.e., load-carrying, pressure, the film must be convergent in the direction of relative motion
© 2001 by CRC Press LLC
TABLE 11.2
Advantages/Disadvantages of Flow Restrictors
Compensating Elements
Capillary
Orifice
Valve
Initial cost
Cost to fabricate, install
Space requirement
Reliability
Useful life
Availability
Tendency to clog
Serviceability
Adjustability
2
2
2
1
1
2
1
2
3
1
3
1
2
2
3
2
1
2
3
1
3
3
3
1
3
3
1
(Figure 11.1). No outside agency is required to create and maintain a load-carrying film, provided that
adequate lubricant is made available. Hydrodynamic films are easy to obtain; in fact they often occur
even when their presence is deemed undesirable, e.g., in hydroplaning of automobile tires on wet
pavement.
Prototypes of conformal hydrodynamic bearings are journal bearings and thrust bearings; these bearings might also be called “thick film” bearings. A journal bearing at load per projected bearing area of
1.36 MPa, speed 60 rps, and lubricated with an ASTM Grade 315 oil at 52°C would have a minimum
film thickness of the order of 88.4 µm. This film is thick in comparison to film found in counterformal
bearings, such as ball and roller bearings.
Journal bearings are designed to support radial loads on rotating shafts, while thrust bearings, as their
name implies, support axial or thrust loads. Although the mode of lubrication is identical in these two
bearing types, their geometry is sufficiently distinct for us to discuss them under separate headings; in
journal bearings, in general, the clearance geometry is convergent–divergent, and film rupture occurs in
the divergent part. Conventional thrust bearings, on the other hand, are purely convergent and their film
remains continuous throughout the clearance.
The third main heading of this section introduces the idea of lubricant film instability in the dynamic
sense; in most cases of application, the thermomechanical load on the bearing varies with time and, as
a 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 supports
the rotating shaft, as in Figure 11.7. The clearance space between bearing and shaft is filled with a
lubricant, usually a petroleum oil.
Under zero load and negligible body weight, the rotating journal is concentric with its bearing, but as
the external load on the journal is increased, the shaft center sinks below the center of the bearing. For
FIGURE 11.7 Schematic of a full journal bearing. (From Szeri, A.Z. (1998), Fluid Film Lubrication, Theory and
Design, Cambridge University Press. With permission.)
© 2001 by CRC Press LLC
FIGURE 11.8 Shaft trajectory during static loading. (From Szeri, A.Z. (1998), Fluid Film Lubrication, Theory and
Design, Cambridge University Press. With permission.)
isothermal operations, somewhat of a rarity in practice, the loading conditions on the shaft can be
characterized by a single dimensionless group, the Sommerfeld number S, defined by
S=
µN  R 
 
P C
2
where N is the rotation and R is the radius of the shaft, C is the radial clearance between bearing and
shaft, 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 used
even 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. On
increasing the external load or decreasing the speed, i.e., on decreasing S, the journal will move away
from its concentric position, the journal trajectory approximating a semicircular arc (Figure 11.8). Under
extreme load or vanishing speed, S → 0, metal-to-metal contact occurs at the point where the load line
cuts the bearing.
In most applications there is considerable heat generation, and the viscosity of the lubricant does not
remain uniform throughout the film. In such cases we need more than a single dimensionless group to
characterize journal bearing operations. In fact, under nonisothermal conditions the number of characterizing 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, the
radius of the shaft by Rs and the radius of the cylinder by Rb , so that C = Rb – Rs , then in normal design
practice (C/R) = O(10–3). This signifies that the lubrication approximation to Equations 11.1 and 11.2
is valid, and we can employ the Reynolds equation for evaluating bearing performance. The small value
of C/R further signifies that the curvature of the film can be neglected to the same order, thus the analysis
may be performed in an orthogonal Cartesian coordinate system. The (x,z) plane of this coordinate
system lies in the surface of the bearing, so that x is the “circumferential” coordinate and z is parallel to
the axis of the shaft. The y coordinate is normal to the “plane” of the bearing, i.e., it points toward the
center of the bearing arc, yet to the approximation involved here the y arrays appear parallel to one
© 2001 by CRC Press LLC
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 film
thickness between eccentric cylinders is well approximated by the formula
(
h = C + e cos θ = C 1 + ε cos θ
)
(11.46)
where e is the eccentricity, h the film thickness, and θ the angle measured from the line of centers in the
direction 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
[ (
h = C + e cos Ξ − φ + ψ
)]
(11.47)
Here φ is the angle between load-line and line of centers, the so-called attitude angle, and ψ defines the
load-line relative to the fixed position = 0. Equation 11.47 can be used to evaluate the right-hand side
of the Reynolds equation, Equation 11.19, to obtain (Szeri, 1998)
∂  h 3 ∂p  ∂  h 3 ∂p 
∂h
+ 
= 6 Rω + 12 e˙ cos θ + e φ˙ + ω W sin θ



∂x  µ ∂x  ∂z  µ ∂z 
∂x
[
(
)
]
(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/R
·
· φ,
provided that e,
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 amenable
to analytical solutions. These simplifications, known as the short-bearing and the long-bearing approximations to the Reynolds equation, must be used with great caution, however, as they may yield incorrect
performance parameter values. Before discussing these approximations to Equation 11.49 we shall make
the equation nondimensional.
The Reynolds equation is nondimensionalized via the substitutions
2
 R
L
x = Rθ, z = z , h = CH = C 1 + ε cos θ , p = µN   p
2
C
(
)
(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
2
∂  3 ∂p   D  ∂  3 ∂p 
∂H
H
 + 
H
 = 12π
∂θ 
∂θ   L  ∂z 
∂z 
∂θ
The individual terms on the left-hand side of Equation 11.50
2
 D  ∂  3 ∂p 
∂  3 ∂p 
H
 and  
H

∂θ 
∂θ 
∂z 
 L  ∂z 
© 2001 by CRC Press LLC
(11.50)
represent the rate of average (across the film thickness) pressure flow in the circumferential and axial
directions respectively, the sum of which is balanced by the shear flow
12π
∂H
∂θ
By the normalizing transformation, Equation 11.49, each of the variable terms of Equation 11.50 are of
the same order of magnitude; thus for long-bearings for which (D/L)2 → 0 the following approximation
is acceptable
d  3 d p
dH
 = 12π
H
dθ 
dθ 
∂θ
(11.51)
whereas for short-bearings (L/D)2 → 0, and we may approximate Equation 11.50 in the form
2
 L  ∂H
∂  3 ∂p 
H
 = 12π 
∂z 
∂z 
 D  ∂θ
(11.52)
When solved with zero pressure boundary conditions, the pressure distributions specified by
Equation 11.50 or its approximations, Equations 11.51 and 11.52, are 2π-periodic functions, antisymmetric with respect to the position of minimum film thickness, θ = π. They yield negative pressures of
the 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 short
distance downstream from the position of the minimum film thickness. Though the cavitation zone
might be preceded by a short range of subambient pressures, in most performance calculations this region
of subambient pressures is disregarded.
The boundary condition at film-cavity interface is complicated, particularly when dynamic loading
conditions prevail, and is still under investigation. Most computer calculations are based on the so-called
Swift–Stieber boundary conditions
p=
∂p
= 0, at θ = θcav
∂θ
(11.53)
–
that preserve flow continuity at the film-cavity interface. Here θcav = θcav (z)
denotes the angular position
–
of 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,
–
and only with some difficulty in the long-bearing
as the latter is governed by a differential equation in z,
approximation. The closest we can come to Equation 11.53 when using the short-bearing approximation
is to disregard negative (below ambient) pressures in calculating bearing performance, i.e., assume the
film to cavitate at the position of minimum film thickness. The conditions
()
p(θ) = 0,
p θ ≥ 0,
0≤θ≤π
π ≤ θ ≤ 2π
(11.54)
are used extensively for performance calculations in both short-bearing and long-bearing approximations
and are known as the Gümbel boundary conditions. We will evaluate bearing performance, for both shortbearing and long-bearing approximations, under the conditions specified in Equation 11.54, but for finite
bearings we use the Swift–Stieber conditions, Equation 11.35. Figure 11.9 displays pressure distribution
under Sommerfeld, Gümbel, and Swift–Stieber boundary conditions.
© 2001 by CRC Press LLC
FIGURE 11.9 Journal bearing pressure distribution under (a) Sommerfeld, Gümbel, and (b) Swift–Stieber boundary conditions. (From Szeri, A.Z. (1998), Fluid Film Lubrication, Theory and Design, Cambridge University Press.
With permission.)
Bearing performance is calculated from the pressure distribution by substitution into the following
formulas
Load capacity:
The force balance (Figure 11.10)
W cos φ + FR = 0
(11.55)
−W sin φ + FT = 0
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
( ) ∫ ∫
Rθcav
( ) ∫ ∫
Rθcav
2
FR ≡ f R LDµN R C =
2
FT ≡ fT LDµN R C =
L 2
−L 2 0
L 2
−L 2 0
p cosθdxdz
(11.56a)
p sinθdxdz
(11.56b)
Equations 11.56 define the he nondimensional force components fR , fT . The Sommerfeld number
reemerges here as the inverse of the nondimensional pressure force
© 2001 by CRC Press LLC
FIGURE 11.10 Journal bearing nomenclature. (From Szeri, A.Z. (1998), Fluid Film Lubrication, Theory and Design,
Cambridge University Press. With permission.)
2
µN  R 
2
2
S≡
  = f R + fT
P C
(
)
−1 2
(11.57)
Attitude angle:
The attitude angle, i.e., the angle between the load vector and the line of centers is given by
fT
fR
φ = arctan
(11.58)
Friction variable:
The friction force exerted on the shaft is found from
C
Fµ ≡ c µ   W =
 R
L
∫∫
0
2 πR
0
τ xy h ( x ) dxdz
(11.59)
where τxyh(x) is the shear stress on the shaft and cµ is the friction variable, which is the conventional
friction coefficient scaled with (C/R) to numerically convenient values.
Lubricant flow:
The rate of inflow can be calculated from
Qi ≡ qi NRLC = 2
()
∫ ∫ u(0, y, z )dydz
2 πR
0
h x
0
where u is given by Equation 11.14 and qi is the dimensionless inflow variable.
© 2001 by CRC Press LLC
(11.60)
TABLE 11.3
Performance of Short- and Long-Bearings
Short-Bearing Approximation
(Gümbel condition)
2
Long-Bearing Approximation
(Gümbel condition)
Equation for pressure, p
∂p  3 dp 
 L  ∂H
H
 = 12π  
 D  ∂θ
∂z 
dz 
∂  3 dp 
∂H
H
 = 12π
∂θ  dθ 
∂θ
Boundary condition
p = pa at z = ±1
p(0) = p(2π) = pi
–
2
Pressure, p
 L  1 ∂H 2
6 π 
z − 1 + pa
 D  H 3 ∂θ
Radial force, fR
 L  4πε 2
− 
 D 1− ε
2
Tangential force, fT
π 2ε
 L
 
 D  (1 − ε )3 2
2
Attitude angle, φ
 π 1 − ε2
arctan
 4ε

Sommerfeld number, S
1− ε
 D
 
 L  πε π 2 1 − ε 2 + 16ε 2
–
(
)
12πε sin θ(2 + ε cos θ)
(2 + ε )(1 + ε cos θ)
2
2
(
)
−
2
(
(
Friction variable, cµ
Flow variable, qi
2 π 2S
(1 − ε )
2
2πε
+ pi
12πε 2
(2 + ε )(1 − ε )
2
2
6π 2ε
2
2
2
(2 + ε )(1 − ε )
2



2
12
 π 1 − ε2
arctan
 2ε

2
)
)
2



(2 + ε )(1 − ε )
6πε 4ε + π (1 − ε )
2
2
ε sin φ +
2
2
2
4π 2 S
1 − ε2
—
Table 11.3 lists the performance characteristics for both short- and long-bearings. These calculations
are 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 no
useful purpose. This leads to the idea of employing partial arc pads instead of 360°, or full, bearings to
support the shaft. Different types of fixed pad partial journal bearings, each of arc β, are shown schematically in Figure 11.11.
FIGURE 11.11 Fixed type journal bearings: (a) full 360° bearing, (b) centrally loaded partial bearing, (c) offset
loaded partial bearing (offset parameter α/β). (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.)
© 2001 by CRC Press LLC
11.3.1.1 Finite Journal Bearings
The pressure distribution in a finite journal bearing of length L, diameter D and angular extent β, is
governed by the Reynolds equation
2
∂  3 ∂p   D  ∂  3 ∂p 
∂H
H
 + 
H
 = 12π
∂θ 
∂θ   L  ∂z 
∂z 
∂θ
(11.61)
–
and the boundary conditions (considering p as gauge pressure)
p = 0 at z = ±1
(11.62a)
p = 0 at θ = θ1, θ1 + β
(11.62b)
If (θ1 + β) > π, Equation 11.61 yields both positive and negative pressures, leading to film rupture as
previously discussed. At the film-cavity interface the Swift–Stieber conditions, Equation 11.35, are usually
applied.
Equation 11.61 and its boundary conditions, Equation 11.62, contain three parameters, (L/D), θ1, and
β. The only additional parameter of the problem, ε, appears in the definition of the lubricant film
geometry

x
H = 1 + ε cos θ = 1 + ε cos θ1 +  , θ1 ≤ θ ≤ β
R

(11.63)
Thus the journal bearing problem is uniquely characterized by the parameter set
{L D, β, ε, θ }
1
(11.64)
The first two parameters of this set define bearing geometry, while the last two characterize the geometry
of the film. Having selected parameter set (11.64), we can find the pressure by solving the system consisting
of Equations 11.61, 11.62, and 11.63. The nondimensional lubricant force components are obtained from
(note the limits of integration on z)
fR =
1
2
∫∫
θcav
fT =
1
2
1
θcav
1
θ1
0
∫∫
0
θ1
p cosθdθdz
(11.65a)
p sinθdθdz
(11.65b)
and the attitude angle from
φ = arctan
fT
fR
(11.66)
Knowledge of the force components {fR, fT } thus enables us to determine both the magnitude, f =
f R + f T , and the direction, φ, of the load the lubricant film will support under the specified conditions.
2
2
Instead of characterizing the oil-film force this way, however, it has been customary to employ an alternate
representation of the Sommerfeld number
© 2001 by CRC Press LLC
S =1 f
(11.67)
and the offset parameter (α/β)
[ (
α 1
= π − θ1 + φ
β 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 the
parameters L/D and β from the list.)
What is known at the design stage are the pad geometry and the performance requirements, i.e., the
magnitude and direction of the external load, the shaft speed and the viscosity of the lubricant, and it
is easy for the designer to define the couple {S, α/β}; but the designer has no way of determining the
couple {ε, θ1} that is required in order to compute the minimum film thickness, often the controlling
parameter.
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 the
designer. The task for the analyst is then to find that particular {ε, θ1} that yields
( )
S − Ω ε, θ1 = 0
(11.69a)
α
− Ψ ε, θ1 = 0
β
(11.69b)
(
)
We can solve this pair of nonlinear equations for the unknowns {ε, θ1} by iteration, e.g., using Newton’s
method
 ∂Ω
 ∂ε

 ∂Ψ
 ∂ε

∂Ω 


n
n −1
∂θ1   ε( ) − ε( )   Ω − S 
α , n = 1, 2, 3, …
 (n ) (n−1)  =
∂Ψ  θ − θ  Ψ − 
1
 
 1
β
∂θ1 
(11.70)
The performance curves in Figure 11.12, taken from Raimondi and Szeri (1984), are for centrally
loaded fixed-pad partial bearings of L/D = 1, β = 160° and various values of the Reynolds number. The
turbulent 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
(
)
β = 2.79rad 160°
−3
C R = 2 × 10
D = L = 0.508 m
(
N = 40 sec, ω = 251rad sec
ISO VG 32, ρ = 831kg m
W = 355.84 kN
3
To start the design process, we assume an effective viscosity
µ = 3.447 × 10−2 Pa ⋅ s ;
© 2001 by CRC Press LLC
ν = 0.4148 cm2 s
)
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.)
For Reynolds and Sommerfeld numbers, respectively, we obtain
Re =
S=
© 2001 by CRC Press LLC
25.4 × 251 × 0.0508
= 781 < 1000, the flow is laminar
0.4148
3.447 × 10−2 × 40
1.3789 × 10
6
(500) = 0.25
2
FIGURE 11.12 (continued)
© 2001 by CRC Press LLC
FIGURE 11.12 (continued)
Entering Figure 11.12c, d, and e in succession we find
H = 3.05 × 2π × 355.84 × 103 × 40 × 5.08 × 10−4 = 0.1386 MW
Q = 3.2 × 0.254 × 5.08 × 10−4 × 40 × 0.508 = 8.39 × 10−3 m 3 s
Qs = 1.32 × 2.6219 × 10−3 = 3.46 × 10−3 m 3 s
The temperature rise across the bearing is calculated from a simple heat balance, Equation 11.28,
∆T =
1.386 × 105
(
)
1.39 × 106 8.39 − 3.46 2 × 10−3
= 14.97 C
and, assuming an inlet temperature of Ti = 45 C, yields the operating temperature
Ts = Ti + 0.5 × ∆T = 52.5C
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
Re = 3904 > 1000, turbulent flow
S = 0.05
Q = 8.39 × 10−3 m 3 s
© 2001 by CRC Press LLC
)
FIGURE 11.13 ISO viscosity grade for lubricating oils. Points A, B, and the line drawn through them, refer to
Example 2. (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.)
Q s = 4.93 × 10−3 m 3 s
H = 0.123 MW
∆T = 14.94 C
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
Ts = 52.5 C and µ s = 1.7 × 10−2 Pa ⋅ s
giving the effective temperature and effective viscosity that is consistent with the operating conditions
of the bearing under the assumption that a single viscosity can portray bearing performance. The final
Reynolds number, Sommerfeld number, and minimum film thickness are
Re = 1583, S = 0.123, hn = 0.4 × 0.0508 = 0.0203 cm
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 rotor
vibrations. This becomes a problem especially when attempting to operate the rotor in the neighborhood
of a system critical speed. For this reason the pads are often pivoted in one point or along a line. Pivoted
© 2001 by CRC Press LLC
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, in
CRC Handbook of Lubrication, Vol. 2, Booser, E.R. (Ed.), CRC Press, Boca Raton, FL. With permission.)
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 that
the clearance can be closely controlled by making the pivots adjust radially, thus enabling operation with
smaller clearances than considered appropriate for a plain journal bearing, and the pads can be preloaded
to 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, the
film thickness is uniform and equal to C, and the pad develops no hydrodynamic force. If the pad is now
moved inward the distance (C – C′) into position 2, the film thickness will no longer be uniform; the
resulting hydrodynamic force preloads the pad. The degree of preload is indicated by the preload coefficient
m = (C – C′)/C, the value of which varies between m = 0 for no preload, to m = 1, for metal-to-metal
contact between pad and shaft.
Figure 11.15 is a schematic of the geometrical relationships in a pivoted-pad journal bearing. The
symbols OB and OJ mark the positions of the bearing center and the instantaneous position of the shaft
center. The center of the pad is at Ono when the pad is unloaded and moves to On when it is loaded. The
pivot point P is located at angle ψ relative to the vertical load line WB . The eccentricity of the journal
relative to the bearing center, OB , is e0 = C′ε0 and its attitude angle φ0. Relative to the instantaneous pad
center, OJ , the journal eccentricity is e = Cε and the attitude angle φ, the latter measured from the load
line that, by necessity, intersects the pivot P. The radii of journal, pad, and pivot circle are R, O n P , and
O B P , respectively. From geometric consideration we have (Lund, 1964)
ε n cos φn = 1 −
C′
− ε 0 cos ψ n − φ0 = m − ε 0 cos ψ n − φ0
C
(
)
(
)
(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 to
calculate individual pad performance that can then be summed to yield the performance characteristics
of the bearing. Unfortunately (ε0,φ0) is not known at the design stage, and the best the designer can do
is assume ε0 and use the condition that WB is purely vertical to calculate the corresponding φ0. This
procedure is, at least, tedious. If, however, the pivots are arranged symmetrically with respect to the loadline, the pads are centrally pivoted and are identical, the journal will move along the load line WB and
© 2001 by CRC Press LLC
FIGURE 11.15 Geometry of a pivoted-pad journal bearing. (From Lund, J.W. (1964), Spring and damping coefficients for the tilting pad journal bearing, ASLE Trans., 7, 342. With permission.)
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
( )
β = 1.05 rad 60°
W = 11.12 kN
D = 12.7 cm
L = 6.35 cm
C = C ′ = 0.0127 cm
µ = 1.379 × 10 −2 Pa ⋅ s
v = 0.1658 cm2 s
N = 60 r sec
Bearing performance is calculated as follows:
Re =
6.35 × 2π × 60 × 0.0127
= 183 < 1000 laminar
0.1658
(
)
2
S=
1.379 × 10 −2 × 60  6.35 

 = 0.15
1.3789 × 106  0.0127 
hn = 0.26 × 0.127 = 0.0033 cm, from Figure 11.16a
H = 3.9 × 2π × 11.12 × 10 3 × 60 × 1.27 × 10 −4 = 2.08 kW , from Figure 11.16b
ε ′0 = 0.67, from Figure 11.16c
The normalized bearing eccentricity ratio ε′0 = ε0/1.236 will be used to obtain stiffness and damping
coefficients.
© 2001 by CRC Press LLC
FIGURE 11.16 Performance of a five-pad pivoted-pad journal bearing (L/D) = 1, β = 160°: (a) minimum film
thickness, (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.)
© 2001 by CRC Press LLC
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.)
11.3.2 Thrust Bearings
Thrust bearings in their most elementary form consist of two inclined plane surfaces in relative motion
to one another. The geometry of the bearing surfaces is commonly rectangular, to accommodate a linear
motion, 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
2
∂  3 ∂p   B  ∂  3 ∂p 
x
 + 4 
x
 = −1
∂x  ∂x   L  ∂z  ∂z 
© 2001 by CRC Press LLC
(11.72a)
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
0.20
0.25
0.50
0.60
0.80
1.00
1.25
1.67
2.00
2.50
4.00
5.00
10.0
0.0458
0.0700
0.0825
0.1155
0.1205
0.1238
0.1225
0.1179
0.1081
0.1005
0.0901
0.0676
0.0577
0.0330
0.0424
—
0.0755
0.1047
—
—
0.1110
—
—
0.0903
—
0.0607
0.0518
0.0296
0.0360
—
0.0623
0.0845
—
—
0.0879
—
—
0.0715
—
0.0479
0.0409
0.0234
0.0303
0.0441
0.0509
0.0675
0.0697
—
0.0692
0.0662
0.0604
0.0559
0.0500
0.0374
0.0319
0.0182
0.0216
0.0303
0.0344
0.0337
0.0447
0.0448
0.0437
0.0415
0.0377
0.0348
0.0311
0.0232
0.0198
0.0113
0.0161
0.0218
0.0243
0.0298
0.0303
0.0301
0.0291
0.0276
0.0249
0.0230
0.0205
0.0153
0.0130
0.074
0.0099
0.0127
0.0139
—
0.0162
0.0159
0.0152
0.0144
0.0129
0.0119
0.0106
0.0078
0.0067
0.0038
Source: Szeri, A.Z. and Powers, D. (1970), Pivoted plane pad bearings: a variational solution, ASME Trans., Ser. F., 92, 466-72.
(
)
p x, ±1 = 0
( ) (
(11.72b)
)
p x1, z = p x2 , z = 0
(11.72c)
Here
x = Bx , y = mBy , z =
L
6µU
z, p =
p
2
Bm2
(11.72d)
The nondimensional force f is calculated from
f≡
Fh12
= 6 x12
µULB2
1
∫∫
0
x1 +1
( )
p x , z dxdz
x1
(11.73)
Its values at various values of x– 1 and aspect ratio (L/B) are displayed in Table 11.4a.
The center of pressure, xp, is calculated from
Fx p =
–
∫ ∫ xp(x, z )dxdz
L 2
x2
− L 2 x1
(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
δ = − x1 +
6 x12
f
1
∫∫
0
x1 +1
x1
( )
xp x , z dx dz
(11.75)
Table 11.4b lists pivot position δ at various values of x– 1 and aspect ratio (L/B). The nondimensional flow
variable at inlet, x2, is listed in Table 11.4c.
© 2001 by CRC Press LLC
TABLE 11.4b
–
Optimum Pivot Position: δ = (x– p – x–1)
x1/(L/B)
8/3
2
4/3
1
2/3
1/2
1/3
0.125
0.20
0.25
0.50
0.60
0.80
1.00
1.25
1.67
2.00
2.50
4.00
5.00
10.0
0.2900
0.3236
0.3397
0.3876
0.3992
0.4161
0.4281
0.4388
0.4510
0.4576
0.4649
0.4568
0.4811
0.4883
0.2858
—
0.3363
0.3851
—
—
0.4264
—
—
0.4567
—
0.4763
0.4806
0.4878
0.2766
—
0.3286
0.3793
—
—
0.4226
—
—
0.4544
—
0.4750
0.4796
0.4868
0.2663
0.3021
0.3196
0.3725
0.3854
—
0.4180
0.4302
0.4440
0.4516
0.4599
0.4736
0.4784
0.4854
0.2452
0.2825
0.3009
0.3579
0.3721
0.3931
0.4081
0.4217
0.4372
0.4456
0.4549
0.4704
0.4758
0.4820
0.2268
0.2649
0.2841
0.3444
0.3597
0.3825
0.3988
0.4137
0.4307
0.4399
0.4503
0.4674
0.4735
0.4780
0.2000
0.2388
0.2588
—
0.3408
0.3662
0.3844
0.4013
0.4207
0.4311
0.4431
0.4631
0.4701
—
Source: Szeri, A.Z. and Powers, D. (1970), Pivoted plane pad bearings: a variational 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
0.20
0.25
0.50
0.60
0.80
1.00
1.25
1.67
2.00
2.50
4.00
5.00
10.0
1.6990
1.3390
1.2059
0.9168
0.8621
0.7879
0.7396
0.6990
0.6550
0.6322
0.6083
0.5702
0.5569
0.5293
1.9662
—
1.3318
0.9717
—
—
0.7638
—
—
0.6427
—
0.5750
0.5607
0.5310
2.4632
—
1.5588
1.0735
—
—
0.8078
—
—
0.6621
—
0.5838
0.5676
0.5343
2.8866
2.0402
1.7544
1.1584
1.0558
—
0.8449
0.7791
0.7123
0.6783
0.6439
0.5912
0.5733
0.5370
3.4626
2.3825
2.0156
1.2728
1.1485
0.9905
0.8948
0.8176
0.7393
0.7004
0.6611
0.6014
0.5813
0.5409
3.7744
2.5670
2.1584
1.3360
1.1997
1.0270
0.9229
0.8393
0.7549
0.7130
0.6709
0.6072
0.5859
0.5431
4.0710
2.7362
2.2904
—
1.2789
1.0625
0.9505
0.8608
0.7704
0.7260
0.6809
0.6132
0.5906
0.5454
Source: Szeri, A.Z. and Powers, D. (1970), Pivoted plane pad bearings: a variational
solution, ASME Trans., Ser. F, 92, 466-72.
Example 4. Plane Slider
Calculate the performance of a fixed-pad slider bearing if the following data are specified
W = 16.013 kN ,
L = 20.32 cm,
U = 30.5 m s ,
µ = 0.04137 Pa ⋅ s
B = 7.62 cm,
From Equation 11.73 we obtain the relationship between the external load and the dimensionless lubricant force
f=
© 2001 by CRC Press LLC
16.013 × 10 3 × h12
= 1.0756 × 107 h12 m
0.04137 × 30.5 × 0.2032 × 0.07622
( )
For (L/B) = 8/3, the dimensionless force has its maximum value of f = 0.1238 at x– 1 = 0.8, yielding the
largest value of the exit film thickness at which this bearing can carry the assigned load.
 0.1238 
h1 = 
7
1.0756 × 10 
12
= 1.073 × 10 −4 m
The corresponding surface slope is calculated from
m=
1.073 × 10 −4
= 0.00176
0.8 × 7.62 × 10 −2
The film thickness at inlet is h2 = 0.01073 + 0.00176 × 7.62 = 0.024 cm. The lubricant flow-rate at inlet
is 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 annular
configuration 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. 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.)
© 2001 by CRC Press LLC
FIGURE 11.19 Pivoted-pad thrust bearing. Source: Raimondi, A. A. and Szeri, A. Z. 1984. Journal and thrust
bearings. (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.)
Approximate performance calculations of this bearing can be made by relating the rectangular slider
bearing (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 the
pivot fixes the location of the center of pressure, on loading a pad will swivel until it occupies a position
that places the center of pressure over the pivot. While performance of a pivoted pad is identical to that
of 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 varying
speed and load, and (c) having the capability of operating in either direction of rotation. Theoretically, the
pivoted pad can be optimized for all speeds and loads by judicious pivot positioning, whereas the fixedpad bearing can have optimum performance only for one operating condition. Although pivoted-pad
bearings 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
∂  3 ∂p  1 ∂  3 ∂p 
∂h
 rh
+
h
 = 6µωr
∂r 
∂r  r ∂θ  ∂θ 
∂θ
(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 slope
parameter mr . Load capacity, minimum film thickness, power loss, flow and pivot location (if pivotedpad 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 sector
angular length of 40°. While this angle corresponds to seven sectors to form a full thrust bearing, the
results should generally give a preliminary indication of performance of other geometries with the same
surface 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:
© 2001 by CRC Press LLC
(
β = 40°
γ r = 0 no radial tilt
R2 = 13.97 cm
γ θ = 5.82 × 10−4 rad
)
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.)
R1 = 6.985 cm
µ = 1.379 × 10−2 Pa ⋅ s
hc = 50.8 µm, pad center
N = 50 r sec ω = 314 rad sec
(
)
Calculating first
(
mθ = R1 hc
)
(
)
γ θ = 6.985 × 10 −2 50.8 × 10 −6 × 5.82 × 10 −4 = 0.80
Enter Figures 11.20a to 11.20d with mθ = 0.8 and mr = 0.0
(
) (50.8 × 10 )
W = 0.058 × 6 × 1.379 × 10 −2 × 314 × 0.1397 − 0.06985
hmin = 0.45 × 50.8 × 10 −6 = 22.86 µm
© 2001 by CRC Press LLC
4
−6
2
= 13.9 kN