SIST ISO 7902-1:2015
Hydrodynamic plain journal bearings under steady-state conditions - Circular cylindrical bearings - Part 1: Calculation procedure
Hydrodynamic plain journal bearings under steady-state conditions - Circular cylindrical bearings - Part 1: Calculation procedure
This part of ISO 7902 specifies a calculation procedure for oil-lubricated hydrodynamic plain bearings,
with complete separation of the shaft and bearing sliding surfaces by a film of lubricant, used for
designing plain bearings that are reliable in operation.
It deals with circular cylindrical bearings having angular spans, Ω, of 360°, 180°, 150°, 120°, and 90°,
the arc segment being loaded centrally. Their clearance geometry is constant except for negligible
deformations resulting from lubricant film pressure and temperature.
The calculation procedure serves to dimension and optimize plain bearings in turbines, generators,
electric motors, gear units, rolling mills, pumps, and other machines. It is limited to steady-state
operation, i.e. under continuously driven operating conditions, with the magnitude and direction
of loading as well as the angular speeds of all rotating parts constant. It can also be applied if a full
plain bearing is subjected to a constant force rotating at any speed. Dynamic loadings, i.e. those whose
magnitude and direction vary with time, such as can result from vibration effects and instabilities of
rapid-running rotors, are not taken into account.
Paliers lisses hydrodynamiques radiaux fonctionnant en régime stabilisé - Paliers circulaires cylindriques - Partie 1: Méthode de calcul
Hidrodinamični radialni drsni ležaji za neprekinjeno obratovanje - Valjasti ležaji - 1. del: Postopek dimenzioniranja
Ta del standarda ISO 7902 določa postopek dimenzioniranja hidrodinamičnih radialnih drsnih ležajev z oljnim mazanjem s popolno ločitvijo grede in drsečih površin ležaja s plastjo maziva, ki se uporabljajo za načrtovanje radialnih ležajev z zanesljivim delovanjem.
Zajema valjaste ležaje s kotnim razponom, Ω, 360°, 180°, 150°, 120° in 90° ter sredinsko umeščenim obločnim delom. Geometrija razdalj je stalna z izjemo zanemarljivih deformacij, ki so posledica pritiska in temperature plasti maziva.
Postopek izračuna služi izmeri in optimizaciji drsnih ležajev v turbinah, generatorjih, električnih motorjih, menjalnikih, valjarnah, črpalkah in drugih strojih. Omejen je na neprekinjeno delovanje, t. j. pri neprekinjenih delovnih pogojih, s konstantno magnitudo, smerjo obremenitve in konstantnimi kotnimi hitrostmi vseh vrtljivih delov. Uporabiti ga je mogoče tudi, če je drsni ležaj izpostavljen stalni sili s katero koli hitrostjo vrtenja. Dinamične obremenitve, npr. tiste, pri katerih se magnituda in smer s časom spreminjata ter so lahko posledica učinkov vibracij in nestabilnosti hitro delujočih motorjev, niso zajete.
General Information
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Buy Standard
Standards Content (Sample)
SLOVENSKI STANDARD
SIST ISO 7902-1:2015
01-marec-2015
1DGRPHãþD
SIST ISO 7902-1:2002
+LGURGLQDPLþQLUDGLDOQLGUVQLOHåDML]DQHSUHNLQMHQRREUDWRYDQMH9DOMDVWLOHåDML
GHO 3RVWRSHNGLPHQ]LRQLUDQMD
Hydrodynamic plain journal bearings under steady-state conditions - Circular cylindrical
bearings - Part 1: Calculation procedure
Paliers lisses hydrodynamiques radiaux fonctionnant en régime stabilisé - Paliers
circulaires cylindriques - Partie 1: Méthode de calcul
Ta slovenski standard je istoveten z: ISO 7902-1:2013
ICS:
21.100.10 Drsni ležaji Plain bearings
SIST ISO 7902-1:2015 en
2003-01.Slovenski inštitut za standardizacijo. Razmnoževanje celote ali delov tega standarda ni dovoljeno.
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SIST ISO 7902-1:2015
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SIST ISO 7902-1:2015
INTERNATIONAL ISO
STANDARD 7902-1
Second edition
2013-11-01
Hydrodynamic plain journal bearings
under steady-state conditions —
Circular cylindrical bearings —
Part 1:
Calculation procedure
Paliers lisses hydrodynamiques radiaux fonctionnant en régime
stabilisé — Paliers circulaires cylindriques —
Partie 1: Méthode de calcul
Reference number
ISO 7902-1:2013(E)
©
ISO 2013
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SIST ISO 7902-1:2015
ISO 7902-1:2013(E)
COPYRIGHT PROTECTED DOCUMENT
© ISO 2013
All rights reserved. Unless otherwise specified, no part of this publication may be reproduced or utilized otherwise in any form
or by any means, electronic or mechanical, including photocopying, or posting on the internet or an intranet, without prior
written permission. Permission can be requested from either ISO at the address below or ISO’s member body in the country of
the requester.
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Tel. + 41 22 749 01 11
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Published in Switzerland
ii © ISO 2013 – All rights reserved
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Contents Page
Foreword .iv
1 Scope . 1
2 Normative references . 1
3 Basis of calculation, assumptions, and preconditions . 1
4 Calculation procedure . 3
5 Symbols and units . 5
6 Definition of symbols . 6
6.1 Load-carrying capacity . 6
6.2 Frictional power loss . 9
6.3 Lubricant flow rate .10
6.4 Heat balance .11
6.5 Minimum lubricant film thickness and specific bearing load .13
6.6 Operational conditions.14
6.7 Further influencing factors .15
Annex A (normative) Calculation examples .17
Bibliography .32
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Foreword
ISO (the International Organization for Standardization) is a worldwide federation of national standards
bodies (ISO member bodies). The work of preparing International Standards is normally carried out
through ISO technical committees. Each member body interested in a subject for which a technical
committee has been established has the right to be represented on that committee. International
organizations, governmental and non-governmental, in liaison with ISO, also take part in the work.
ISO collaborates closely with the International Electrotechnical Commission (IEC) on all matters of
electrotechnical standardization.
The procedures used to develop this document and those intended for its further maintenance are
described in the ISO/IEC Directives, Part 1. In particular the different approval criteria needed for the
different types of ISO documents should be noted. This document was drafted in accordance with the
editorial rules of the ISO/IEC Directives, Part 2. www.iso.org/directives.
Attention is drawn to the possibility that some of the elements of this document may be the subject of
patent rights. ISO shall not be held responsible for identifying any or all such patent rights. Details of any
patent rights identified during the development of the document will be in the Introduction and/or on
the ISO list of patent declarations received. www.iso.org/patents.
Any trade name used in this document is information given for the convenience of users and does not
constitute an endorsement.
The committee responsible for this document is ISO/TC 123, Plain bearings, Subcommittee SC 4, Methods
of calculation of plain bearings.
This second edition cancels and replaces the first edition (ISO 7902-1:1998), which has been
technically revised.
ISO 7902 consists of the following parts, under the general title Hydrodynamic plain journal bearings
under steady-state conditions — Circular cylindrical bearings:
— Part 1: Calculation procedure
— Part 2: Functions used in the calculation procedure
— Part 3: Permissible operational parameters
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INTERNATIONAL STANDARD ISO 7902-1:2013(E)
Hydrodynamic plain journal bearings under steady-state
conditions — Circular cylindrical bearings —
Part 1:
Calculation procedure
1 Scope
This part of ISO 7902 specifies a calculation procedure for oil-lubricated hydrodynamic plain bearings,
with complete separation of the shaft and bearing sliding surfaces by a film of lubricant, used for
designing plain bearings that are reliable in operation.
It deals with circular cylindrical bearings having angular spans, Ω, of 360°, 180°, 150°, 120°, and 90°,
the arc segment being loaded centrally. Their clearance geometry is constant except for negligible
deformations resulting from lubricant film pressure and temperature.
The calculation procedure serves to dimension and optimize plain bearings in turbines, generators,
electric motors, gear units, rolling mills, pumps, and other machines. It is limited to steady-state
operation, i.e. under continuously driven operating conditions, with the magnitude and direction
of loading as well as the angular speeds of all rotating parts constant. It can also be applied if a full
plain bearing is subjected to a constant force rotating at any speed. Dynamic loadings, i.e. those whose
magnitude and direction vary with time, such as can result from vibration effects and instabilities of
rapid-running rotors, are not taken into account.
2 Normative references
The following documents, in whole or in part, are normatively referenced in this document and are
indispensable for its application. For dated references, only the edition cited applies. For undated
references, the latest edition of the referenced document (including any amendments) applies.
ISO 3448, Industrial liquid lubricants — ISO viscosity classification
ISO 7902-2:1998, Hydrodynamic plain journal bearings under steady-state conditions — Circular cylindrical
bearings — Part 2: Functions used in the calculation procedure
ISO 7902-3, Hydrodynamic plain journal bearings under steady-state conditions — Circular cylindrical
bearings — Part 3: Permissible operational parameters
3 Basis of calculation, assumptions, and preconditions
3.1 The basis of calculation is the numerical solution to Reynolds’ differential equation for a finite bearing
length, taking into account the physically correct boundary conditions for the generation of pressure:
∂ ∂p ∂ ∂p ∂h
33
h + h =+6η uu (1)
()
JB
∂x ∂xx∂ ∂z ∂x
The symbols are given in Clause 5.
See References [1] to [3] and References [11] to [14] for the derivation of Reynolds’ differential equation
and References [4] to [6], [12], and [13] for its numerical solution.
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3.2 The following idealizing assumptions and preconditions are made, the permissibility of which has
been sufficiently confirmed both experimentally and in practice.
a) The lubricant corresponds to a Newtonian fluid.
b) All lubricant flows are laminar.
c) The lubricant adheres completely to the sliding surfaces.
d) The lubricant is incompressible.
e) The lubricant clearance gap in the loaded area is completely filled with lubricant. Filling up of the
unloaded area depends on the way the lubricant is supplied to the bearing.
f) Inertia effects, gravitational and magnetic forces of the lubricant are negligible.
g) The components forming the lubrication clearance gap are rigid or their deformation is negligible;
their surfaces are ideal circular cylinders.
h) The radii of curvature of the surfaces in relative motion are large in comparison with the lubricant
film thicknesses.
i) The lubricant film thickness in the axial direction (z-coordinate) is constant.
j) Fluctuations in pressure within the lubricant film normal to the bearing surfaces (y-coordinate)
are negligible.
k) There is no motion normal to the bearing surfaces ( y-coordinate).
l) The lubricant is isoviscous over the entire lubrication clearance gap.
m) The lubricant is fed in at the start of the bearing liner or where the lubrication clearance gap is
widest; the magnitude of the lubricant feed pressure is negligible in comparison with the lubricant
film pressures.
3.3 The boundary conditions for the generation of lubricant film pressure fulfil the following
continuity conditions:
— at the leading edge of the pressure profile: pzϕ , =0 ;
()
1
— at the bearing rim: pzϕ, =±B 20= ;
()
— at the trailing edge of the pressure profile: pzϕ ,z =0 ;
()
2
— ∂∂pzϕϕ ,z =0 .
()
2
For some types and sizes of bearing, the boundary conditions may be specified.
In partial bearings, if Formula (2) is satisfied:
π
ϕπ−−β < (2)
()
2
2
then the trailing edge of the pressure profile lies at the outlet end of the bearing:
pzϕϕ= , =0 (3)
()
2
3.4 The numerical integration of the Reynolds’ differential equation is carried out (possibly by
applying transformation of pressure as suggested in References [3], [11], and [12]) by a transformation
to a differential formula which is applied to a grid system of supporting points, and which results in a
system of linear formulae. The number of supporting points is significant to the accuracy of the numerical
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integration; the use of a non-equidistant grid as given in References [6] and [13] is advantageous. After
substituting the boundary conditions at the trailing edge of the pressure profile, integration yields the
pressure distribution in the circumferential and axial directions.
The application of the similarity principle to hydrodynamic plain bearing theory results in dimensionless
magnitudes of similarity for parameters of interest, such as load-carrying capacity, frictional behaviour,
lubricant flow rate, and relative bearing length. The application of magnitudes of similarity reduces the
number of numerical solutions required of Reynolds’ differential equation specified in ISO 7902-2. Other
solutions may also be applied, provided they fulfil the conditions laid down in ISO 7902-2 and are of a
similar numerical accuracy.
3.5 ISO 7902-3 includes permissible operational parameters towards which the result of the calculation
shall be oriented in order to ensure correct functioning of the plain bearings.
In special cases, operational parameters deviating from ISO 7902-3 may be agreed upon for specific
applications.
4 Calculation procedure
4.1 Calculation is understood to mean determination of correct operation by computation using actual
operating parameters (see Figure 1), which can be compared with operational parameters. The operating
parameters determined under varying operating conditions shall therefore lie within the range of
permissibility as compared with the operational parameters. To this end, all operating conditions during
continuous operation shall be investigated.
4.2 Freedom from wear is guaranteed only if complete separation of the mating bearing parts is
achieved by the lubricant. Continuous operation in the mixed friction range results in failure. Short-time
operation in the mixed friction range, for example starting up and running down machines with plain
bearings, is unavoidable and does not generally result in bearing damage. When a bearing is subjected to
heavy load, an auxiliary hydrostatic arrangement may be necessary for starting up and running down at a
slow speed. Running-in and adaptive wear to compensate for deviations of the surface geometry from the
ideal are permissible as long as they are limited in area and time and occur without overloading effects.
In certain cases, a specific running-in procedure may be beneficial, depending on the choice of materials.
4.3 The limits of mechanical loading are a function of the strength of the bearing material. Slight permanent
deformations are permissible as long as they do not impair correct functioning of the plain bearing.
4.4 The limits of thermal loading result not only from the thermal stability of the bearing material but
also from the viscosity-temperature relationship and by degradation of the lubricant.
4.5 A correct calculation for plain bearings presupposes that the operating conditions are known for
all cases of continuous operation. In practice, however, additional influences frequently occur, which
are unknown at the design stage and cannot always be predicted. The application of an appropriate
safety margin between the actual operating parameters and permissible operational parameters is
recommended. Influences include, for example:
— spurious forces (out-of-balance, vibrations, etc.);
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yes
Figure 1 — Outline of calculation
— deviations from the ideal geometry (machining tolerances, deviations during assembly, etc.);
— lubricants contaminated by dirt, water, air, etc.;
— corrosion, electrical erosion, etc.
Data on other influencing factors are given in 6.7.
4.6 The Reynolds number shall be used to verify that ISO 7902-2, for which laminar flow in the
lubrication clearance gap is a necessary condition, can be applied:
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C C
Re,,ff Reff
ρU πDN
J J
D
22
Re== ≤41,3 (4)
η v C
Re, ff
In the case of plain bearings with Re>41,3 DC (for example as a result of high peripheral speed),
R,eff
higher loss coefficients and bearing temperatures shall be expected. Calculations for bearings with
turbulent flow cannot be carried out in accordance with this part of ISO 7902.
4.7 The plain bearing calculation takes into account the following factors (starting with the known
bearing dimensions and operational data):
— the relationship between load-carrying capacity and lubricant film thickness;
— the frictional power rate;
— the lubricant flow rate;
— the heat balance.
All these factors are mutually dependent.
The solution is obtained using an iterative method; the sequence is outlined in the flow chart in Figure 1.
For optimization of individual parameters, parameter variation can be applied; modification of the
calculation sequence is possible.
5 Symbols and units
See Figure 2 and Table 1.
Minimum lubricant film thickness, h :
min
DD−
J
h = −=eD0,51ψε− (5)
()
min
2
where the relative eccentricity, ε, is given by
e
ε = (6)
DD−
J
2
If
π
ϕπ−−()β < (7)
2
2
then
hD=+05,(ψε1 cos)ϕ (8)
min 2
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6 Definition of symbols
6.1 Load-carrying capacity
A characteristic parameter for the load-carrying capacity is the dimensionless Sommerfeld number, So:
2
Fψ
B
eff
So==So ε,,Ω (9)
DBηω D
effh
Values of So as a function of the relative eccentricity, ε, the relative bearing length, B/D, and the angular
span of bearing segment, Ω, are given in ISO 7902-2. The variables ω , η , and ϕ take into account the
h eff eff
thermal effects and the angular velocities of shaft, bearing, and bearing force (see 6.4 and 6.7).
The relative eccentricity, ε, together with the attitude angle, β (see ISO 7902-2), describes the magnitude
and position of the minimum thickness of lubricant film. For a full bearing (Ω = 360°), the oil should be
introduced at the greatest lubricant clearance gap or, with respect to the direction of rotation, shortly
before it. For this reason, it is useful to know the attitude angle, β.
Figure 2 — Illustration of symbols
Table 1 — Symbols and their designations
Symbol Designation Unit
2
A Area of heat-emitting surface (bearing housing) m
b Width of oil groove m
G
B Nominal bearing width m
c Specific heat capacity of the lubricant J/(kg·K)
C Nominal bearing clearance m
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Table 1 (continued)
Symbol Designation Unit
C Effective bearing radial clearance m
R,eff
d Oil hole diameter m
L
D Nominal bearing diameter (inside diameter) m
D Nominal shaft diameter m
J
D , Maximum value of D m
J max J
D , Minimum value of D m
J min J
D Maximum value of D m
max
D Minimum value of D m
min
e Eccentricity between the axis of the shaft and the bearing axis m
E Modulus of elasticity 1
f Coefficient of friction 1
F Bearing force (nominal load) N
F Friction force in the loaded area of the lubricant film N
f
'
Frictional force in the unloaded area of the lubricant film N
F
f
G Shear modulus 1
h Local lubricant film thickness m
h Minimum permissible lubricant film thickness m
lim
h Minimum lubricant film thickness m
min
h Waviness of sliding surface m
wav
h Effective waviness of sliding surface m
wav,eff
h Maximum permissible effective waviness m
wav,eff,lim
k Outer heat transmission coefficient w/(m2·K)
A
l Length of oil groove m
G
l Length of oil pocket m
P
L Length of bearing housing Rotational m
H
−1
N Frequency of the bearing Rotational s
B
−1
N Frequency of the bearing force Rotational s
F
−1
N Frequency of the shaft s
J
p Local lubricant film pressure Pa
p Specific bearing load Pa
P Lubricant feed pressure Pa
en
p Maximum permissible lubricant film pressure Pa
lim
p Maximum permissible specific bearing load Pa
lim
P Frictional power W
f
P Heat flow rate W
th
P Heat flow rate to the ambient W
th,amb
P Heat flow rate due to frictional power W
th,f
P Heat flow rate in the lubricant W
th,L
3
Q Lubricant flow rate m /s
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Table 1 (continued)
Symbol Designation Unit
3
Q Lubricant flow rate at the inlet to clearance gap m /s
1
3
Q Lubricant flow rate at the outlet to clearance gap m /s
2
3
Q Lubricant flow rate due to hydrodynamic pressure m /s
3
*
Lubricant flow rate parameter due to hydrodynamic pressure 1
Q
3
3
Q Lubricant flow rate due to feed pressure m /s
p
*
Lubricant flow rate parameter due to feed pressure 1
Q
p
Rz Average peak-to-valley height of bearing sliding surface m
B
Rz Average peak-to-valley height of shaft mating surface m
J
Re Reynolds number 1
So Sommerfeld number 1
T Ambient temperature °C
amb
T Bearing temperature °C
B
T Assumed initial bearing temperature °C
B,0
T Calculated bearing temperature resulting from iteration procedure °C
B,1
T Lubricant temperature at bearing entrance °C
en
T Lubricant temperature at bearing exit °C
ex
T Assumed initial lubricant temperature at bearing exit °C
ex,0
T Calculated lubricant temperature at bearing exit °C
ex,1
T Shaft temperature °C
J
T Maximum permissible bearing temperature °C
lim
Mean lubricant temperature °C
T
L
U Linear velocity (peripheral speed) of bearing m/s
B
U Linear velocity (peripheral speed) of shaft m/s
J
V Air ventilating velocity m/s
a
x Coordinate parallel to the sliding surface in the circumferential direction m
y Coordinate perpendicular to the sliding surface m
z Coordinate parallel to the sliding surface in the axial direction m
−1
α Linear heat expansion coefficient of the bearing K
l,B
−1
α Linear heat expansion coefficient of the shaft K
l,J
β Attitude angle (angular position of the shaft eccentricity related to the direction °
of load)
δ Angle of misalignment of the shaft rad
J
ε Relative eccentricity 1
η Dynamic viscosity of the lubricant Pa·s
η Effective dynamic viscosity of the lubricant Pa·s
eff
v Kinematic viscosity of the lubricant Pa·s
ξ
Coefficient of resistance to rotation in the loaded area of the lubricant film 1
ξ '
Coefficient of resistance to rotation in the unloaded area of the lubricant film 1
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Table 1 (continued)
Symbol Designation Unit
ξ Coefficient of resistance to rotation in the area of circumferential groove 1
G
Coefficient of resistance to rotation in the area of the pocket 1
ξ
P
3
ρ Density of lubricant kg/m
φ Angular coordinate in the circumferential direction rad
φ Angular coordinate of pressure leading edge rad
1
φ Angular coordinate of pressure trailing edge rad
2
ψ
Relative bearing clearance 1
ψ Mean relative bearing clearance 1
ψ Effective relative bearing clearance 1
eff
ψ Maximum relative bearing clearance 1
max
Minimum relative bearing clearance 1
ψ
min
−1
ω Angular velocity of bearing s
B
−1
ω Hydrodynamic angular velocity s
h
−1
ω Angular velocity of shaft s
J
Ω Angular span of bearing segment °
Ω Angular span of lubrication groove °
G
Ω Angular span of lubrication pocket °
P
6.2 Frictional power loss
Friction in a hydrodynamic plain bearing due to viscous shear stress is given by the coefficient of friction
f = F /F and the derived non-dimensional characteristics of frictional power loss ξ and f /ψ
f
eff
Fψ
feff
ξ = (10)
DBηω
effh
f ξ
= (11)
ψ So
eff
They are applied if the frictional power loss is encountered only in the loaded area of the lubricant film.
It is still necessary to calculate frictional power loss in both the loaded and unloaded areas then the values
f
fF,,ξ,
f
ψ
eff
are substituted by:
f '
'
fF', ,'ξ
f
ψ
eff
in Formulae (10) and (11). This means that the whole of the clearance gap is filled with lubricant.
′
The values of f ψ and f ψ for various values of ε, B/D, and Ω are given in ISO 7902-2. It also gives
eff eff
the approximation formulae, based on Reference [15], which are used to determine frictional power loss
values in the bearings, taking account of the influence of lubricating pockets and grooves.
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The frictional power in a bearing or the amount of heat generated is given by:
D
PP== fF ω (12)
fth,fh
2
D
'
P = fF' ω (13)
fh
2
6.3 Lubricant flow rate
The lubricant fed to the bearing forms a film of lubricant separating the sliding surfaces. The pressure
build-up in this film forces lubricant out of the ends of the bearing. This is the proportion Q of the
lubricant flow rate, resulting from the build-up of hydrodynamic pressure.
3 *
QD= ψω Q (14)
3 effh 3
**
where QQ= εΩ,,BD is given in ISO 7902-2.
()
33 1
There is also a flow of lubricant in the peripheral direction through the narrowest clearance gap into the
diverging, pressure-free gap. For increased loading and with a small lubrication gap clearance; however,
this proportion of the lubricant flow is negligible.
The lubricant feed pressure, p , forces additional lubricant out of the ends of the plain bearing. This is
en
the amount Q of the lubricant flow rate resulting from feed pressure:
p
33
Dpψ
effen *
Q = Q (15)
p p
η
eff
**
where QQ= εΩ,,BD is given in ISO 7902-2.
()
pp
6.3.1 Lubricant feed elements are lubrication holes, lubrication grooves, and lubrication pockets. The
lubricant feed pressure, p , should be markedly less than the specific bearing load, p , to avoid additional
en
hydrostatic loads. Usually, p lies between 0,05 MPa and 0,2 MPa. The depth of the lubrication grooves
en
and lubrication pockets is considerably greater than the bearing clearance.
6.3.2 Lubrication grooves are elements designed to distribute lubricant in the circumferential direction.
The recesses machined into the sliding surface run circumferentially and are kept narrow in the axial
direction. If lubrication grooves are located in the vicinity of pressure rise, the pressure distribution
is split into two independent pressure “hills” and the load-carrying capacity is markedly reduced (see
Figure 3). In this case, the calculation shall be carried out for half the load applied to each half bearing.
However, because of the build-up of hydrodynamic pressure, Q , only half of the lubricant flow rate shall
3
be taken into account when balancing heat losses (see 6.4), since the return into the lubrication groove
plays no part in dissipating heat. It is more advantageous, for a full bearing, to arrange the lubrication
groove in the unloaded part. The entire lubricant flow amount, Q , goes into the heat balance.
p
6.3.3 Lubrication pockets are elements for distributing the lubricant over the length of the bearing.
The recesses machined into the sliding surface are oriented in the axial direction and should be as short
as possible in the circumferential direction. Relative pocket lengths should be such shall b /B < 0,7.
p
Although larger values increase the oil flow rate, the oil emerging over the narrow, restricting webs at the
ends plays no part in dissipating heat. This is even more true if the end webs are penetrated axially. For
full bearings (Ω = 360°), a lubrication pocket opposite to the direction of load as well as two lubrication
pockets normal to the direction of loading are machined in. Since the lubricant flow rate, even in the
unloaded part of the bearing, provides for the dissipation of frictional heat arising from shearing, the
lubricating pockets shall be fully taken into account in the heat balance. For shell segments (Ω < 360°),
the lubricant flow rate due to feed pressure through lubrication pockets at the inlet or outlet of the shell
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segment makes practically no contribution to heat dissipation, since the lubrication pockets are scarcely
restricted at the segment ends and the greater proportion of this lubricant flow emerges directly.
If the lubricant fills the loaded area of the bearing and there is no lubricant in the unloaded part, then
the heat dissipation counts as lubricant flow rate in the loaded part only.
Key
1 lubrication hole
2 lubrication groove
Figure 3
The influence of the type and the arrangement of the lubricant feed elements on the lubricant flow rate
are dealt with in ISO 7902-2.
The overall lubricant flow rate is given by:
QQ= (16)
3
for lubricant filling only the loaded area of the bearing;
QQ=+ Q (17)
3p
for lubricant filling the whole circular lubrication clearance gap including unloaded part, i.e. 2π.
6.4 Heat balanc
...
INTERNATIONAL ISO
STANDARD 7902-1
Second edition
2013-11-01
Hydrodynamic plain journal bearings
under steady-state conditions —
Circular cylindrical bearings —
Part 1:
Calculation procedure
Paliers lisses hydrodynamiques radiaux fonctionnant en régime
stabilisé — Paliers circulaires cylindriques —
Partie 1: Méthode de calcul
Reference number
ISO 7902-1:2013(E)
©
ISO 2013
---------------------- Page: 1 ----------------------
ISO 7902-1:2013(E)
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© ISO 2013
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Published in Switzerland
ii © ISO 2013 – All rights reserved
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ISO 7902-1:2013(E)
Contents Page
Foreword .iv
1 Scope . 1
2 Normative references . 1
3 Basis of calculation, assumptions, and preconditions . 1
4 Calculation procedure . 3
5 Symbols and units . 5
6 Definition of symbols . 6
6.1 Load-carrying capacity . 6
6.2 Frictional power loss . 9
6.3 Lubricant flow rate .10
6.4 Heat balance .11
6.5 Minimum lubricant film thickness and specific bearing load .13
6.6 Operational conditions.14
6.7 Further influencing factors .15
Annex A (normative) Calculation examples .17
Bibliography .32
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ISO 7902-1:2013(E)
Foreword
ISO (the International Organization for Standardization) is a worldwide federation of national standards
bodies (ISO member bodies). The work of preparing International Standards is normally carried out
through ISO technical committees. Each member body interested in a subject for which a technical
committee has been established has the right to be represented on that committee. International
organizations, governmental and non-governmental, in liaison with ISO, also take part in the work.
ISO collaborates closely with the International Electrotechnical Commission (IEC) on all matters of
electrotechnical standardization.
The procedures used to develop this document and those intended for its further maintenance are
described in the ISO/IEC Directives, Part 1. In particular the different approval criteria needed for the
different types of ISO documents should be noted. This document was drafted in accordance with the
editorial rules of the ISO/IEC Directives, Part 2. www.iso.org/directives.
Attention is drawn to the possibility that some of the elements of this document may be the subject of
patent rights. ISO shall not be held responsible for identifying any or all such patent rights. Details of any
patent rights identified during the development of the document will be in the Introduction and/or on
the ISO list of patent declarations received. www.iso.org/patents.
Any trade name used in this document is information given for the convenience of users and does not
constitute an endorsement.
The committee responsible for this document is ISO/TC 123, Plain bearings, Subcommittee SC 4, Methods
of calculation of plain bearings.
This second edition cancels and replaces the first edition (ISO 7902-1:1998), which has been
technically revised.
ISO 7902 consists of the following parts, under the general title Hydrodynamic plain journal bearings
under steady-state conditions — Circular cylindrical bearings:
— Part 1: Calculation procedure
— Part 2: Functions used in the calculation procedure
— Part 3: Permissible operational parameters
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INTERNATIONAL STANDARD ISO 7902-1:2013(E)
Hydrodynamic plain journal bearings under steady-state
conditions — Circular cylindrical bearings —
Part 1:
Calculation procedure
1 Scope
This part of ISO 7902 specifies a calculation procedure for oil-lubricated hydrodynamic plain bearings,
with complete separation of the shaft and bearing sliding surfaces by a film of lubricant, used for
designing plain bearings that are reliable in operation.
It deals with circular cylindrical bearings having angular spans, Ω, of 360°, 180°, 150°, 120°, and 90°,
the arc segment being loaded centrally. Their clearance geometry is constant except for negligible
deformations resulting from lubricant film pressure and temperature.
The calculation procedure serves to dimension and optimize plain bearings in turbines, generators,
electric motors, gear units, rolling mills, pumps, and other machines. It is limited to steady-state
operation, i.e. under continuously driven operating conditions, with the magnitude and direction
of loading as well as the angular speeds of all rotating parts constant. It can also be applied if a full
plain bearing is subjected to a constant force rotating at any speed. Dynamic loadings, i.e. those whose
magnitude and direction vary with time, such as can result from vibration effects and instabilities of
rapid-running rotors, are not taken into account.
2 Normative references
The following documents, in whole or in part, are normatively referenced in this document and are
indispensable for its application. For dated references, only the edition cited applies. For undated
references, the latest edition of the referenced document (including any amendments) applies.
ISO 3448, Industrial liquid lubricants — ISO viscosity classification
ISO 7902-2:1998, Hydrodynamic plain journal bearings under steady-state conditions — Circular cylindrical
bearings — Part 2: Functions used in the calculation procedure
ISO 7902-3, Hydrodynamic plain journal bearings under steady-state conditions — Circular cylindrical
bearings — Part 3: Permissible operational parameters
3 Basis of calculation, assumptions, and preconditions
3.1 The basis of calculation is the numerical solution to Reynolds’ differential equation for a finite bearing
length, taking into account the physically correct boundary conditions for the generation of pressure:
∂ ∂p ∂ ∂p ∂h
33
h + h =+6η uu (1)
()
JB
∂x ∂xx∂ ∂z ∂x
The symbols are given in Clause 5.
See References [1] to [3] and References [11] to [14] for the derivation of Reynolds’ differential equation
and References [4] to [6], [12], and [13] for its numerical solution.
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ISO 7902-1:2013(E)
3.2 The following idealizing assumptions and preconditions are made, the permissibility of which has
been sufficiently confirmed both experimentally and in practice.
a) The lubricant corresponds to a Newtonian fluid.
b) All lubricant flows are laminar.
c) The lubricant adheres completely to the sliding surfaces.
d) The lubricant is incompressible.
e) The lubricant clearance gap in the loaded area is completely filled with lubricant. Filling up of the
unloaded area depends on the way the lubricant is supplied to the bearing.
f) Inertia effects, gravitational and magnetic forces of the lubricant are negligible.
g) The components forming the lubrication clearance gap are rigid or their deformation is negligible;
their surfaces are ideal circular cylinders.
h) The radii of curvature of the surfaces in relative motion are large in comparison with the lubricant
film thicknesses.
i) The lubricant film thickness in the axial direction (z-coordinate) is constant.
j) Fluctuations in pressure within the lubricant film normal to the bearing surfaces (y-coordinate)
are negligible.
k) There is no motion normal to the bearing surfaces ( y-coordinate).
l) The lubricant is isoviscous over the entire lubrication clearance gap.
m) The lubricant is fed in at the start of the bearing liner or where the lubrication clearance gap is
widest; the magnitude of the lubricant feed pressure is negligible in comparison with the lubricant
film pressures.
3.3 The boundary conditions for the generation of lubricant film pressure fulfil the following
continuity conditions:
— at the leading edge of the pressure profile: pzϕ , =0 ;
()
1
— at the bearing rim: pzϕ, =±B 20= ;
()
— at the trailing edge of the pressure profile: pzϕ ,z =0 ;
()
2
— ∂∂pzϕϕ ,z =0 .
()
2
For some types and sizes of bearing, the boundary conditions may be specified.
In partial bearings, if Formula (2) is satisfied:
π
ϕπ−−β < (2)
()
2
2
then the trailing edge of the pressure profile lies at the outlet end of the bearing:
pzϕϕ= , =0 (3)
()
2
3.4 The numerical integration of the Reynolds’ differential equation is carried out (possibly by
applying transformation of pressure as suggested in References [3], [11], and [12]) by a transformation
to a differential formula which is applied to a grid system of supporting points, and which results in a
system of linear formulae. The number of supporting points is significant to the accuracy of the numerical
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ISO 7902-1:2013(E)
integration; the use of a non-equidistant grid as given in References [6] and [13] is advantageous. After
substituting the boundary conditions at the trailing edge of the pressure profile, integration yields the
pressure distribution in the circumferential and axial directions.
The application of the similarity principle to hydrodynamic plain bearing theory results in dimensionless
magnitudes of similarity for parameters of interest, such as load-carrying capacity, frictional behaviour,
lubricant flow rate, and relative bearing length. The application of magnitudes of similarity reduces the
number of numerical solutions required of Reynolds’ differential equation specified in ISO 7902-2. Other
solutions may also be applied, provided they fulfil the conditions laid down in ISO 7902-2 and are of a
similar numerical accuracy.
3.5 ISO 7902-3 includes permissible operational parameters towards which the result of the calculation
shall be oriented in order to ensure correct functioning of the plain bearings.
In special cases, operational parameters deviating from ISO 7902-3 may be agreed upon for specific
applications.
4 Calculation procedure
4.1 Calculation is understood to mean determination of correct operation by computation using actual
operating parameters (see Figure 1), which can be compared with operational parameters. The operating
parameters determined under varying operating conditions shall therefore lie within the range of
permissibility as compared with the operational parameters. To this end, all operating conditions during
continuous operation shall be investigated.
4.2 Freedom from wear is guaranteed only if complete separation of the mating bearing parts is
achieved by the lubricant. Continuous operation in the mixed friction range results in failure. Short-time
operation in the mixed friction range, for example starting up and running down machines with plain
bearings, is unavoidable and does not generally result in bearing damage. When a bearing is subjected to
heavy load, an auxiliary hydrostatic arrangement may be necessary for starting up and running down at a
slow speed. Running-in and adaptive wear to compensate for deviations of the surface geometry from the
ideal are permissible as long as they are limited in area and time and occur without overloading effects.
In certain cases, a specific running-in procedure may be beneficial, depending on the choice of materials.
4.3 The limits of mechanical loading are a function of the strength of the bearing material. Slight permanent
deformations are permissible as long as they do not impair correct functioning of the plain bearing.
4.4 The limits of thermal loading result not only from the thermal stability of the bearing material but
also from the viscosity-temperature relationship and by degradation of the lubricant.
4.5 A correct calculation for plain bearings presupposes that the operating conditions are known for
all cases of continuous operation. In practice, however, additional influences frequently occur, which
are unknown at the design stage and cannot always be predicted. The application of an appropriate
safety margin between the actual operating parameters and permissible operational parameters is
recommended. Influences include, for example:
— spurious forces (out-of-balance, vibrations, etc.);
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ISO 7902-1:2013(E)
yes
Figure 1 — Outline of calculation
— deviations from the ideal geometry (machining tolerances, deviations during assembly, etc.);
— lubricants contaminated by dirt, water, air, etc.;
— corrosion, electrical erosion, etc.
Data on other influencing factors are given in 6.7.
4.6 The Reynolds number shall be used to verify that ISO 7902-2, for which laminar flow in the
lubrication clearance gap is a necessary condition, can be applied:
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ISO 7902-1:2013(E)
C C
Re,,ff Reff
ρU πDN
J J
D
22
Re== ≤41,3 (4)
η v C
Re, ff
In the case of plain bearings with Re>41,3 DC (for example as a result of high peripheral speed),
R,eff
higher loss coefficients and bearing temperatures shall be expected. Calculations for bearings with
turbulent flow cannot be carried out in accordance with this part of ISO 7902.
4.7 The plain bearing calculation takes into account the following factors (starting with the known
bearing dimensions and operational data):
— the relationship between load-carrying capacity and lubricant film thickness;
— the frictional power rate;
— the lubricant flow rate;
— the heat balance.
All these factors are mutually dependent.
The solution is obtained using an iterative method; the sequence is outlined in the flow chart in Figure 1.
For optimization of individual parameters, parameter variation can be applied; modification of the
calculation sequence is possible.
5 Symbols and units
See Figure 2 and Table 1.
Minimum lubricant film thickness, h :
min
DD−
J
h = −=eD0,51ψε− (5)
()
min
2
where the relative eccentricity, ε, is given by
e
ε = (6)
DD−
J
2
If
π
ϕπ−−()β < (7)
2
2
then
hD=+05,(ψε1 cos)ϕ (8)
min 2
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ISO 7902-1:2013(E)
6 Definition of symbols
6.1 Load-carrying capacity
A characteristic parameter for the load-carrying capacity is the dimensionless Sommerfeld number, So:
2
Fψ
B
eff
So==So ε,,Ω (9)
DBηω D
effh
Values of So as a function of the relative eccentricity, ε, the relative bearing length, B/D, and the angular
span of bearing segment, Ω, are given in ISO 7902-2. The variables ω , η , and ϕ take into account the
h eff eff
thermal effects and the angular velocities of shaft, bearing, and bearing force (see 6.4 and 6.7).
The relative eccentricity, ε, together with the attitude angle, β (see ISO 7902-2), describes the magnitude
and position of the minimum thickness of lubricant film. For a full bearing (Ω = 360°), the oil should be
introduced at the greatest lubricant clearance gap or, with respect to the direction of rotation, shortly
before it. For this reason, it is useful to know the attitude angle, β.
Figure 2 — Illustration of symbols
Table 1 — Symbols and their designations
Symbol Designation Unit
2
A Area of heat-emitting surface (bearing housing) m
b Width of oil groove m
G
B Nominal bearing width m
c Specific heat capacity of the lubricant J/(kg·K)
C Nominal bearing clearance m
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ISO 7902-1:2013(E)
Table 1 (continued)
Symbol Designation Unit
C Effective bearing radial clearance m
R,eff
d Oil hole diameter m
L
D Nominal bearing diameter (inside diameter) m
D Nominal shaft diameter m
J
D , Maximum value of D m
J max J
D , Minimum value of D m
J min J
D Maximum value of D m
max
D Minimum value of D m
min
e Eccentricity between the axis of the shaft and the bearing axis m
E Modulus of elasticity 1
f Coefficient of friction 1
F Bearing force (nominal load) N
F Friction force in the loaded area of the lubricant film N
f
'
Frictional force in the unloaded area of the lubricant film N
F
f
G Shear modulus 1
h Local lubricant film thickness m
h Minimum permissible lubricant film thickness m
lim
h Minimum lubricant film thickness m
min
h Waviness of sliding surface m
wav
h Effective waviness of sliding surface m
wav,eff
h Maximum permissible effective waviness m
wav,eff,lim
k Outer heat transmission coefficient w/(m2·K)
A
l Length of oil groove m
G
l Length of oil pocket m
P
L Length of bearing housing Rotational m
H
−1
N Frequency of the bearing Rotational s
B
−1
N Frequency of the bearing force Rotational s
F
−1
N Frequency of the shaft s
J
p Local lubricant film pressure Pa
p Specific bearing load Pa
P Lubricant feed pressure Pa
en
p Maximum permissible lubricant film pressure Pa
lim
p Maximum permissible specific bearing load Pa
lim
P Frictional power W
f
P Heat flow rate W
th
P Heat flow rate to the ambient W
th,amb
P Heat flow rate due to frictional power W
th,f
P Heat flow rate in the lubricant W
th,L
3
Q Lubricant flow rate m /s
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ISO 7902-1:2013(E)
Table 1 (continued)
Symbol Designation Unit
3
Q Lubricant flow rate at the inlet to clearance gap m /s
1
3
Q Lubricant flow rate at the outlet to clearance gap m /s
2
3
Q Lubricant flow rate due to hydrodynamic pressure m /s
3
*
Lubricant flow rate parameter due to hydrodynamic pressure 1
Q
3
3
Q Lubricant flow rate due to feed pressure m /s
p
*
Lubricant flow rate parameter due to feed pressure 1
Q
p
Rz Average peak-to-valley height of bearing sliding surface m
B
Rz Average peak-to-valley height of shaft mating surface m
J
Re Reynolds number 1
So Sommerfeld number 1
T Ambient temperature °C
amb
T Bearing temperature °C
B
T Assumed initial bearing temperature °C
B,0
T Calculated bearing temperature resulting from iteration procedure °C
B,1
T Lubricant temperature at bearing entrance °C
en
T Lubricant temperature at bearing exit °C
ex
T Assumed initial lubricant temperature at bearing exit °C
ex,0
T Calculated lubricant temperature at bearing exit °C
ex,1
T Shaft temperature °C
J
T Maximum permissible bearing temperature °C
lim
Mean lubricant temperature °C
T
L
U Linear velocity (peripheral speed) of bearing m/s
B
U Linear velocity (peripheral speed) of shaft m/s
J
V Air ventilating velocity m/s
a
x Coordinate parallel to the sliding surface in the circumferential direction m
y Coordinate perpendicular to the sliding surface m
z Coordinate parallel to the sliding surface in the axial direction m
−1
α Linear heat expansion coefficient of the bearing K
l,B
−1
α Linear heat expansion coefficient of the shaft K
l,J
β Attitude angle (angular position of the shaft eccentricity related to the direction °
of load)
δ Angle of misalignment of the shaft rad
J
ε Relative eccentricity 1
η Dynamic viscosity of the lubricant Pa·s
η Effective dynamic viscosity of the lubricant Pa·s
eff
v Kinematic viscosity of the lubricant Pa·s
ξ
Coefficient of resistance to rotation in the loaded area of the lubricant film 1
ξ '
Coefficient of resistance to rotation in the unloaded area of the lubricant film 1
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ISO 7902-1:2013(E)
Table 1 (continued)
Symbol Designation Unit
ξ Coefficient of resistance to rotation in the area of circumferential groove 1
G
Coefficient of resistance to rotation in the area of the pocket 1
ξ
P
3
ρ Density of lubricant kg/m
φ Angular coordinate in the circumferential direction rad
φ Angular coordinate of pressure leading edge rad
1
φ Angular coordinate of pressure trailing edge rad
2
ψ
Relative bearing clearance 1
ψ Mean relative bearing clearance 1
ψ Effective relative bearing clearance 1
eff
ψ Maximum relative bearing clearance 1
max
Minimum relative bearing clearance 1
ψ
min
−1
ω Angular velocity of bearing s
B
−1
ω Hydrodynamic angular velocity s
h
−1
ω Angular velocity of shaft s
J
Ω Angular span of bearing segment °
Ω Angular span of lubrication groove °
G
Ω Angular span of lubrication pocket °
P
6.2 Frictional power loss
Friction in a hydrodynamic plain bearing due to viscous shear stress is given by the coefficient of friction
f = F /F and the derived non-dimensional characteristics of frictional power loss ξ and f /ψ
f
eff
Fψ
feff
ξ = (10)
DBηω
effh
f ξ
= (11)
ψ So
eff
They are applied if the frictional power loss is encountered only in the loaded area of the lubricant film.
It is still necessary to calculate frictional power loss in both the loaded and unloaded areas then the values
f
fF,,ξ,
f
ψ
eff
are substituted by:
f '
'
fF', ,'ξ
f
ψ
eff
in Formulae (10) and (11). This means that the whole of the clearance gap is filled with lubricant.
′
The values of f ψ and f ψ for various values of ε, B/D, and Ω are given in ISO 7902-2. It also gives
eff eff
the approximation formulae, based on Reference [15], which are used to determine frictional power loss
values in the bearings, taking account of the influence of lubricating pockets and grooves.
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ISO 7902-1:2013(E)
The frictional power in a bearing or the amount of heat generated is given by:
D
PP== fF ω (12)
fth,fh
2
D
'
P = fF' ω (13)
fh
2
6.3 Lubricant flow rate
The lubricant fed to the bearing forms a film of lubricant separating the sliding surfaces. The pressure
build-up in this film forces lubricant out of the ends of the bearing. This is the proportion Q of the
lubricant flow rate, resulting from the build-up of hydrodynamic pressure.
3 *
QD= ψω Q (14)
3 effh 3
**
where QQ= εΩ,,BD is given in ISO 7902-2.
()
33 1
There is also a flow of lubricant in the peripheral direction through the narrowest clearance gap into the
diverging, pressure-free gap. For increased loading and with a small lubrication gap clearance; however,
this proportion of the lubricant flow is negligible.
The lubricant feed pressure, p , forces additional lubricant out of the ends of the plain bearing. This is
en
the amount Q of the lubricant flow rate resulting from feed pressure:
p
33
Dpψ
effen *
Q = Q (15)
p p
η
eff
**
where QQ= εΩ,,BD is given in ISO 7902-2.
()
pp
6.3.1 Lubricant feed elements are lubrication holes, lubrication grooves, and lubrication pockets. The
lubricant feed pressure, p , should be markedly less than the specific bearing load, p , to avoid additional
en
hydrostatic loads. Usually, p lies between 0,05 MPa and 0,2 MPa. The depth of the lubrication grooves
en
and lubrication pockets is considerably greater than the bearing clearance.
6.3.2 Lubrication grooves are elements designed to distribute lubricant in the circumferential direction.
The recesses machined into the sliding surface run circumferentially and are kept narrow in the axial
direction. If lubrication grooves are located in the vicinity of pressure rise, the pressure distribution
is split into two independent pressure “hills” and the load-carrying capacity is markedly reduced (see
Figure 3). In this case, the calculation shall be carried out for half the load applied to each half bearing.
However, because of the build-up of hydrodynamic pressure, Q , only half of the lubricant flow rate shall
3
be taken into account when balancing heat losses (see 6.4), since the return into the lubrication groove
plays no part in dissipating heat. It is more advantageous, for a full bearing, to arrange the lubrication
groove in the unloaded part. The entire lubricant flow amount, Q , goes into the heat balance.
p
6.3.3 Lubrication pockets are elements for distributing the lubricant over the length of the bearing.
The recesses machined into the sliding surface are oriented in the axial direction and should be as short
as possible in the circumferential direction. Relative pocket lengths should be such shall b /B < 0,7.
p
Although larger values increase the oil flow rate, the oil emerging over the narrow, restricting webs at the
ends plays no part in dissipating heat. This is even more true if the end webs are penetrated axially. For
full bearings (Ω = 360°), a lubrication pocket opposite to the direction of load as well as two lubrication
pockets normal to the direction of loading are machined in. Since the lubricant flow rate, even in the
unloaded part of the bearing, provides for the dissipation of frictional heat arising from shearing, the
lubricating pockets shall be fully taken into account in the heat balance. For shell segments (Ω < 360°),
the lubricant flow rate due to feed pressure through lubrication pockets at the inlet or outlet of the shell
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ISO 7902-1:2013(E)
segment makes practically no contribution to heat dissipation, since the lubrication pockets are scarcely
restricted at the segment ends and the greater proportion of this lubricant flow emerges directly.
If the lubricant fills the loaded area of the bearing and there is no lubricant in the unloaded part, then
the heat dissipation counts as lubricant flow rate in the loaded part only.
Key
1 lubrication hole
2 lubrication groove
Figure 3
The influence of the type and the arrangement of the lubricant feed elements on the lubricant flow rate
are dealt with in ISO 7902-2.
The overall lubricant flow rate is given by:
QQ= (16)
3
for lubricant filling only the loaded area of the bearing;
QQ=+ Q (17)
3p
for lubricant filling the whole circular lubrication clearance gap including unloaded part, i.e. 2π.
6.4 Heat balance
The thermal condition of the plain bearing can be obtained from the heat balance. The heat flow, P ,
th,f
arising from frictional power in the bearing, P , is dissipated via the bearing housing to the environment
f
and the lubricant emerging from the bearing. In practice, one or other of the two types of heat dissipation
dominates. By neglecting the other, an additional safety margin is obtained during the design stage. The
following assumptions can be made:
a) Pressureless-lubricated bearings (for example ring lubrication) dissipate heat mainly through
convection to the environment: P = P
th,f th,amb
b) Pressure-lubricated bearings dissipate heat mainly via the lubricant: P = P
th,f th,L
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ISO 7902-1:2013(E)
6.4.1 Heat dissipation by convection
Heat dissipation by convection takes place by thermal conduction in the bearing housing and radiation
and convection from the surface of the housing to the environment. The complex processes during the
heat transfer can be summed up by:
Pk=−AT T (18)
()
th,amb A Bamb
where
2
()
k =⋅15to20W/ mK
A
or, by ventilating the bearin
...
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