ISO 7902-1:2020

Hydrodynamic plain journal bearings under steady-state conditions - Circular cylindrical bearings - Part 1: Calculation procedure

This document 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 provide dimensions 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 those that can result from vibration effects and instabilities of rapid-running rotors, are not taken into account. NOTE Equivalent calculation procedures exist that enable operating conditions to be estimated and checked against acceptable conditions. The use of them is equally admissible.

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

General Information

• Status: Published
• Publication Date: 09-Jun-2020
• ICS: 21.100.10 - Plain bearings
• Technical Committee: ISO/TC 123/SC 8 - Calculation methods for plain bearings and their applications
• Drafting Committee: ISO/TC 123/SC 8 - Calculation methods for plain bearings and their applications
• Current Stage: 9092 - International Standard to be revised
• Start Date: 03-Nov-2025
• Completion Date: 13-Dec-2025
• Ref Project: SIST ISO 7902-1:2021 - Hydrodynamic plain journal bearings under steady-state conditions - Circular cylindrical bearings - Part 1: Calculation procedure

Relations

• Revises: ISO 7902-1:2013 - Hydrodynamic plain journal bearings under steady-state conditions - Circular cylindrical bearings - Part 1: Calculation procedure
• Effective Date: 23-Apr-2020

Overview

ISO 7902-1:2020 defines a standardized calculation procedure for oil-lubricated, hydrodynamic plain journal bearings operating in steady-state conditions. The standard covers circular cylindrical bearings with angular spans of 360°, 180°, 150°, 120° and 90° (centrally loaded arc segments) and assumes complete separation of shaft and bearing sliding surfaces by a lubricant film. It is the authoritative guide for sizing and optimizing plain bearings where clearance geometry is effectively constant (except for negligible deformations from film pressure and temperature). The third edition (2020) updates the calculation method and annexed examples for practical design use.

Key Topics

• Calculation basis: Numerical solution of the Reynolds equation for finite bearing length with physically correct pressure boundary conditions.
• Assumptions & scope: Oil-lubricated, full-film hydrodynamic operation under steady-state (constant load magnitude/direction and constant rotational speeds). Dynamic or time-varying loads (e.g., rotor vibration, transient events) are excluded.
• Geometry covered: Circular cylindrical bearings with specified angular spans and centrally loaded arc segments.
• Performance criteria & limits:
  • Freedom from wear and minimum lubricant film thickness requirements
  • Mechanical and thermal loading limits and permissible operational parameters
  • Frictional power loss and heat balance (including convection and lubricant heat flow)
• Hydrodynamic and hydraulic factors:
  • Reynolds number, Sommerfeld number, lubricant flow rates (hydrodynamic, feed, pocket/groove effects)
  • Calculation factors for resistance to rotation and lubricant supply
• Symbols, units and definitions: Complete lists for load-carrying capacity, frictional losses, lubricant flow, temperatures, clearances, and more
• Practical tools: Iterative heat-balance approach and worked examples (Annex A) to converge bearing temperature and lubricant conditions

Applications

ISO 7902-1 is intended for engineers and designers who need reliable plain bearing designs in rotating equipment, including:

• Turbines and generators
• Electric motors and gear units
• Pumps and compressors
• Rolling mills and heavy machinery
• OEMs, tribologists, rotating equipment engineers, maintenance and reliability teams

Use cases: bearing dimensioning, optimization for minimal wear and acceptable temperature rise, estimation of frictional losses and lubricant flow needs, and pre-checking operating conditions against permissible parameters.

Related Standards

• ISO 7902-2 - Functions used in the calculation procedure (normative reference)
• ISO 7902-3 - Permissible operational parameters (normative reference)

Note: Equivalent calculation procedures that yield comparable operating-condition estimates are also admissible under the scope of this standard.

Standards Content (Sample)

SLOVENSKI STANDARD
01-oktober-2021
Nadomešča:
SIST ISO 7902-1:2015
Hidrodinamični radialni drsni ležaji za neprekinjeno obratovanje - Valjasti ležaji - 1.
del: Postopek dimenzioniranja
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 7902-1:2020
ICS:
21.100.10 Drsni ležaji Plain bearings
2003-01.Slovenski inštitut za standardizacijo. Razmnoževanje celote ali delov tega standarda ni dovoljeno.

INTERNATIONAL ISO
STANDARD 7902-1
Third edition
2020-06
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 2020
© ISO 2020
All rights reserved. Unless otherwise specified, or required in the context of its implementation, 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.
ISO copyright office
CP 401 • Ch. de Blandonnet 8
CH-1214 Vernier, Geneva
Phone: [phone]
Fax: [phone]
Email: [e-mail]
Website: www.iso.org
Published in Switzerland
ii © ISO 2020 – All rights reserved

Contents Page
Foreword .iv
1 Scope . 1
2 Normative references . 1
3 Terms and definitions . 1
4 Symbols and units . 1
5 Basis of calculation, assumptions, and preconditions . 5
5.1 Reynolds equation . 5
5.2 Assumptions and preconditions . 5
5.3 Boundary conditions . 5
5.4 Basis of calculation . 6
5.5 Permissible operational parameters . 6
6 Calculation procedure . 6
6.1 General . 6
6.2 Freedom from wear . 6
6.3 The limits of mechanical loading. 7
6.4 The limits of thermal loading . 7
6.5 Influencing factors . 7
6.6 Reynolds number . 7
6.7 Calculation factors . 7
7 Definition of symbols . 9
7.1 Load-carrying capacity . 9
7.2 Frictional power loss . 9
7.3 Lubricant flow rate .10
7.3.1 General.10
7.3.2 Lubricant feed elements .10
7.3.3 Lubrication grooves .10
7.3.4 Lubrication pockets .10
7.3.5 Lubricant flow rate .11
7.4 Heat balance .11
7.4.1 General.11
7.4.2 Heat dissipation by convection .12
7.4.3 Heat dissipation via the lubricant .12
7.5 Minimum lubricant film thickness and specific bearing load .13
7.6 Operational conditions.14
7.7 Further influencing factors .14
Annex A (informative) Calculation examples .17
Bibliography .32
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 (see 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 (see 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.
For an explanation of the voluntary nature of standards, the meaning of ISO specific terms and
expressions related to conformity assessment, as well as information about ISO's adherence to the
World Trade Organization (WTO) principles in the Technical Barriers to Trade (TBT), see www .iso .org/
iso/ foreword .html.
This document was prepared by Technical Committee ISO/TC 123, Plain bearings, Subcommittee SC 8,
Calculation methods for plain bearings and their applications.
This third edition cancels and replaces the second edition (ISO 7902-1:2013), which has been technically
revised.
The main changes compared to the previous edition are as follows:
— subclause titles have been added;
— symbols have been corrected and added in Table 1;
— calculation values in Annex A have been corrected;
— adjustments have been made to ISO/IEC Directives, Part 2:2018;
— typographical errors have been corrected.
A list of all parts in the ISO 7902 series can be found on the ISO website.
Any feedback or questions on this document should be directed to the user’s national standards body. A
complete listing of these bodies can be found at www .iso .org/ members .html.
iv © ISO 2020 – All rights reserved

INTERNATIONAL STANDARD ISO 7902-1:2020(E)
Hydrodynamic plain journal bearings under steady-state
conditions — Circular cylindrical bearings —
Part 1:
Calculation procedure
1 Scope
This document 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 provide dimensions 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 those that can result from vibration effects and
instabilities of rapid-running rotors, are not taken into account.
NOTE Equivalent calculation procedures exist that enable operating conditions to be estimated and checked
against acceptable conditions. The use of them is equally admissible.
2 Normative references
The following documents are referred to in the text in such a way that some or all of their content
constitutes requirements of this document. For dated references, only the edition cited applies. For
undated references, the latest edition of the referenced document (including any amendments) applies.
ISO 7902-2, 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 Terms and definitions
No terms and definitions are listed in this document.
ISO and IEC maintain terminological databases for use in standardization at the following addresses:
— ISO Online browsing platform: available at https:// www .iso .org/ obp
— IEC Electropedia: available at http:// www .electropedia .org/
4 Symbols and units
Symbols and units are defined in Figure 1 and Table 1.
Figure 1 — Illustration of symbols
Table 1 — Symbols and their designations
Symbol Designation Unit
A Area of heat-emitting surface (bearing housing) m
b Width of lubrication groove m
G
b Width of lubrication pocket m
P
B Nominal bearing width m
B Length of the axial housing m
H
c Specific heat capacity of the lubricant J/(kg·K)
p
C Nominal bearing clearance m
C Effective bearing radial clearance m
R,eff
d Lubrication hole diameter m
L
D Nominal bearing diameter (inside diameter) m
D Length of the outside diameter of the housing m
H
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
f Coefficient of friction in the loaded area of the lubricant film ( f = F /F) 1
f
f′ Coefficient of friction in both the loaded and unloaded area of the lubricant film 1
F Bearing force (nominal load) N
F Friction force in the loaded area of the lubricant film N
f
2 © ISO 2020 – All rights reserved

Table 1 (continued)
Symbol Designation Unit
′ Frictional force in both the loaded and the unloaded area of the lubricant film N
F
f
h Local lubricant film thickness m
h Effective lubricant film thickness m
eff
h Depth of lubrication groove m
G
h Minimum permissible lubricant film thickness m
lim
h Minimum lubricant film thickness m
min
h Depth of lubrication pocket m
P
H Length of the total height of the pedestal bearing m
k Outer heat transmission coefficient W/(m ·K)
A
−1
N Rotational frequency of the bearing s
B
−1
N Rotational 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
Maximum permissible specific bearing load Pa
p
lim
P Frictional power W
f
P ′ Frictional power in both the loaded and the unloaded area of the lubricant film 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
q Coefficient related to lubricant flow rate due to feed pressure 1
L
q Coefficient related to lubricant flow rate from pocket 1
P
Q Lubricant flow rate m /s
Q Lubricant flow rate due to hydrodynamic pressure m /s
* Lubricant flow rate parameter due to hydrodynamic pressure 1
Q
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
So Transition Sommerfeld number 1
u
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 Effective lubricant temperature °C
eff
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
Table 1 (continued)
Symbol Designation Unit
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
Attitude angle (angular position of the shaft eccentricity related to the direction °
β
of load)
ε Relative eccentricity [ε = 2e/(D – D )] 1
J
ε Transition eccentricity 1
u
η Dynamic viscosity of the lubricant Pa·s
η Effective dynamic viscosity of the lubricant Pa·s
eff
v Kinematic viscosity of the lubricant m /s
ξ
Coefficient of resistance to rotation in the loaded area of the lubricant film 1
Coefficient of resistance to rotation in both the loaded and the unloaded area of 1
′
ξ
the lubricant film
Coefficient of resistance to rotation in the area of circumferential groove 1
ξ
G
Coefficient of resistance to rotation in the area of the pocket 1
ξ
P
ρ Density of lubricant kg/m
φ Angular coordinate in the circumferential direction rad
φ Angular coordinate of pressure leading edge rad
φ Angular coordinate of pressure trailing edge rad
ψ
Relative bearing clearance 1
ψ
Mean relative bearing clearance 1
Effective relative bearing clearance 1
ψ
eff
Maximum relative bearing clearance 1
ψ
max
Minimum relative bearing clearance 1
ψ
min
−1
ω Angular velocity of bearing s
B
−1
ω Angular velocity of rotating force s
F
−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
4 © ISO 2020 – All rights reserved

5 Basis of calculation, assumptions, and preconditions
5.1 Reynolds equation
The basis of calculation is the numerical solution to Reynolds equation for a finite bearing length,
taking into account the physically correct boundary conditions for the generation of pressure. Reynolds
equation is defined as Formula (1).
∂ ∂p ∂ ∂p ∂h
   
h + h =6η UU+ (1)
    ()
JB
∂x  ∂xz ∂  ∂z  ∂x
See References [3] to [5] and References [13] to [16] for the derivation of Reynolds equation and
References [6] to [8], [14] and [15] for its numerical solution.
5.2 Assumptions and preconditions
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.
5.3 Boundary conditions
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 ;
()
— at the bearing rim:pzϕ,/=±B 20= ;
()
— at the trailing edge of the pressure profile:pzϕ (),z =0 ;
[]
— ∂∂pz/,ϕϕ () z =0 .
[]
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)
Then the trailing edge of the pressure profile lies at the outlet end of the bearing is:
pzϕϕ= , =0
()
5.4 Basis of calculation
The numerical integration of the Reynolds equation is carried out (possibly by applying transformation
of pressure as suggested in References [5], [13] and [14]) 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 integration; the use
of a non-equidistant grid as given in References [8] and [15] 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 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.
5.5 Permissible operational parameters
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.
6 Calculation procedure
6.1 General
Calculation is understood to mean determination of correct operation by computation using actual
operating parameters (see Figure 2), 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.
6.2 Freedom from wear
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
6 © ISO 2020 – All rights reserved

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.
6.3 The limits of mechanical loading
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.
6.4 The limits of thermal loading
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.
6.5 Influencing factors
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 (e.g. out-of-balance, vibrations);
— deviations from the ideal geometry (e.g. machining tolerances, deviations during assembly);
— lubricants contaminated by, for example, dirt, water, air;
— corrosion, electrical erosion.
Data on other influencing factors are given in 7.7.
6.6 Reynolds number
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:
C C
Re,,ff Reff
ρU πDN
J J
D
Re== ≤41,3 (3)
η v C
Re, ff
In the case of plain bearings with Re>41,3 DC/ (e.g. as a result of high peripheral speed), higher
R,eff
loss coefficients and bearing temperatures shall be expected. Calculations for bearings with turbulent
flow cannot be carried out in accordance with this document.
6.7 Calculation factors
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 2.
For optimization of individual parameters, parameter variation can be applied; modification of the
calculation sequence is possible.
Figure 2 — Outline of calculation
8 © ISO 2020 – All rights reserved

7 Definition of symbols
7.1 Load-carrying capacity
A characteristic parameter for the load-carrying capacity is the dimensionless Sommerfeld number, So:
Fψ
B
 
eff
So==So ε ,,Ω (4)
 
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 ψ take into account
h eff
eff
the thermal effects and the angular velocities of shaft, bearing, and bearing force (see 7.4 and 7.7).
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.
7.2 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= (5)
DBηω
effh
f ξ
= (6)
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
f f′
values, fF,,ξ,and , are substituted by fF′′,,ξ′, and , respectively in Formulae (5) and (6).
f f
ψ ψ
eff eff
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 [17], which are used to determine frictional power
loss values in the bearings, taking account of the influence of lubrication pockets and grooves.
The frictional power in a bearing or the amount of heat generated is given by Formulae (7) and (8).
D
PP== fF ω (7)
fthf, h
D
′ ′
Pf= F ω (8)
fh
7.3 Lubricant flow rate
7.3.1 General
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 (9)
3 effh 3
**
where QQ= ()εΩ,/BD, is given in ISO 7902-2.
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 as defined by Formula (10).
p
Dpψ
eff en
*
Q = Q (10)
pp
η
eff
**
where QQ= εΩ,/BD, is given in ISO 7902-2.
()
pp
7.3.2 Lubricant feed elements
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
en
additional hydrostatic loads. Usually, p lies between 0,05 MPa and 0,2 MPa. The depth of the lubrication
en
grooves and lubrication pockets is considerably greater than the bearing clearance.
7.3.3 Lubrication grooves
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 be taken into account when balancing heat losses (see 7.4), 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
p
heat balance.
7.3.4 Lubrication pockets
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 as b /B < 0,7. Although
p
larger values increase the lubricant 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
lubrication 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
10 © ISO 2020 – All rights reserved

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.
Key
1 lubrication hole
2 lubrication groove
Figure 3 — Lubricant film pressure in bearings with lubrication groove
7.3.5 Lubricant flow rate
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.
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
— for lubricant filling only in the loaded area of the bearing:
QQ= (11)
— for lubricant filling in the whole circular lubrication clearance gap including unloaded part, i.e. 2π:
QQ=+Q (12)
3p
7.4 Heat balance
7.4.1 General
The thermal condition of the plain bearing can be obtained from the heat balance. The heat flow,
P , arising from frictional power in the bearing, P , is dissipated via the bearing housing to the
th,f f
environment 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 (e.g. 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
Calculation examples are introduced in Annex A.
7.4.2 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 (13)
()
th,ambA Bamb
where
k =()15 to20 Wm/ ⋅K
()
A
or, by ventilating the bearing housing with air at a velocity of V > 1,2 m/s
a
kV=+712
Aa
See References [5] and [16].
Should the area of the heat-emitting surface, A, of the bearing housing not be known exactly, the
following can be used as an approximation
— for cylindrical housings:
π
AD=−2 DD+π B
()
HH H
— for pedestal bearings:
H
 
AH=+π B
 
H
 2 
— for bearings in the machine structure:
AD=()15to20 B
7.4.3 Heat dissipation via the lubricant
In the case of force-feed lubrication, heat dissipation is via the lubricant:
Pc=−ρ QT T (14)
()
th,Lep xen
For mineral lubricants, the volume-specific heat is given by:
ρ×=c 18,/×⋅10 Jm K
()
p
From the heat balance, it follows that PP= for pressureless-lubricated bearings and PP=
th,,fthamb th,,fthL
for pressure-lubricated bearings.
This gives bearing temperature, T (see Reference [17]), and lubricant outlet temperature, T
B ex
(see Reference [17]). The effective film lubricant temperature with reference to the lubricant viscosity is
a) in the case of pure convection: T = T , and
eff B
12 © ISO 2020 – All rights reserved

b) in the case of heat dissipation via the lubricant: T =T = 0,5 (T + T ).
eff en ex
L
At high peripheral speed, it is possible to select, instead of these mean values, a temperature which lies
nearer to the lubricant outlet temperature.
The values calculated for T and T shall be checked for their permissibility by comparison with the
B ex
permissible operational parameters, T , given in ISO 7902-3.
lim
In the sequence of calculations, at first only the operational data T or T are known, but not the
amb en
effective temperature, T , which is required at the start of the calculation. The solution is obtained by
eff
first starting the calculation using an estimated temperature rise, i.e.
a) T − T = 20 K
B,0 amb
b) T − T = 20 K
ex,0 en
and the corresponding operating temperatures, T . From the heat balance, corrected temperatures,
eff
T or T , are obtained, which, by averaging with the temperatures previously assumed (T or T ),
B,1 ex,1 B,0 ex,0
are iteratively improved until the difference between the values with index 0 and 1 becomes negligibly
small, for example 2 K. The condition then attained corresponds to the steady condition. During the
iterative steps, the influencing factors given in 7.7 shall be taken into account. As a rule, the iteration
converges rapidly. It can also be replaced by graphical interpolation in which, for calculating P and
th,f
P or P , several temperature differences are assumed. If the heat flows Pf= ()T or
th,amb th,L
th,ambB
Pf= T are plotted, then the steady condition is given by the intersection of the two curves (see
()
th,Lex
Figure A.1).
7.5 Minimum lubricant film thickness and specific bearing load
The clearance gap, h, in a circular cylindrical journal bearing with the shaft offset is a function given by:
hD=+05, ψε1 cosϕ
()
eff
starting with ϕϕ= , in the widest clearance gap (see Figure 1).
The minimum lubricant film thickness
hD=−05, ψε1 (15)
()
mineff
shall be compared with the permissible operational parameter, h , specified in ISO 7902-3.
lim
The specific bearing load:
F
p= (16)
DB
shall be compared with the permissible operational parameter, p , specified in ISO 7902-3.
lim
In partial bearings, if the follow formula is satisfied:
π
ϕπ−−()β <
then
hD=+05, ψε1 cosϕ
()
min 2
7.6 Operational conditions
Should the plain bearing be operated under several, varying sets of operating conditions over lengthy
periods, then they shall be checked for the most unfavourable p , h , and T . First, a decision shall be
min B
reached as to whether or not the bearing can be lubricated without pressure and whether or not the
heat dissipation by convection suffices. The most unfavourable thermal case shall be investigated,
which, as a rule, corresponds to an operating condition at high rotary frequency together with heavy
loading. If, for pure convection, excessive bearing temperatures occur, which even by increasing the
dimensions of the bearing or of the surface area of the housing to their greatest possible extent cannot
be lowered to permissible values, then force-feed lubrication and oil cooling are necessary.
If an operating condition under high thermal loading (low dynamic lubricant viscosity) is followed
directly by one with high specific bearing load and low rotary frequency, this new operating condition
should be investigated while keeping the thermal condition from the preceding operating point.
The transition to mixed friction is due to contact of the roughness peaks of the shaft and bearing under
the criteria for h specified in ISO 7902-3, whereby deformation is also to be taken into account. A
lim
transition eccentricity:
h
lim
ε =−1
u
D
ψ
eff
and a transition Sommerfeld number:
Fψ
B
 
eff
So == f ε ,,Ω
 
u u
DBηω D
 
effh
(see ISO 7902-2) can be assigned to this value. Thus, the individual transition conditions (load,
viscosity, and rotary speed) can be determined. The transition condition can be described by just three
coexistent parameters. In order to be able to determine one of them, the two others shall be substituted
in the manner appropriate to this condition. For rapid run-down of the machine, the thermal state
corresponds mostly to the previous continuously driven operating condition of high thermal loading. If
cooling is shut off immediately when the machine is switched off, this can result in an accumulation of
heat in the bearing, so that a yet more unfavourable value shall be selected for h . If the machine runs
eff
down slowly, lowering of the temperature of the lubricant or bearing is to be expected.
7.7 Further influencing factors
The calculation procedure applies to steady-state operation, in particular for loading that is constant in
magnitude and direction and in which it is possible for the shaft and the bearing to rotate with uniform
speed. The effective angular velocity is given by:
ωω=+ω (17)
hJ B
The calculation procedure, however, also applies for the case of a constant load which rotates at an
angular velocity ω . In this case, the angular velocity is given by:
F
ωω=+ωω−2
hJ BF
For an out-of-balance force rotating with the shaft (ω = ω ), then:
F J
ωω=− +ω
hJ B
The absolute value of ω shall be used to calculate the Sommerfeld number. It shall be borne in mind
h
that in the case where ω < 0, the shaft eccentricity is at the angle −β (see Figure 4).
h
NOTE All rotary motions and angular directions are positive with respect to the direction of shaft rotation.
14 © ISO 2020 – All rights reserved

The dynamic viscosity is strongly dependent on temperature. It is thus necessary to know the temperature
dependence of the lubricant and its specification (see ISO 3448). The effective dynamic viscosity, η , is
eff
determined by means of the effective lubricant film temperature, T , that is η results from averaging
eff eff
temperatures T and T and not from averaging the dynamic viscosities η(T ) and η(T ).
en ex en ex
The dynamic viscosity is also pressure-dependent, but to a lesser degree. For bearings in steady-state
conditions and under the usual specific bearing loads, p , the pressure dependence can, however, be
neglected. This neglecting of pressure dependency represents an additional design factor of safety.
For non-Newtonian lubricants (intrinsically viscous oils, multi-range oils), reversible and irreversible
fluctuations in viscosity occur as a function of the shear loading within the lubricant clearance gap
[7]
and of the service life. These effects are investigated only for a few lubricants and are not taken into
account in all parts of ISO 7902.
The opera
...

INTERNATIONAL ISO
STANDARD 7902-1
Third edition
2020-06
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 2020
© ISO 2020
All rights reserved. Unless otherwise specified, or required in the context of its implementation, 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.
ISO copyright office
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Phone: [phone]
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Email: [e-mail]
Website: www.iso.org
Published in Switzerland
ii © ISO 2020 – All rights reserved

Contents Page
Foreword .iv
1 Scope . 1
2 Normative references . 1
3 Terms and definitions . 1
4 Symbols and units . 1
5 Basis of calculation, assumptions, and preconditions . 5
5.1 Reynolds equation . 5
5.2 Assumptions and preconditions . 5
5.3 Boundary conditions . 5
5.4 Basis of calculation . 6
5.5 Permissible operational parameters . 6
6 Calculation procedure . 6
6.1 General . 6
6.2 Freedom from wear . 6
6.3 The limits of mechanical loading. 7
6.4 The limits of thermal loading . 7
6.5 Influencing factors . 7
6.6 Reynolds number . 7
6.7 Calculation factors . 7
7 Definition of symbols . 9
7.1 Load-carrying capacity . 9
7.2 Frictional power loss . 9
7.3 Lubricant flow rate .10
7.3.1 General.10
7.3.2 Lubricant feed elements .10
7.3.3 Lubrication grooves .10
7.3.4 Lubrication pockets .10
7.3.5 Lubricant flow rate .11
7.4 Heat balance .11
7.4.1 General.11
7.4.2 Heat dissipation by convection .12
7.4.3 Heat dissipation via the lubricant .12
7.5 Minimum lubricant film thickness and specific bearing load .13
7.6 Operational conditions.14
7.7 Further influencing factors .14
Annex A (informative) Calculation examples .17
Bibliography .32
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 (see 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 (see 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.
For an explanation of the voluntary nature of standards, the meaning of ISO specific terms and
expressions related to conformity assessment, as well as information about ISO's adherence to the
World Trade Organization (WTO) principles in the Technical Barriers to Trade (TBT), see www .iso .org/
iso/ foreword .html.
This document was prepared by Technical Committee ISO/TC 123, Plain bearings, Subcommittee SC 8,
Calculation methods for plain bearings and their applications.
This third edition cancels and replaces the second edition (ISO 7902-1:2013), which has been technically
revised.
The main changes compared to the previous edition are as follows:
— subclause titles have been added;
— symbols have been corrected and added in Table 1;
— calculation values in Annex A have been corrected;
— adjustments have been made to ISO/IEC Directives, Part 2:2018;
— typographical errors have been corrected.
A list of all parts in the ISO 7902 series can be found on the ISO website.
Any feedback or questions on this document should be directed to the user’s national standards body. A
complete listing of these bodies can be found at www .iso .org/ members .html.
iv © ISO 2020 – All rights reserved

INTERNATIONAL STANDARD ISO 7902-1:2020(E)
Hydrodynamic plain journal bearings under steady-state
conditions — Circular cylindrical bearings —
Part 1:
Calculation procedure
1 Scope
This document 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 provide dimensions 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 those that can result from vibration effects and
instabilities of rapid-running rotors, are not taken into account.
NOTE Equivalent calculation procedures exist that enable operating conditions to be estimated and checked
against acceptable conditions. The use of them is equally admissible.
2 Normative references
The following documents are referred to in the text in such a way that some or all of their content
constitutes requirements of this document. For dated references, only the edition cited applies. For
undated references, the latest edition of the referenced document (including any amendments) applies.
ISO 7902-2, 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 Terms and definitions
No terms and definitions are listed in this document.
ISO and IEC maintain terminological databases for use in standardization at the following addresses:
— ISO Online browsing platform: available at https:// www .iso .org/ obp
— IEC Electropedia: available at http:// www .electropedia .org/
4 Symbols and units
Symbols and units are defined in Figure 1 and Table 1.
Figure 1 — Illustration of symbols
Table 1 — Symbols and their designations
Symbol Designation Unit
A Area of heat-emitting surface (bearing housing) m
b Width of lubrication groove m
G
b Width of lubrication pocket m
P
B Nominal bearing width m
B Length of the axial housing m
H
c Specific heat capacity of the lubricant J/(kg·K)
p
C Nominal bearing clearance m
C Effective bearing radial clearance m
R,eff
d Lubrication hole diameter m
L
D Nominal bearing diameter (inside diameter) m
D Length of the outside diameter of the housing m
H
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
f Coefficient of friction in the loaded area of the lubricant film ( f = F /F) 1
f
f′ Coefficient of friction in both the loaded and unloaded area of the lubricant film 1
F Bearing force (nominal load) N
F Friction force in the loaded area of the lubricant film N
f
2 © ISO 2020 – All rights reserved

Table 1 (continued)
Symbol Designation Unit
′ Frictional force in both the loaded and the unloaded area of the lubricant film N
F
f
h Local lubricant film thickness m
h Effective lubricant film thickness m
eff
h Depth of lubrication groove m
G
h Minimum permissible lubricant film thickness m
lim
h Minimum lubricant film thickness m
min
h Depth of lubrication pocket m
P
H Length of the total height of the pedestal bearing m
k Outer heat transmission coefficient W/(m ·K)
A
−1
N Rotational frequency of the bearing s
B
−1
N Rotational 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
Maximum permissible specific bearing load Pa
p
lim
P Frictional power W
f
P ′ Frictional power in both the loaded and the unloaded area of the lubricant film 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
q Coefficient related to lubricant flow rate due to feed pressure 1
L
q Coefficient related to lubricant flow rate from pocket 1
P
Q Lubricant flow rate m /s
Q Lubricant flow rate due to hydrodynamic pressure m /s
* Lubricant flow rate parameter due to hydrodynamic pressure 1
Q
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
So Transition Sommerfeld number 1
u
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 Effective lubricant temperature °C
eff
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
Table 1 (continued)
Symbol Designation Unit
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
Attitude angle (angular position of the shaft eccentricity related to the direction °
β
of load)
ε Relative eccentricity [ε = 2e/(D – D )] 1
J
ε Transition eccentricity 1
u
η Dynamic viscosity of the lubricant Pa·s
η Effective dynamic viscosity of the lubricant Pa·s
eff
v Kinematic viscosity of the lubricant m /s
ξ
Coefficient of resistance to rotation in the loaded area of the lubricant film 1
Coefficient of resistance to rotation in both the loaded and the unloaded area of 1
′
ξ
the lubricant film
Coefficient of resistance to rotation in the area of circumferential groove 1
ξ
G
Coefficient of resistance to rotation in the area of the pocket 1
ξ
P
ρ Density of lubricant kg/m
φ Angular coordinate in the circumferential direction rad
φ Angular coordinate of pressure leading edge rad
φ Angular coordinate of pressure trailing edge rad
ψ
Relative bearing clearance 1
ψ
Mean relative bearing clearance 1
Effective relative bearing clearance 1
ψ
eff
Maximum relative bearing clearance 1
ψ
max
Minimum relative bearing clearance 1
ψ
min
−1
ω Angular velocity of bearing s
B
−1
ω Angular velocity of rotating force s
F
−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
4 © ISO 2020 – All rights reserved

5 Basis of calculation, assumptions, and preconditions
5.1 Reynolds equation
The basis of calculation is the numerical solution to Reynolds equation for a finite bearing length,
taking into account the physically correct boundary conditions for the generation of pressure. Reynolds
equation is defined as Formula (1).
∂ ∂p ∂ ∂p ∂h
   
h + h =6η UU+ (1)
    ()
JB
∂x  ∂xz ∂  ∂z  ∂x
See References [3] to [5] and References [13] to [16] for the derivation of Reynolds equation and
References [6] to [8], [14] and [15] for its numerical solution.
5.2 Assumptions and preconditions
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.
5.3 Boundary conditions
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 ;
()
— at the bearing rim:pzϕ,/=±B 20= ;
()
— at the trailing edge of the pressure profile:pzϕ (),z =0 ;
[]
— ∂∂pz/,ϕϕ () z =0 .
[]
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)
Then the trailing edge of the pressure profile lies at the outlet end of the bearing is:
pzϕϕ= , =0
()
5.4 Basis of calculation
The numerical integration of the Reynolds equation is carried out (possibly by applying transformation
of pressure as suggested in References [5], [13] and [14]) 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 integration; the use
of a non-equidistant grid as given in References [8] and [15] 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 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.
5.5 Permissible operational parameters
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.
6 Calculation procedure
6.1 General
Calculation is understood to mean determination of correct operation by computation using actual
operating parameters (see Figure 2), 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.
6.2 Freedom from wear
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
6 © ISO 2020 – All rights reserved

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.
6.3 The limits of mechanical loading
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.
6.4 The limits of thermal loading
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.
6.5 Influencing factors
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 (e.g. out-of-balance, vibrations);
— deviations from the ideal geometry (e.g. machining tolerances, deviations during assembly);
— lubricants contaminated by, for example, dirt, water, air;
— corrosion, electrical erosion.
Data on other influencing factors are given in 7.7.
6.6 Reynolds number
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:
C C
Re,,ff Reff
ρU πDN
J J
D
Re== ≤41,3 (3)
η v C
Re, ff
In the case of plain bearings with Re>41,3 DC/ (e.g. as a result of high peripheral speed), higher
R,eff
loss coefficients and bearing temperatures shall be expected. Calculations for bearings with turbulent
flow cannot be carried out in accordance with this document.
6.7 Calculation factors
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 2.
For optimization of individual parameters, parameter variation can be applied; modification of the
calculation sequence is possible.
Figure 2 — Outline of calculation
8 © ISO 2020 – All rights reserved

7 Definition of symbols
7.1 Load-carrying capacity
A characteristic parameter for the load-carrying capacity is the dimensionless Sommerfeld number, So:
Fψ
B
 
eff
So==So ε ,,Ω (4)
 
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 ψ take into account
h eff
eff
the thermal effects and the angular velocities of shaft, bearing, and bearing force (see 7.4 and 7.7).
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.
7.2 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= (5)
DBηω
effh
f ξ
= (6)
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
f f′
values, fF,,ξ,and , are substituted by fF′′,,ξ′, and , respectively in Formulae (5) and (6).
f f
ψ ψ
eff eff
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 [17], which are used to determine frictional power
loss values in the bearings, taking account of the influence of lubrication pockets and grooves.
The frictional power in a bearing or the amount of heat generated is given by Formulae (7) and (8).
D
PP== fF ω (7)
fthf, h
D
′ ′
Pf= F ω (8)
fh
7.3 Lubricant flow rate
7.3.1 General
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 (9)
3 effh 3
**
where QQ= ()εΩ,/BD, is given in ISO 7902-2.
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 as defined by Formula (10).
p
Dpψ
eff en
*
Q = Q (10)
pp
η
eff
**
where QQ= εΩ,/BD, is given in ISO 7902-2.
()
pp
7.3.2 Lubricant feed elements
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
en
additional hydrostatic loads. Usually, p lies between 0,05 MPa and 0,2 MPa. The depth of the lubrication
en
grooves and lubrication pockets is considerably greater than the bearing clearance.
7.3.3 Lubrication grooves
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 be taken into account when balancing heat losses (see 7.4), 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
p
heat balance.
7.3.4 Lubrication pockets
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 as b /B < 0,7. Although
p
larger values increase the lubricant 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
lubrication 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
10 © ISO 2020 – All rights reserved

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.
Key
1 lubrication hole
2 lubrication groove
Figure 3 — Lubricant film pressure in bearings with lubrication groove
7.3.5 Lubricant flow rate
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.
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
— for lubricant filling only in the loaded area of the bearing:
QQ= (11)
— for lubricant filling in the whole circular lubrication clearance gap including unloaded part, i.e. 2π:
QQ=+Q (12)
3p
7.4 Heat balance
7.4.1 General
The thermal condition of the plain bearing can be obtained from the heat balance. The heat flow,
P , arising from frictional power in the bearing, P , is dissipated via the bearing housing to the
th,f f
environment 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 (e.g. 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
Calculation examples are introduced in Annex A.
7.4.2 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 (13)
()
th,ambA Bamb
where
k =()15 to20 Wm/ ⋅K
()
A
or, by ventilating the bearing housing with air at a velocity of V > 1,2 m/s
a
kV=+712
Aa
See References [5] and [16].
Should the area of the heat-emitting surface, A, of the bearing housing not be known exactly, the
following can be used as an approximation
— for cylindrical housings:
π
AD=−2 DD+π B
()
HH H
— for pedestal bearings:
H
 
AH=+π B
 
H
 2 
— for bearings in the machine structure:
AD=()15to20 B
7.4.3 Heat dissipation via the lubricant
In the case of force-feed lubrication, heat dissipation is via the lubricant:
Pc=−ρ QT T (14)
()
th,Lep xen
For mineral lubricants, the volume-specific heat is given by:
ρ×=c 18,/×⋅10 Jm K
()
p
From the heat balance, it follows that PP= for pressureless-lubricated bearings and PP=
th,,fthamb th,,fthL
for pressure-lubricated bearings.
This gives bearing temperature, T (see Reference [17]), and lubricant outlet temperature, T
B ex
(see Reference [17]). The effective film lubricant temperature with reference to the lubricant viscosity is
a) in the case of pure convection: T = T , and
eff B
12 © ISO 2020 – All rights reserved

b) in the case of heat dissipation via the lubricant: T =T = 0,5 (T + T ).
eff en ex
L
At high peripheral speed, it is possible to select, instead of these mean values, a temperature which lies
nearer to the lubricant outlet temperature.
The values calculated for T and T shall be checked for their permissibility by comparison with the
B ex
permissible operational parameters, T , given in ISO 7902-3.
lim
In the sequence of calculations, at first only the operational data T or T are known, but not the
amb en
effective temperature, T , which is required at the start of the calculation. The solution is obtained by
eff
first starting the calculation using an estimated temperature rise, i.e.
a) T − T = 20 K
B,0 amb
b) T − T = 20 K
ex,0 en
and the corresponding operating temperatures, T . From the heat balance, corrected temperatures,
eff
T or T , are obtained, which, by averaging with the temperatures previously assumed (T or T ),
B,1 ex,1 B,0 ex,0
are iteratively improved until the difference between the values with index 0 and 1 becomes negligibly
small, for example 2 K. The condition then attained corresponds to the steady condition. During the
iterative steps, the influencing factors given in 7.7 shall be taken into account. As a rule, the iteration
converges rapidly. It can also be replaced by graphical interpolation in which, for calculating P and
th,f
P or P , several temperature differences are assumed. If the heat flows Pf= ()T or
th,amb th,L
th,ambB
Pf= T are plotted, then the steady condition is given by the intersection of the two curves (see
()
th,Lex
Figure A.1).
7.5 Minimum lubricant film thickness and specific bearing load
The clearance gap, h, in a circular cylindrical journal bearing with the shaft offset is a function given by:
hD=+05, ψε1 cosϕ
()
eff
starting with ϕϕ= , in the widest clearance gap (see Figure 1).
The minimum lubricant film thickness
hD=−05, ψε1 (15)
()
mineff
shall be compared with the permissible operational parameter, h , specified in ISO 7902-3.
lim
The specific bearing load:
F
p= (16)
DB
shall be compared with the permissible operational parameter, p , specified in ISO 7902-3.
lim
In partial bearings, if the follow formula is satisfied:
π
ϕπ−−()β <
then
hD=+05, ψε1 cosϕ
()
min 2
7.6 Operational conditions
Should the plain bearing be operated under several, varying sets of operating conditions over lengthy
periods, then they shall be checked for the most unfavourable p , h , and T . First, a decision shall be
min B
reached as to whether or not the bearing can be lubricated without pressure and whether or not the
heat dissipation by convection suffices. The most unfavourable thermal case shall be investigated,
which, as a rule, corresponds to an operating condition at high rotary frequency together with heavy
loading. If, for pure convection, excessive bearing temperatures occur, which even by increasing the
dimensions of the bearing or of the surface area of the housing to their greatest possible extent cannot
be lowered to permissible values, then force-feed lubrication and oil cooling are necessary.
If an operating condition under high thermal loading (low dynamic lubricant viscosity) is followed
directly by one with high specific bearing load and low rotary frequency, this new operating condition
should be investigated while keeping the thermal condition from the preceding operating point.
The transition to mixed friction is due to contact of the roughness peaks of the shaft and bearing under
the criteria for h specified in ISO 7902-3, whereby deformation is also to be taken into account. A
lim
transition eccentricity:
h
lim
ε =−1
u
D
ψ
eff
and a transition Sommerfeld number:
Fψ
B
 
eff
So == f ε ,,Ω
 
u u
DBηω D
 
effh
(see ISO 7902-2) can be assigned to this value. Thus, the individual transition conditions (load,
viscosity, and rotary speed) can be determined. The transition condition can be described by just three
coexistent parameters. In order to be able to determine one of them, the two others shall be substituted
in the manner appropriate to this condition. For rapid run-down of the machine, the thermal state
corresponds mostly to the previous continuously driven operating condition of high thermal loading. If
cooling is shut off immediately when the machine is switched off, this can result in an accumulation of
heat in the bearing, so that a yet more unfavourable value shall be selected for h . If the machine runs
eff
down slowly, lowering of the temperature of the lubricant or bearing is to be expected.
7.7 Further influencing factors
The calculation procedure applies to steady-state operation, in particular for loading that is constant in
magnitude and direction and in which it is possible for the shaft and the bearing to rotate with uniform
speed. The effective angular velocity is given by:
ωω=+ω (17)
hJ B
The calculation procedure, however, also applies for the case of a constant load which rotates at an
angular velocity ω . In this case, the angular velocity is given by:
F
ωω=+ωω−2
hJ BF
For an out-of-balance force rotating with the shaft (ω = ω ), then:
F J
ωω=− +ω
hJ B
The absolute value of ω shall be used to calculate the Sommerfeld number. It shall be borne in mind
h
that in the case where ω < 0, the shaft eccentricity is at the angle −β (see Figure 4).
h
NOTE All rotary motions and angular directions are positive with respect to the direction of shaft rotation.
14 © ISO 2020 – All rights reserved

The dynamic viscosity is strongly dependent on temperature. It is thus necessary to know the temperature
dependence of the lubricant and its specification (see ISO 3448). The effective dynamic viscosity, η , is
eff
determined by means of the effective lubricant film temperature, T , that is η results from averaging
eff eff
temperatures T and T and not from averaging the dynamic viscosities η(T ) and η(T ).
en ex en ex
The dynamic viscosity is also pressure-dependent, but to a lesser degree. For bearings in steady-state
conditions and under the usual specific bearing loads, p , the pressure dependence can, however, be
neglected. This neglecting of pressure dependency represents an additional design factor of safety.
For non-Newtonian lubricants (intrinsically viscous oils, multi-range oils), reversible and irreversible
fluctuations in viscosity occur as a function of the shear loading within the lubricant clearance gap
[7]
and of the service life. These effects are investigated only for a few lubricants and are not taken into
account in all parts of ISO 7902.
The operation bearing clearance results from the fit and the thermal expansion behaviour of bearing
and shaft. In the installed condition (20 °C), the relative bearing clearance is given by:
DD−
maxJ,min
ψ = (18)
max
D
DD−
minJ,max
ψ = (19)
min
D
ψ =+05, ψψ (20)
()
maxmin
The deciding factor in the calculation is the effective relative bearing clearance, ψ , at the effective
eff
lubricant film temperature, T , which can be regarded (subject to the assumptions in 5.5) as the mean
eff
temperature of bearing and shaft. Insofar as the coefficients of linear expansion of the shaft, α , and
l,J
of the bearing, α , do not differ, the clearance when cold (20 °C) is equal to the clearance when hot
l,B
(T ). Should shaft and bearing (bearing liner with housing) show different temperatures (T , T ) due
eff J B
to external influences, then this shall be taken into account [see Formula (22)]. The linear expansion of
the thin bearing layer can be neglected.
For coefficients of linear expansion which differ for shaft and bearing, the thermal change of the relative
bearing clearance is given by:
Δψ =−αα T −°20 C (21)
()
()
ll,,BJ eff
Δψ =−ααTT20°CC−−20° (22)
() ()
ll,,BB JJ
ψ =+ψψΔ (23)
eff
Permissible operational values for the bearing clea
...