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
07-Sep-2021
Technical Committee
Current Stage
6060 - National Implementation/Publication (Adopted Project)
Start Date
02-Sep-2021
Due Date
07-Nov-2021
Completion Date
08-Sep-2021

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SLOVENSKI STANDARD
SIST ISO 7902-1:2021
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
SIST ISO 7902-1:2021 en

2003-01.Slovenski inštitut za standardizacijo. Razmnoževanje celote ali delov tega standarda ni dovoljeno.

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SIST ISO 7902-1:2021
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SIST ISO 7902-1:2021
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 7902-1:2020(E)
ISO 2020
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SIST ISO 7902-1:2021
ISO 7902-1:2020(E)
COPYRIGHT PROTECTED DOCUMENT
© 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: +41 22 749 01 11
Fax: +41 22 749 09 47
Email: copyright@iso.org
Website: www.iso.org
Published in Switzerland
ii © ISO 2020 – All rights reserved
---------------------- Page: 4 ----------------------
SIST ISO 7902-1:2021
ISO 7902-1:2020(E)
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

© ISO 2020 – All rights reserved iii
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SIST ISO 7902-1:2021
ISO 7902-1:2020(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 (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
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SIST ISO 7902-1:2021
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.
© ISO 2020 – All rights reserved 1
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SIST ISO 7902-1:2021
ISO 7902-1:2020(E)
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
b Width of lubrication pocket m
B Nominal bearing width m
B Length of the axial housing m
c Specific heat capacity of the lubricant J/(kg·K)
C Nominal bearing clearance m
C Effective bearing radial clearance m
R,eff
d Lubrication hole diameter m
D Nominal bearing diameter (inside diameter) m
D Length of the outside diameter of the housing m
D Nominal shaft diameter m
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′ 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
2 © ISO 2020 – All rights reserved
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SIST ISO 7902-1:2021
ISO 7902-1:2020(E)
Table 1 (continued)
Symbol Designation Unit

′ Frictional force in both the loaded and the unloaded area of the lubricant film N

h Local lubricant film thickness m
h Effective lubricant film thickness m
eff
h Depth of lubrication groove m
h Minimum permissible lubricant film thickness m
lim
h Minimum lubricant film thickness m
min
h Depth of lubrication pocket m
H Length of the total height of the pedestal bearing m
k Outer heat transmission coefficient W/(m ·K)
N Rotational frequency of the bearing s
N Rotational frequency of the shaft s
p Local lubricant film pressure Pa
Specific bearing load Pa
p Lubricant feed pressure Pa
p Maximum permissible lubricant film pressure Pa
lim
Maximum permissible specific bearing load Pa
lim
P Frictional power W

P ′ Frictional power in both the loaded and the unloaded area of the lubricant film W

P Heat flow rate W
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
q Coefficient related to lubricant flow rate from pocket 1
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 Lubricant flow rate due to feed pressure m /s
Lubricant flow rate parameter due to feed pressure 1
Rz Average peak-to-valley height of bearing sliding surface m
Rz Average peak-to-valley height of shaft mating surface m
Re Reynolds number 1
So Sommerfeld number 1
So Transition Sommerfeld number 1
T Ambient temperature °C
amb
T Bearing temperature °C
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
T Lubricant temperature at bearing exit °C
T Assumed initial lubricant temperature at bearing exit °C
ex,0
T Calculated lubricant temperature at bearing exit °C
ex,1
© ISO 2020 – All rights reserved 3
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SIST ISO 7902-1:2021
ISO 7902-1:2020(E)
Table 1 (continued)
Symbol Designation Unit
T Shaft temperature °C
T Maximum permissible bearing temperature °C
lim
Mean lubricant temperature °C
U Linear velocity (peripheral speed) of bearing m/s
U Linear velocity (peripheral speed) of shaft m/s
V Air ventilating velocity m/s
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
α Linear heat expansion coefficient of the bearing K
l,B
α Linear heat expansion coefficient of the shaft K
l,J

Attitude angle (angular position of the shaft eccentricity related to the direction °

of load)
ε Relative eccentricity [ε = 2e/(D – D )] 1
ε Transition 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 m /s
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

the lubricant film
Coefficient of resistance to rotation in the area of circumferential groove 1
Coefficient of resistance to rotation in the area of the pocket 1
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
Relative bearing clearance 1
Mean relative bearing clearance 1
Effective relative bearing clearance 1
eff
Maximum relative bearing clearance 1
max
Minimum relative bearing clearance 1
min
ω Angular velocity of bearing s
ω Angular velocity of rotating force s
ω Hydrodynamic angular velocity s
ω Angular velocity of shaft s
Ω Angular span of bearing segment °
Ω Angular span of lubrication groove °
Ω Angular span of lubrication pocket °
4 © ISO 2020 – All rights reserved
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SIST ISO 7902-1:2021
ISO 7902-1:2020(E)
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)
    ()
∂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= ;
© ISO 2020 – All rights reserved 5
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SIST ISO 7902-1:2021
ISO 7902-1:2020(E)
— 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:
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
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SIST ISO 7902-1:2021
ISO 7902-1:2020(E)

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
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.
© ISO 2020 – All rights reserved 7
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SIST ISO 7902-1:2021
ISO 7902-1:2020(E)
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
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SIST ISO 7902-1:2021
ISO 7902-1:2020(E)
7 Definition of symbols
7.1 Load-carrying capacity

A characteristic parameter for the load-carrying capacity is the dimensionless Sommerfeld number, So:

 
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 give
...

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 7902-1:2020(E)
ISO 2020
---------------------- Page: 1 ----------------------
ISO 7902-1:2020(E)
COPYRIGHT PROTECTED DOCUMENT
© 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: +41 22 749 01 11
Fax: +41 22 749 09 47
Email: copyright@iso.org
Website: www.iso.org
Published in Switzerland
ii © ISO 2020 – All rights reserved
---------------------- Page: 2 ----------------------
ISO 7902-1:2020(E)
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

© ISO 2020 – All rights reserved iii
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ISO 7902-1:2020(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.

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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
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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.
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ISO 7902-1:2020(E)
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
b Width of lubrication pocket m
B Nominal bearing width m
B Length of the axial housing m
c Specific heat capacity of the lubricant J/(kg·K)
C Nominal bearing clearance m
C Effective bearing radial clearance m
R,eff
d Lubrication hole diameter m
D Nominal bearing diameter (inside diameter) m
D Length of the outside diameter of the housing m
D Nominal shaft diameter m
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′ 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
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ISO 7902-1:2020(E)
Table 1 (continued)
Symbol Designation Unit

′ Frictional force in both the loaded and the unloaded area of the lubricant film N

h Local lubricant film thickness m
h Effective lubricant film thickness m
eff
h Depth of lubrication groove m
h Minimum permissible lubricant film thickness m
lim
h Minimum lubricant film thickness m
min
h Depth of lubrication pocket m
H Length of the total height of the pedestal bearing m
k Outer heat transmission coefficient W/(m ·K)
N Rotational frequency of the bearing s
N Rotational frequency of the shaft s
p Local lubricant film pressure Pa
Specific bearing load Pa
p Lubricant feed pressure Pa
p Maximum permissible lubricant film pressure Pa
lim
Maximum permissible specific bearing load Pa
lim
P Frictional power W

P ′ Frictional power in both the loaded and the unloaded area of the lubricant film W

P Heat flow rate W
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
q Coefficient related to lubricant flow rate from pocket 1
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 Lubricant flow rate due to feed pressure m /s
Lubricant flow rate parameter due to feed pressure 1
Rz Average peak-to-valley height of bearing sliding surface m
Rz Average peak-to-valley height of shaft mating surface m
Re Reynolds number 1
So Sommerfeld number 1
So Transition Sommerfeld number 1
T Ambient temperature °C
amb
T Bearing temperature °C
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
T Lubricant temperature at bearing exit °C
T Assumed initial lubricant temperature at bearing exit °C
ex,0
T Calculated lubricant temperature at bearing exit °C
ex,1
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ISO 7902-1:2020(E)
Table 1 (continued)
Symbol Designation Unit
T Shaft temperature °C
T Maximum permissible bearing temperature °C
lim
Mean lubricant temperature °C
U Linear velocity (peripheral speed) of bearing m/s
U Linear velocity (peripheral speed) of shaft m/s
V Air ventilating velocity m/s
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
α Linear heat expansion coefficient of the bearing K
l,B
α Linear heat expansion coefficient of the shaft K
l,J

Attitude angle (angular position of the shaft eccentricity related to the direction °

of load)
ε Relative eccentricity [ε = 2e/(D – D )] 1
ε Transition 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 m /s
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

the lubricant film
Coefficient of resistance to rotation in the area of circumferential groove 1
Coefficient of resistance to rotation in the area of the pocket 1
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
Relative bearing clearance 1
Mean relative bearing clearance 1
Effective relative bearing clearance 1
eff
Maximum relative bearing clearance 1
max
Minimum relative bearing clearance 1
min
ω Angular velocity of bearing s
ω Angular velocity of rotating force s
ω Hydrodynamic angular velocity s
ω Angular velocity of shaft s
Ω Angular span of bearing segment °
Ω Angular span of lubrication groove °
Ω Angular span of lubrication pocket °
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ISO 7902-1:2020(E)
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)
    ()
∂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= ;
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ISO 7902-1:2020(E)
— 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:
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

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ISO 7902-1:2020(E)

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
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.
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ISO 7902-1:2020(E)
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
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ISO 7902-1:2020(E)
7 Definition of symbols
7.1 Load-carrying capacity

A characteristic parameter for the load-carrying capacity is the dimensionless Sommerfeld number, So:

 
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 :

eff
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).

PP== fF ω (7)
fthf, h
′ ′
Pf= F ω (8)
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