Plain bearings — Hydrodynamic plain journal bearings under steady-state conditions — Part 1: Calculation of multi-lobed and tilting pad journal bearings

This document specifies the general principles, assumptions and preconditions for the calculation of multi-lobed and tilting-pad journal bearings by means of an easy-to-use calculation procedure based on numerous simplifying assumptions. For a reliable evaluation of the results of this calculation method, it is indispensable to consider the physical implications of these assumptions as well as practical experiences for instance from temperature measurements carried out on real machinery under typical operating conditions. Applied in this sense, this document presents a simple way to predict the approximate performance of plain journal bearings for those unable to access more complex and accurate calculation techniques. The calculation method serves for the design and optimisation of plain bearings, for example in turbines, compressors, generators, electric motors, gears and pumps. It is restricted to steady-state operation, i.e. in continuous operating states the load according to size and direction and the angular velocity of the rotor are constant. Unsteady operating states are not recorded. The stiffness and damping coefficients of the plain journal bearings required for the linear vibration and stability investigations are indicated in ISO/TS 31657-2 and ISO/TS 31657-3.

titre manque — Partie 1: titre manque

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Publication Date
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6060 - International Standard published
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TECHNICAL ISO/TS
SPECIFICATION 31657-1
First edition
2020-06
Plain bearings — Hydrodynamic plain
journal bearings under steady-state
conditions —
Part 1:
Calculation of multi-lobed and tilting
pad journal bearings
Reference number
ISO/TS 31657-1:2020(E)
ISO 2020
---------------------- Page: 1 ----------------------
ISO/TS 31657-1:2020(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
Email: copyright@iso.org
Website: www.iso.org
Published in Switzerland
ii © ISO 2020 – All rights reserved
---------------------- Page: 2 ----------------------
ISO/TS 31657-1:2020(E)
Contents Page

Foreword ........................................................................................................................................................................................................................................iv

Introduction ..................................................................................................................................................................................................................................v

1 Scope ................................................................................................................................................................................................................................. 1

2 Normative references ...................................................................................................................................................................................... 1

3 Terms and definitions ..................................................................................................................................................................................... 1

4 Symbols and units ............................................................................................................................................................................................... 1

5 General principles, assumptions and preconditions ...................................................................................................... 7

6 Calculation method ............................................................................................................................................................................................ 9

6.1 General ........................................................................................................................................................................................................... 9

6.2 Load carrying capacity ..................................................................................................................................................................11

6.3 Frictional power..................................................................................................................................................................................11

6.4 Lubricant flow rate ...........................................................................................................................................................................12

6.5 Heat balance ...........................................................................................................................................................................................13

6.6 Maximum lubricant film temperature .............................................................................................................................14

6.7 Maximum lubricant film pressure .......................................................................................................................................15

6.8 Operating states ..................................................................................................................................................................................15

6.9 Further influencing parameters ............................................................................................................................................15

6.10 Stiffness and damping coefficients .....................................................................................................................................16

7 Figures ..........................................................................................................................................................................................................................18

Annex A (informative) Calculation examples ...........................................................................................................................................23

Bibliography .............................................................................................................................................................................................................................37

© ISO 2020 – All rights reserved iii
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ISO/TS 31657-1:2020(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.
A list of all parts in the ISO/TS 31657 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
---------------------- Page: 4 ----------------------
ISO/TS 31657-1:2020(E)
Introduction

The aim of this document is the operationally-safe design of plain journal bearings for medium or high

journal circumferential velocities, U , up to approximately 90 m/s by applying a calculation method for

oil-lubricated hydrodynamic plain bearings with complete separation of journal and bearing sliding

surfaces by a lubricating film.

For low circumferential velocities up to approximately 30 m/s usually circular cylindrical bearings

are applied. For these bearings a similar calculation method is given in ISO 7902-1, ISO 7902-2 and

ISO 7902-3.

Based on practical experience the calculation procedure is usable for application cases where specific

bearing load times circumferential speed, pU⋅ , does not exceed approximately 200 MPa·m/s.

This document discusses multi-lobed journal bearings with two, three and four equal, symmetrical

sliding surfaces, which are separated by laterally-closed lubrication pockets, and symmetrically-loaded

tilting-pad journal bearings with four and five pads. Here, the curvature radii, R , of the sliding

surfaces are usually chosen larger than half the bearing diameter, D, so that an increased bearing

clearance results at the pad ends.

The calculation method described here can also be used for other gap forms, for example asymmetrical

multi-lobed journal bearings like offset-halves bearings, pressure-dam bearings or other tilting-pad

journal bearing designs, if the numerical solutions of the basic formulas are available for these designs.

© ISO 2020 – All rights reserved v
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TECHNICAL SPECIFICATION ISO/TS 31657-1:2020(E)
Plain bearings — Hydrodynamic plain journal bearings
under steady-state conditions —
Part 1:
Calculation of multi-lobed and tilting pad journal bearings
1 Scope

This document specifies the general principles, assumptions and preconditions for the calculation of

multi-lobed and tilting-pad journal bearings by means of an easy-to-use calculation procedure based on

numerous simplifying assumptions. For a reliable evaluation of the results of this calculation method,

it is indispensable to consider the physical implications of these assumptions as well as practical

experiences for instance from temperature measurements carried out on real machinery under

typical operating conditions. Applied in this sense, this document presents a simple way to predict

the approximate performance of plain journal bearings for those unable to access more complex and

accurate calculation techniques.

The calculation method serves for the design and optimisation of plain bearings, for example in

turbines, compressors, generators, electric motors, gears and pumps. It is restricted to steady-state

operation, i.e. in continuous operating states the load according to size and direction and the angular

velocity of the rotor are constant.

Unsteady operating states are not recorded. The stiffness and damping coefficients of the plain journal

bearings required for the linear vibration and stability investigations are indicated in ISO/TS 31657-2

and ISO/TS 31657-3.
2 Normative references
There are no normative references in this document.
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
Table 1 contains the symbols used in the ISO 31657 series.
Table 1 — Symbols and units
Symbol Description Unit
B Bearing width m
Relative bearing width, width ratio as given by: B = 1
© ISO 2020 – All rights reserved 1
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ISO/TS 31657-1:2020(E)
Table 1 (continued)
Symbol Description Unit
b Width of lubricant pocket m
* P
b 1
Relative width of lubricant pocket, as given by: b =
Bearing radial clearance, as given by: CR=−R
C m
Effective radial bearing clearance m
R,eff
c Stiffness coefficient of lubricant film (i,k = 1,2) N/m
Non-dimensional stiffness coefficient of lubricant film, as given by:
* ψ
c 1
eff
c = ⋅=ci(),,k 12
ik ik
2⋅⋅Bηω⋅
eff
J/(kg
Specific heat capacity (p = constant)
D Nominal bearing diameter (inside diameter of journal bearing) m
D Maximum value of D m
max
Minimum value of D m
min

Journal diameter (diameter of the shaft section located inside of a journal bearing) m

D Maximum value of D
Jm, ax J
D Minimum value of D
Jm, in J
Damping coefficient of lubricant film (i,k = 1,2) N s/m
Non-dimensional damping coefficient of lubricant film, as given by:
* 3
ψ 1
ik eff
d = ⋅⋅ω di(),,k=12
ik ik
2⋅⋅Bηω⋅
eff
e Eccentricity (distance between journal and bearing axis) m

Eccentricity of the bearing sliding surfaces (pads) of a multi-lobed or tilting-pad journal

e m
bearing
f Bearing force, bearing load, nominal bearing load, load-carrying capacity N
ΔF Component of additional dynamic force in x-direction N
Component of additional dynamic force in y-direction N
Component of additional dynamic force parameter in x-direction, as given by:
* ΔF ⋅ψ
ΔF x
eff
ΔF =
BD⋅⋅ηω⋅
eff
Component of additional dynamic force parameter in y-direction, as given by:
ΔF ⋅ψ
ΔF 1
y eff
ΔF =
BD⋅⋅ηω⋅
eff
Friction force, as given by: Ff=⋅F
Friction force parameter, as given by: F =⋅So
F 1
eff
Bearing force at transition to mixed friction N
F Coefficient of friction 1
Journal deflection m
h(φ) Local lubricant film thickness m
2 © ISO 2020 – All rights reserved
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ISO/TS 31657-1:2020(E)
Table 1 (continued)
Symbol Description Unit
h()ϕ
Relative local lubricant film thickness, as given by: h ()ϕ =
h ()ϕ
Minimum admissible lubricant film thickness at transition to mixed friction m
lim,tr

Minimum admissible relative lubricant film thickness at transition to mixed friction, as

lim,tr
h 1
lim,tr
given by: h =
lim,tr
R,eff
Minimum lubricant film thickness, minimum gap m
min
min

Minimum relative lubricant film thickness, minimum relative gap, as given by: h =

h 1
min
min
R,eff
Minimum lubricant film thickness at transition to mixed friction m
min,tr

Minimum relative lubricant film thickness at transition to mixed friction, as given by:

min,tr
h 1
min,tr
h =
min,tr
R,eff
h ϕ
() Local gap at ε=0 , gap function m
h ()ϕ
Relative local gap at ε=0 , profile function, as given by: h ()ϕ =
h ()ϕ 1
0 R
Maximum gap at ε=0 m
0,max
0,max
Maximum relative gap at ε=0 , gap ratio, as given by: h =
0,max
h 1
0,max

Profile factor (relative difference between lobe or pad bore radius and journal radius), as

B 1
given by: K ==
Cm1−
Effective profile factor 1
P,eff
Profile factor at 20 °C 1
P,20
M Mixing factor 1
m Preload factor, preload of bearing or pad sliding surface 1

N Rotational speed (rotational frequency) of the rotor (revolutions per time unit) s

Critical speed (critical rotational frequency) s

Rotational speed (rotational frequency) at the stability speed limit of the rotor supported

lim
by plain bearings

Resonance speed (resonance rotational frequency) of the rotor supported by plain bearings s

rsn

Rotational speed (rotational frequency) at transition to mixed friction, transition rotational

N s
speed, transition rotational frequency
O Centreline of plain bearing 1
Centreline of sliding surface No. i 1
Centreline of journal 1
P Frictional power, as given by: PF=⋅U
f ff J
Heat flow via the lubricant W
th,L
p Lubricant film pressure, local lubricant film pressure Pa
© ISO 2020 – All rights reserved 3
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ISO/TS 31657-1:2020(E)
Table 1 (continued)
Symbol Description Unit
p Pa
Specific bearing load, as given by: p=
BD⋅
p Lubricant supply pressure Pa
p ⋅ψ
eff
* *
p Lubricant supply pressure parameter, as given by: p = 1
en en
ηω⋅
eff
p Maximum admissible lubricant film pressure Pa
lim
Maximum admissible specific bearing load at transition to mixed friction Pa
lim,tr
p Maximum lubricant film pressure Pa
max
max
* *
p Maximum lubricant film pressure parameter, as given by: p = 1
max max
p Pa
Specific bearing load at transition to mixed friction, as given by: p =
BD⋅
Lubricant flow rate, as given by: QQ=+Q
Q m /s
3 p
Q Minimum admissible lubricant flow rate m /s
lim
Lubricant flow rate due to supply pressure m /s
* *
Lubricant flow rate parameter due to supply pressure, as given by: Q = 1
p p
pQ⋅
en 0
3 3
Reference value of Q, as given by: QR=⋅ωψ⋅ m /s
0 eff

Q Lubricant flow rate at the entrance into the lubrication gap (circumferential direction) m /s

Lubricant flow rate at the exit of the lubrication gap (circumferential direction), as

m /s
given by: QQ=−Q
21 3

Lubricant flow rate parameter at the exit of the lubrication gap (circumferential direction),

Q 2 1
2 as given by: Q =

Q Lubricant flow rate due to hydrodynamic pressure build-up (side flow rate) m /s

Lubricant flow rate parameter due to hydrodynamic pressure build-up (side flow parame-

* Q
3 1
Q *
ter), as given by: Q =
R m
Journal bearing inside radius, as given by: R=
R Lobe or pad bore radius of a multi-lobed or tilting-pad journal bearing m

Difference between lobe or pad bore radius and journal radius, as given by: ΔRR=−R

BB J

Journal radius (radius of the shaft section located inside of a journal bearing), as given by:

R D
R =
Surface finish ten-point average of bearing sliding surface m
z,B
Surface finish ten-point average of journal sliding surface m
z,J
ρω⋅⋅RC⋅
R,eff
Re Reynolds number, as given by: Re= 1
eff
Critical Reynolds number 1
4 © ISO 2020 – All rights reserved
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ISO/TS 31657-1:2020(E)
Table 1 (continued)
Symbol Description Unit
F⋅ψ
eff
So Sommerfeld number, as given by: So= 1
BD⋅⋅ηω⋅
eff
So Sommerfeld number at transition to mixed friction 1
S Displacement amplitude of the rotor (mechanical oscillation) m
T Temperature °C
Heating of lubricant between bearing entrance and exit, as given by: ΔTT=−T
ΔT K
ex en
ΔT Maximum admissible heating of lubricant between bearing entrance and exit K
lim
Bearing temperature °C
T Effective temperature of lubricant film °C
eff
Lubricant temperature at the bearing entrance °C
T Lubricant temperature at the bearing exit °C
Journal temperature °C
T Maximum admissible bearing temperature °C
lim
Maximum temperature of lubricant film °C
max

Difference between maximum temperature of lubricant film and lubricant temperature in

max
the lubricant pocket, as given by: ΔTT=−T
maxmax 1

Non-dimensional difference between maximum temperature of lubricant film and lubricant

* ρψ⋅⋅c
ΔT p eff 1
max
temperature in the lubricant pocket, as given by: ΔΔT = ⋅ T
max max
pf⋅

T Lubricant temperature at the entrance into the lubrication gap (circumferential direction) °C

Difference between lubricant temperature at the entrance into the lubrication gap and

lubricant temperature at the bearing entrance, as given by: ΔTT=−T
11 en

Lubricant temperature at pressure profile trailing edge (circumferential direction) °C

Difference between lubricant temperature at pressure profile trailing edge and lubricant

ΔT K
temperature at the entrance into the lubrication gap, as given by: ΔTT=−T
22 1
t Time s
Circumferential speed of the journal, sliding velocity
m/s
UR=⋅ω
Circumferential speed at transition to mixed friction m/s
Minimum admissible circumferential speed at transition to mixed friction m/s
lim,tr
u Velocity component in the φ-direction m/s
Average velocity component in the φ-direction m/s
w Velocity component in the z- direction m/s
w Average velocity component in the z-direction m/s
x Coordinate of journal radial motion, normal to direction of load m

Relative coordinate of journal radial motion, normal to direction of load, as given by: x =

Coordinate normal to sliding surface (across the lubricant film, in the radial direction);

y m
coordinate of journal radial motion, in direction of load

Relative coordinate of journal radial motion, in direction of load, as given by: y =

© ISO 2020 – All rights reserved 5
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ISO/TS 31657-1:2020(E)
Table 1 (continued)
Symbol Description Unit
y Coordinate normal to sliding surface (across the lubricant film) m
Z Number of sliding surfaces (pads), number of pockets per bearing 1

Coordinate parallel to the sliding surface, normal to direction of motion (normal to circum-

z m
ferential direction, in the axial direction)
Linear thermal expansion coefficient of bearing material K
lB,
Linear thermal expansion coefficient of journal material K
l ,J

β Attitude angle (angular position of journal eccentricity related to the direction of load) °

Angle between direction of load and position of minimum lubricant film thickness °

h,min
Journal misalignment angle (angular deviation of journal) °
Relative eccentricity: ε=
ε 1
R,eff
η Dynamic viscosity of the lubricant Pa s
Effective dynamic viscosity in the lubricant film Pa s
eff
ρ Density of the lubricant kg/m
φ Angular coordinate in circumferential direction °
Angular coordinate of pivot position of pad (tilting-pad bearing) °
ϕ Angular coordinate of lubricant pocket centreline °

Angular coordinate of bearing sliding surface (segment or pad) centreline at multi-lobed or

ϕ °
tilting-pad journal bearings (with non-tilted pads), see Figure 1, a)
Angular coordinate at the entrance into the gap °
Angular coordinate at the end of the hydrodynamic pressure build-up °
Angular coordinate at the exit of the gap °
ψ ‰
Relative bearing clearance, as given by: ψ=
Tolerance of ψ, as given by: Δψ=−ψψ
Δψ ‰
maxmin
Effective relative bearing clearance ‰
eff
Maximum value of ψ ‰
max
Minimum value of ψ ‰
min
Thermal change of ψ ‰
Relative bearing clearance at 20 °C ‰
Angular span of bearing sliding surface (segment or pad), as given by: Ω=−ϕϕ
Ω °

Angular distance between leading edge and pivot position of pad (tilting-pad bearing), as

given by: Ω =−ϕϕ
FF 1

Relative angular distance between leading edge and pivot position of pad (tilting-pad

Ω 1
bearing), as given by: ΩΩ= /Ω
360°
Angular span of lubricant pocket, as given by: ΩΩ= −
Angular speed of the rotor, as given by: ωπ=⋅2 ⋅N
ω s
Angular speed at transition to mixed friction s
6 © ISO 2020 – All rights reserved
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ISO/TS 31657-1:2020(E)
5 General principles, assumptions and preconditions

The bearing bore form of multi-lobed journal bearings [see Figure 1, a)] and tilting-pad journal bearings

h ϕ

[with non-tilted pads according to Figure 1, b)] is described by the profile function h ()ϕ = in the

case of a centric journal position ε ==0 . The angle φ is counted, starting from the load direction, in

the journal rotational direction.

Formula (1) applies to the shell segment or pad i with the angular length Ω =−ϕϕ :

ii31,,i
ΔΔR R
 
B B
h ()ϕϕ=+ −11⋅−cos( ϕ ),i= ,...,Ζ (1)
 
00,i ,i
C C
 
R R
with the profile factor
RR−
ΔR e
B J
B B
K == =+1 ,
C C C
R R R
minimum clearance
DD−
CR=−R =
R J
and the lubricant film thickness ratio as given by Formula (2):
0,max
* *
hh()ϕ == (2)
00Pi,,max

Here the position of the sliding surface (segment or pad) axis (curvature centre "point") of the shell

segment or pad i is uniquely described by the sliding surface eccentricity e and the associated angle

coordinate ϕ .
0,i
In the case of cylindrical bearings, K = 1 and h ϕ =1 .

NOTE Instead of the profile factor, K the "preload factor", m, is frequently used internationally; the

following relation exists between both variables:
K =
1−m

In the case of an eccentric position of the journal (ε, β), Formula (3) applies to the lubricant film

thickness, h(φ), of the multi-lobed journal bearings [(see Figure 1, c)]:
hC()ϕϕ=⋅hC()=⋅[(h ϕϕ)c−⋅εβos()− ] (3)
RR 0

In the case of tilting-pad journal bearings [see Figure 1, d)], the individual pads automatically adjust

themselves (optimally) so that the lubricant film force F passes through the supporting pad pivot,

[9]

respectively . For a more precise calculation of tilting-pad journal bearings, the elasticities in the pad

support and the elastic and thermal deformations of the pads shall be considered.

© ISO 2020 – All rights reserved 7
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ISO/TS 31657-1:2020(E)

The pressure formation in the lubrication gaps is basically calculated with the numerical solutions of

the Reynolds differential equation for a finite bearing width:
1 ∂  ∂p  ∂ ∂p ∂h
 
⋅ h ⋅ + h ⋅ =⋅6ηω⋅⋅ (4)
 
 
∂ϕϕ∂ ∂z ∂z ∂ϕ
R  
 
with ωπ=⋅2 ⋅N angular speed of the rotor.

For derivation of the Reynolds differential equation, reference is made to Reference [8], for the

numerical solution to Reference [9].

When solving Formula (4), the following idealising assumptions and preconditions are made, whose

[10]
permissibility shall be estimated according to Clause 6, if necessary .
a) The lubricant corresponds to a Newtonian fluid.
b) All flow processes of the lubricant are laminar.
c) The lubricant adheres fully to the sliding surfaces.
d) The lubricant is incompressible.

e) At the leading edge of the segment or pad, the lubrication gap is completely filled with lubricant.

f) Inertia effects, gravitation and magnetic forces of the lubricant are negligible.

g) The components forming the lubrication gap are rigid or their deformation is negligible; the

surfaces of the journal and bearing bore are ideal circular cylinders or cylindrical segments.

h) The curvature radii of the surfaces moving relative to one another are large in comparison to the

lubricant film thicknesses.

i) The lubricant film thickness in an axial direction (z coordinate) is constant.

j) Pressure changes in the lubricant film normal to the sliding surfaces (in the lubricant film thickness

direction) are negligible.

k) A movement normal to the sliding surfaces (in the lubricant film thickness direction) is not

considered here, in contrast to 6.10.
l) The lubricant film is isoviscous in the entire lubrication gap.

m) The lubricant is supplied at the leading edge of the segments or pads respectively; the level of the

supply pressure is negligible compared to the lubricant film pressures themselves.

The boundary conditions for the lubricant film pressure build-up satisfy the continuity condition.

The following applies respectively to the individual segments or pads (see Figures 2 and 4):

— at the lateral bearing edge pzϕ,/=±B 20= ;
— in the lubrication pocket and on the sealing land pz()ϕ, =0 ;
— at the pressure profile trailing edge pz[(ϕ ),z][= ϕ ()zz,]=0 ;
∂p ∂p
— at the beginning of cavitation area pz[(ϕ ),z][= ϕϕ()zz,]= [(zz), ]=0 ;
∂ϕ ∂z
— at the end of cavitation area pz[(ϕ ),z]=0 .
[15][16]

The cavitation theory according to Jakobsson, Floberg and Olsson is used in the cavitation area

and on its edge for fulfilment of the continuity condition.
8 © ISO 2020 – All rights reserved
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ISO/TS 31657-1:2020(E)

The numerical integration of the Reynolds differential equation is done using the transformation of the

pressure proposed in Reference [9] by conversion into a difference formula, which is applied to a grid of

nodal points and which leads to a system of linear formulas.

After specifying the boundary conditions, the integration yields the pressure profile in the

circumferential and axial direction.

The maximum lubricant film temperature is calculated using the numerical solution of the energy

equation averaged by integration with respect to the lubricant film thickness, h
h 22
 
   
u ∂T ∂T η 1 ∂u ∂w
 
⋅ +⋅w = ⋅⋅ + ⋅dy (5)
   
R ∂ϕ ∂zcρ⋅ h ∂y ∂y
 
   
J ph h
00 
[9][13][14]
for the two-dimensional temperature distribution T(φ, z) .
This includes
h h
u =⋅ ud⋅=yw, ⋅⋅wdy
h h
h h

the flow rates averaged over the lubricant film thickness h in the circumferential and axial direction.

When deriving the energy equation, Formula (5), it is also assumed besides the above preconditions

that no heat is dissipated from the lubrication gap by thermal conduction (adiabatic calculation).

When solving Formula (5) the following boundary conditions apply (see Figure 4):
— at the entrance gap Tz(,ϕ )=T ;
— in the axial bearing centre (,ϕ z==00) .

The numerical integration of Formula (5) is carried out similar to the solution of the Reynolds

differential equation, Formula (4), using a suitable difference formula and yields for the specified

boundary conditions the temperature distribution in the circumferential and axial direction.

The application of the similarity principle in the hydrodynamic plain bearing theory leads to

dimensionless similarity variables for the interesting characteristic values (such as load-carrying

capacity, frictional power, lubricant flow rate and relative bearing width). Use of the similarity variables

reduces the number of necessary numerical solutions of the Reynolds differential equation, Formula (4),

and the energy equation, Formula (5), which are summarised in ISO/TS 31657-2 and ISO/TS 31657-3.

As a rule, other solutions can also be used, insofar as they satisfy the conditions indicated in this

document and a corresponding numerical accuracy.

ISO/TS 31657-4 contains operational guide values for checking the calculation results, in order to

...

TECHNICAL ISO/TS
SPECIFICATION 31657-1
First edition
Plain bearings — Hydrodynamic plain
journal bearings under steady-state
conditions —
Part 1:
Calculation of multi-lobed and tilting
pad journal bearings
PROOF/ÉPREUVE
Reference number
ISO/TS 31657-1:2020(E)
ISO 2020
---------------------- Page: 1 ----------------------
ISO/TS 31657-1:2020(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.
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ISO/TS 31657-1:2020(E)
Contents Page

Foreword ........................................................................................................................................................................................................................................iv

Introduction ..................................................................................................................................................................................................................................v

1 Scope ................................................................................................................................................................................................................................. 1

2 Normative references ...................................................................................................................................................................................... 1

3 Terms and definitions ..................................................................................................................................................................................... 1

4 Symbols and units ............................................................................................................................................................................................... 1

5 General principles, assumptions and preconditions ...................................................................................................... 7

6 Calculation method ............................................................................................................................................................................................ 9

6.1 General ........................................................................................................................................................................................................... 9

6.2 Load carrying capacity ..................................................................................................................................................................11

6.3 Frictional power..................................................................................................................................................................................11

6.4 Lubricant flow rate ...........................................................................................................................................................................12

6.5 Heat balance ...........................................................................................................................................................................................13

6.6 Maximum lubricant film temperature .............................................................................................................................14

6.7 Maximum lubricant film pressure .......................................................................................................................................15

6.8 Operating states ..................................................................................................................................................................................15

6.9 Further influencing parameters ............................................................................................................................................15

6.10 Stiffness and damping coefficients .....................................................................................................................................16

7 Figures ..........................................................................................................................................................................................................................18

Annex A (informative) Calculation examples ...........................................................................................................................................23

Bibliography .............................................................................................................................................................................................................................39

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ISO/TS 31657-1:2020(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

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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).

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constitute an endorsement.

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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.
A list of all parts in the ISO/TS 31657 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 PROOF/ÉPREUVE © ISO 2020 – All rights reserved
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ISO/TS 31657-1:2020(E)
Introduction

The aim of this document is the operationally-safe design of plain journal bearings for medium or high

journal circumferential velocities, U , up to approximately 90 m/s by applying a calculation method for

oil-lubricated hydrodynamic plain bearings with complete separation of journal and bearing sliding

surfaces by a lubricating film.

For low circumferential velocities up to approximately 30 m/s usually circular cylindrical bearings

are applied. For these bearings a similar calculation method is given in ISO 7902-1, ISO 7902-2 and

ISO 7902-3.

Based on practical experience the calculation procedure is usable for application cases where specific

bearing load times circumferential speed, pU⋅ , does not exceed approximately 200 MPa·m/s.

This document discusses multi-lobed journal bearings with two, three and four equal, symmetrical

sliding surfaces, which are separated by laterally-closed lubrication pockets, and symmetrically-loaded

tilting-pad journal bearings with four and five pads. Here, the curvature radii, R , of the sliding

surfaces are usually chosen larger than half the bearing diameter, D, so that an increased bearing

clearance results at the pad ends.

The calculation method described here can also be used for other gap forms, for example asymmetrical

multi-lobed journal bearings like offset-halves bearings, pressure-dam bearings or other tilting-pad

journal bearing designs, if the numerical solutions of the basic formulas are available for these designs.

© ISO 2020 – All rights reserved PROOF/ÉPREUVE v
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TECHNICAL SPECIFICATION ISO/TS 31657-1:2020(E)
Plain bearings — Hydrodynamic plain journal bearings
under steady-state conditions —
Part 1:
Calculation of multi-lobed and tilting pad journal bearings
1 Scope

This document specifies the general principles, assumptions and preconditions for the calculation of

multi-lobed and tilting-pad journal bearings by means of an easy-to-use calculation procedure based on

numerous simplifying assumptions. For a reliable evaluation of the results of this calculation method,

it is indispensable to consider the physical implications of these assumptions as well as practical

experiences for instance from temperature measurements carried out on real machinery under

typical operating conditions. Applied in this sense, this document presents a simple way to predict

the approximate performance of plain journal bearings for those unable to access more complex and

accurate calculation techniques.

The calculation method serves for the design and optimisation of plain bearings, for example in

turbines, compressors, generators, electric motors, gears and pumps. It is restricted to steady-state

operation, i.e. in continuous operating states the load according to size and direction and the angular

velocity of the rotor are constant.

Unsteady operating states are not recorded. The stiffness and damping coefficients of the plain journal

bearings required for the linear vibration and stability investigations are indicated in ISO/TS 31657-2

and ISO/TS 31657-3.
2 Normative references
There are no normative references in this document.
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
Table 1 — Symbols and units
Symbol Description Unit
B Bearing width m
* *
Relative bearing width, width ratio as given by: B =
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ISO/TS 31657-1:2020(E)
Table 1 (continued)
Symbol Description Unit
b Width of lubricant pocket m
* P
b 1
Relative width of lubricant pocket, as given by: b =
Bearing radial clearance, as given by: CR=−R
C m
Effective radial bearing clearance m
R,eff
c Stiffness coefficient of lubricant film (i,k = 1,2) N/m
Non-dimensional stiffness coefficient of lubricant film, as given by:
* ψ
c 1
eff
c = ⋅=ci(),,k 12
ik ik
2⋅⋅Bηω⋅
eff
J/(kg
Specific heat capacity (p = constant)
D Nominal bearing diameter (inside diameter of journal bearing) m
D Maximum value of D m
max
Minimum value of D m
min

Journal diameter (diameter of the shaft section located inside of a journal bearing) m

D Maximum value of D
Jm, ax J
D Minimum value of D
Jm, in J
Damping coefficient of lubricant film (i,k = 1,2) N s/m
Non-dimensional damping coefficient of lubricant film, as given by:
* 3
ψ 1
ik eff
d = ⋅⋅ω di(),,k=12
ik ik
2⋅⋅Bηω⋅
eff
E Eccentricity (distance between journal and bearing axis) m

Eccentricity of the bearing sliding surfaces (pads) of a multi-lobed or tilting-pad journal

e m
bearing
F Bearing force, bearing load, nominal bearing load, load-carrying capacity N
ΔF Component of additional dynamic force in x-direction N
Component of additional dynamic force in y-direction N
Component of additional dynamic force parameter in x-direction, as given by:
* ΔF ⋅ψ
ΔF x
eff
ΔF =
BD⋅⋅ηω⋅
eff
Component of additional dynamic force parameter in y-direction, as given by:
ΔF ⋅ψ
ΔF 1
y eff
ΔF =
BD⋅⋅ηω⋅
eff
Friction force, as given by: Ff=⋅F
Friction force parameter, as given by: F =⋅So
F 1
eff
Bearing force at transition to mixed friction N
F Coefficient of friction 1
h(φ) Local lubricant film thickness m
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ISO/TS 31657-1:2020(E)
Table 1 (continued)
Symbol Description Unit
h()ϕ
Relative local lubricant film thickness, as given by: h ()ϕ =
h ()ϕ
Journal deflection 1
Minimum admissible lubricant film thickness at transition to mixed friction m
lim,tr

Minimum admissible relative lubricant film thickness at transition to mixed friction, as

lim,tr
h 1
lim,tr
given by: h =
lim,tr
R,eff
Minimum lubricant film thickness, minimum gap m
min
min

Minimum relative lubricant film thickness, minimum relative gap, as given by: h =

h 1
min
min
R,eff
Minimum lubricant film thickness at transition to mixed friction m
min,tr

Minimum relative lubricant film thickness at transition to mixed friction, as given by:

min,tr
h 1
min,tr
h =
min,tr
R,eff
h ()ϕ Local gap at ε=0 , gap function m
h ϕ
* ()
h ϕ 0 1
() *
0 Relative local gap at ε=0 , profile function, as given by: h ϕ =
Maximum gap at ε=0
0,max
* h
0,max
h 1
0,max
Maximum relative gap at ε=0 , gap ratio, as given by: h =
0,max

Profile factor (relative difference between lobe or pad bore radius and journal radius), as

K B 1
given by: K ==
Cm1−
Effective profile factor 1
P,eff
Profile factor at 20 °C 1
P,20
M Mixing factor 1
m Preload factor, preload of bearing or pad sliding surface 1

N Rotational speed (rotational frequency) of the rotor (revolutions per time unit) s

N Critical speed (critical rotational frequency) s

Rotational speed (rotational frequency) at the stability speed limit of the rotor supported

N s
lim
by plain bearings

Resonance speed (resonance rotational frequency) of the rotor supported by plain bearings s

rsn

Rotational speed (rotational frequency) at transition to mixed friction, transition rotational

speed, transition rotational frequency
Centreline of plain bearing 1
O Centreline of sliding surface No. i 1
Centreline of journal 1
P Frictional power, as given by: PF=⋅U
f ff J
Heat flow via the lubricant W
th,L
p Lubricant film pressure, local lubricant film pressure Pa
Specific bearing load, as given by: p=
BD⋅
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ISO/TS 31657-1:2020(E)
Table 1 (continued)
Symbol Description Unit
Lubricant supply pressure Pa
p ⋅ψ
en eff
* *
p Lubricant supply pressure parameter, as given by: p =
en en
ηω⋅
eff
Maximum admissible lubricant film pressure Pa
lim
Maximum admissible specific bearing load at transition to mixed friction Pa
lim,tr
Maximum lubricant film pressure Pa
max
max
* *
p Maximum lubricant film pressure parameter, as given by: p =
max max
Specific bearing load at transition to mixed friction, as given by: p =
BD⋅
Lubricant flow rate, as given by: QQ=+Q
Q m /s
3 p
Minimum admissible lubricant flow rate m /s
lim
Lubricant flow rate due to supply pressure m /s
Q Lubricant flow rate parameter due to supply pressure, as given by: Q = 1
pQ⋅
en 0
3 3
Q m /s
Reference value of Q, as given by: QR=⋅ωψ⋅
0 eff

Lubricant flow rate at the entrance into the lubrication gap (circumferential direction) m /s

Lubricant flow rate at the exit of the lubrication gap (circumferential direction), as

Q m /s
given by: QQ=−Q
21 3

Lubricant flow rate parameter at the exit of the lubrication gap (circumferential direction),

* Q
2 1
as given by: Q =
Lubricant flow rate due to hydrodynamic pressure build-up (side flow rate) m /s

Lubricant flow rate parameter due to hydrodynamic pressure build-up (side flow parame-

Q 3 1
3 ter), as given by: Q =
R m
Journal bearing inside radius, as given by: R=
Lobe or pad bore radius of a multi-lobed or tilting-pad journal bearing m

Difference between lobe or pad bore radius and journal radius, as given by: ΔRR=−R

ΔR m
BB J

Journal radius (radius of the shaft section located inside of a journal bearing), as given by:

R D
J J
R =
Surface finish ten-point average of bearing sliding surface m
z,B
Surface finish ten-point average of journal sliding surface m
z,J
ρω⋅⋅RC⋅
R,eff
Re Reynolds number, as given by: Re= 1
eff
Re Critical Reynolds number 1
F⋅ψ
eff
So Sommerfeld number, as given by: So= 1
BD⋅⋅ηω⋅
eff
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ISO/TS 31657-1:2020(E)
Table 1 (continued)
Symbol Description Unit
Sommerfeld number at transition to mixed friction 1
S Displacement amplitude of the rotor (mechanical oscillation) m
T Temperature °C
ΔT Heating of lubricant between bearing entrance and exit, as given by: ΔTT=−T K
ex en
Maximum admissible heating of lubricant between bearing entrance and exit K
lim
T Bearing temperature °C
Effective temperature of lubricant film °C
eff
T Lubricant temperature at the bearing entrance °C
Lubricant temperature at the bearing exit °C
Journal temperature °C
Maximum admissible bearing temperature °C
lim
T Maximum temperature of lubricant film °C
max

Difference between maximum temperature of lubricant film and lubricant temperature in

ΔT K
max
the lubricant pocket, as given by: ΔTT=−T
maxmax 1

Non-dimensional difference between maximum temperature of lubricant film and lubricant

* ρψ⋅⋅c
ΔT 1
p eff
max
temperature in the lubricant pocket, as given by: ΔΔT = ⋅ T
max max
pf⋅

Lubricant temperature at the entrance into the lubrication gap (circumferential direction) °C

Difference between lubricant temperature at the entrance into the lubrication gap and

ΔT K
lubricant temperature at the bearing entrance, as given by: ΔTT=−T
11 en

T Lubricant temperature at pressure profile trailing edge (circumferential direction) °C

Difference between lubricant temperature at pressure profile trailing edge and lubricant

temperature at the entrance into the lubrication gap, as given by: ΔTT=−T
22 1
T Time s
Circumferential speed of the journal, sliding velocity
m/s
UR=⋅ω
U Circumferential speed at transition to mixed friction m/s
Minimum admissible circumferential speed at transition to mixed friction m/s
lim,tr
U Velocity component in the φ-direction m/s
u Average velocity component in the φ-direction m/s
W Velocity component in the z- direction m/s
Average velocity component in the z-direction m/s
X Coordinate of journal radial motion, normal to direction of load m

Relative coordinate of journal radial motion, normal to direction of load, as given by: x =

Coordinate normal to sliding surface (across the lubricant film, in the radial direction);

Y m
coordinate of journal radial motion, in direction of load

Relative coordinate of journal radial motion, in direction of load, as given by: y =

y Coordinate normal to sliding surface (across the lubricant film) m
Z Number of sliding surfaces (pads), number of pockets per bearing 1
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ISO/TS 31657-1:2020(E)
Table 1 (continued)
Symbol Description Unit

Coordinate parallel to the sliding surface, normal to direction of motion (normal to circum-

z m
ferential direction, in the axial direction)
Linear thermal expansion coefficient of bearing material K
lB,
Linear thermal expansion coefficient of journal material K
l ,J

β Attitude angle (angular position of journal eccentricity related to the direction of load) °

Angle between direction of load and position of minimum lubricant film thickness °

h,min
Journal misalignment angle (angular deviation of journal) °
Relative eccentricity: ε=
ε 1
R,eff
η Dynamic viscosity of the lubricant Pa s
Effective dynamic viscosity in the lubricant film Pa s
eff
ρ Density of the lubricant kg/m
Φ Angular coordinate in circumferential direction °
ϕ Angular coordinate of pivot position of pad (tilting-pad bearing) °
Angular coordinate of lubricant pocket centreline °

Angular coordinate of bearing sliding surface (segment or pad) centreline at multi-lobed or

tilting-pad journal bearings (with non-tilted pads), see Figure 1, a)
Angular coordinate at the entrance into the gap °
ϕ Angular coordinate at the end of the hydrodynamic pressure build-up °
Angular coordinate at the exit of the gap °
Ψ ‰
Relative bearing clearance, as given by: ψ=
Tolerance of ψ, as given by: Δψ=−ψψ
Δψ ‰
maxmin
ψ Effective relative bearing clearance ‰
eff
Maximum value of ψ ‰
max
ψ Minimum value of ψ ‰
min
Thermal change of ψ ‰
ψ Relative bearing clearance at 20 °C ‰
Angular span of bearing sliding surface (segment or pad), as given by: Ω=−ϕϕ
Ω °

Angular distance between leading edge and pivot position of pad (tilting-pad bearing), as

Ω °
given by: Ω =−ϕϕ
FF 1

Relative angular distance between leading edge and pivot position of pad (tilting-pad

bearing), as given by: ΩΩ= /Ω
360°
Ω °
Angular span of lubricant pocket, as given by: ΩΩ= −
ω Angular speed of the rotor, as given by: ωπ=⋅2 ⋅N s
Angular speed at transition to mixed friction s
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ISO/TS 31657-1:2020(E)
5 General principles, assumptions and preconditions

The bearing bore form of multi-lobed journal bearings [see Figure 1, a)] and tilting-pad journal bearings

h ϕ

[with non-tilted pads according to Figure 1, b)] is described by the profile function h ()ϕ = in the

case of a centric journal position ε ==0 . The angle φ is counted, starting from the load direction, in

the journal rotational direction.

Formula (1) applies to the shell segment or pad i with the angular length Ω =−ϕϕ :

ii31,,i
ΔΔR R
 
B B
h ()φφ=+ −11⋅−cos( φ ),i= ,...,Ζ (1)
 
00,i ,i
C C
 
R R
with the profile factor
RR−
ΔR e
B J
B B
K == =+1 ,
C C C
R R R
minimum clearance
DD−
CR=−R =
R J
and the lubricant film thickness ratio as given by Formula (2):
0,max
* *
hh()φ == (2)
00Pi,,max

Here the position of the sliding surface (segment or pad) axis (curvature centre "point") of the shell

segment or pad i is uniquely described by the sliding surface eccentricity e and the associated angle

coordinate ϕ .
0,i
In the case of cylindrical bearings, K = 1 and h ϕ =1 .

NOTE Instead of the profile factor, K the "preload factor", m, is frequently used internationally; the

following relation exists between both variables:
K =
1−m

In the case of an eccentric position of the journal (ε, β), Formula (3) applies to the lubricant film

thickness, h(φ), of the multi-lobed journal bearings [(see Figure 1, c)]:
hC()φφ=⋅hC()=⋅[(h φε)c−⋅ os()φβ− ] (3)
RR 0

In the case of tilting-pad journal bearings [see Figure 1, d)], the individual pads automatically adjust

themselves (optimally) so that the lubricant film force F passes through the supporting pad pivot,

[9]

respectively . For a more precise calculation of tilting-pad journal bearings, the elasticities in the pad

support and the elastic and thermal deformations of the pads shall be considered.

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ISO/TS 31657-1:2020(E)

The pressure formation in the lubrication gaps is basically calculated with the numerical solutions of

the Reynolds differential equation for a finite bearing width:
1 ∂  ∂p  ∂ ∂p ∂h
 
⋅ h ⋅ + h ⋅ =⋅6ηω⋅⋅ (4)
 
 
∂φφ∂ ∂z ∂z ∂φ
R  
 
with ωπ=⋅2 ⋅N angular speed of the rotor.

For derivation of the Reynolds differential equation, reference is made to Reference [8], for the

numerical solution to Reference [9].

When solving Formula (4), the following idealising assumptions and preconditions are made, whose

[10]
permissibility shall be estimated according to Clause 6, if necessary .
a) The lubricant corresponds to a Newtonian fluid.
b) All flow processes of the lubricant are laminar.
c) The lubricant adheres fully to the sliding surfaces.
d) The lubricant is incompressible.

e) At the leading edge of the segment or pad, the lubrication gap is completely filled with lubricant.

f) Inertia effects, gravitation and magnetic forces of the lubricant are negligible.

g) The components forming the lubrication gap are rigid or their deformation is negligible; the

surfaces of the journal and bearing bore are ideal circular cylinders or cylindrical segments.

h) The curvature radii of the surfaces moving relative to one another are large in comparison to the

lubricant film thicknesses.

i) The lubricant film thickness in an axial direction (z coordinate) is constant.

j) Pressure changes in the lubricant film normal to the sliding surfaces (in the lubricant film thickness

direction) are negligible.

k) A movement normal to the sliding surfaces (in the lubricant film thickness direction) is not

considered here, in contrast to 6.10.
l) The lubricant film is isoviscous in the entire lubrication gap.

m) The lubricant is supplied at the leading edge of the segments or pads respectively; the level of the

supply pressure is negligible compared to the lubricant film pressures themselves.

The boundary conditions for the lubricant film pressure build-up satisfy the continuity condition.

The following applies respectively to the individual segments or pads (see Figures 2 and 4):

— at the lateral bearing edge pzφ,/=±B 20= ;
— in the lubrication pocket and on the sealing land pz()φ, =0 ;
— at the pressure profile trailing edge pz[(φ ),z][= φ ()zz,]=0 ;
∂p ∂p
— at the beginning of cavitation area pz[(φ ),z][= φφ()zz,]= [(zz), ]=0 ;
∂φ ∂z
— at the end of cavitation area pz[(φ ),z]=0 .
[15][16]

The cavitation theory according to Jakobsson, Floberg and Olsson is used in the cavitation area

and on its edge for fulfilment of the continuity condition.
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ISO/TS 31657-1:2020(E)

The numerical integration of the Reynolds differential equation is done using the transformation of the

pressure proposed in Reference [9] by conversion into a difference formula, which is applied to a grid of

nodal points and which leads to a system of linear formulas.

After specifying the boundary conditions, the integration yields the pressure profile in the

circumferential and axial direction.

The maximum lubricant film temperature is calculated using the numerical solution of the energy

equation averaged by integration with respect to the lubricant film thickness, h
h 22
 
   
u ∂T ∂T η 1 ∂u ∂w
 
⋅ +⋅w = ⋅⋅ + ⋅dy (5)
   
R ∂φ ∂zcρ⋅ h ∂y ∂y
 
   
J ph h
00 
[9][13][14]
for the two-dimensional temperature distribution T(φ, z) .
This includes
h h
u =⋅ ud⋅=yw, ⋅⋅wdy
h h
h h

the flow rates averaged over the lubricant film thickness h in the circumferential and axial direction.

When deriving the energy equation, Formula (5), it is also assumed besides the above preconditions

that no heat is dissipated from the lubrication gap by thermal conduction (adiabatic calculation).

When solving Formula (5) the following boundary conditions apply (see Figure 4):
— at the entrance gap Tz(,φ )=T ;
— in the axial bearing centre (,φ z==00) .

The numerical integration of Formula (5) is carried out similar to the solution of the Reynolds

differential equation, Formula (4), using a suitable difference formula and yields for the specified

boundary conditions the temperature distribution in the circumferential and axial direction.

The application of the similarity principle in the hydrodynamic plain bearing theory leads to

dimensionless similarity variables for the interesting characteristic values (such as load-carrying

capacity, frictional power, lubricant flow rate and relative bearing width). Use of the similarity variables

reduces the number of necessary numerical solutions of the Reynolds differential equation, Formula (4),

and the energy equation, Formula (5), which are summarised in ISO/TS 31657-2 and ISO/TS 31657-3.

As a rule, other solutions can also be used, insofar as they satisfy the conditions indicated in this

document and a correspond
...

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