ISO 12167-1:2023

Plain bearings — Hydrostatic plain journal bearings with drainage grooves under steady-state conditions — Part 1: Calculation of oil-lubricated plain journal bearings with drainage grooves

This document specifies the calculation procedures to calculate and assess a given bearing design, as well as to design a bearing as a function of some optional parameters. This document contains the design of the required lubrication system including the calculation of the restrictor data. This document applies to hydrostatic plain journal bearings under steady-state conditions.

Paliers lisses — Paliers lisses radiaux hydrostatiques avec rainures d'écoulement fonctionnant en régime stationnaire — Partie 1: Calcul pour la lubrification des paliers lisses radiaux avec rainures d'écoulement

General Information

• Status: Published
• Publication Date: 13-Jun-2023
• ICS: 21.100.10 - Plain bearings
• Technical Committee: ISO/TC 123/SC 8 - Calculation methods for plain bearings and their applications
• Drafting Committee: ISO/TC 123/SC 8 - Calculation methods for plain bearings and their applications
• Current Stage: 6060 - International Standard published
• Start Date: 14-Jun-2023
• Due Date: 13-Jul-2023
• Completion Date: 14-Jun-2023

Relations

• Revises: ISO 12167-1:2016 - Plain bearings — Hydrostatic plain journal bearings with drainage grooves under steady-state conditions — Part 1: Calculation of oil-lubricated plain journal bearings with drainage grooves
• Effective Date: 06-Jun-2022

Standards Content (Sample)

INTERNATIONAL ISO
STANDARD 12167-1
Third edition
2023-06
Plain bearings — Hydrostatic plain
journal bearings with drainage
grooves under steady-state
conditions —
Part 1:
Calculation of oil-lubricated plain
journal bearings with drainage
grooves
Paliers lisses — Paliers lisses radiaux hydrostatiques avec rainures
d'écoulement fonctionnant en régime stationnaire —
Partie 1: Calcul pour la lubrification des paliers lisses radiaux avec
rainures d'écoulement
Reference number
ISO 12167-1:2023(E)
© ISO 2023

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ISO 12167-1:2023(E)
COPYRIGHT PROTECTED DOCUMENT
© ISO 2023
All rights reserved. Unless otherwise specified, or required in the context of its implementation, no part of this publication may
be reproduced or utilized otherwise in any form or by any means, electronic or mechanical, including photocopying, or posting on
the internet or an intranet, without prior written permission. Permission can be requested from either ISO at the address below
or ISO’s member body in the country of the requester.
ISO copyright office
CP 401 • Ch. de Blandonnet 8
CH-1214 Vernier, Geneva
Phone: [phone]
Email: [e-mail]
Website: www.iso.org
Published in Switzerland
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ISO 12167-1:2023(E)
Contents Page
Foreword .iv
Introduction .v
1 Scope . 1
2 Normative references . 1
3 Terms and definitions . 1
4 Symbols . 1
5 Bases of calculation and boundary conditions . 4
6 Method of calculation .6
6.1 General . 6
6.2 Load-carrying capacity . 6
6.3 Lubricant flow rate and pumping power . 8
6.4 Frictional power . 9
6.5 Optimization . 10
6.6 Temperatures and viscosities . 11
6.7 Minimum pressure in recesses . 11
Annex A (informative) Description of the approximation method for the calculation
of hydrostatic plain journal bearings .13
Annex B (informative) Example of calculation according to the method given in Annex A .23
Bibliography .31
iii
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ISO 12167-1:2023(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 document 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).
ISO draws attention to the possibility that the implementation of this document may involve the use
of (a) patent(s). ISO takes no position concerning the evidence, validity or applicability of any claimed
patent rights in respect thereof. As of the date of publication of this document, ISO had not received
notice of (a) patent(s) which may be required to implement this document. However, implementers are
cautioned that this may not represent the latest information, which may be obtained from the patent
database available at www.iso.org/patents. ISO shall not be held responsible for identifying any or all
such patent rights.
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.
The committee responsible for this document is 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 12167-1:2016), which has been
technically revised.
The main changes are as follows:
— subclause titles have been added;
— symbols have been corrected and added in Table 1;
— calculation values in Annex B have been corrected;
— adjustments have been made according to ISO/IEC Directives, Part 2:2021;
— typographical errors have been corrected.
A list of all parts in the ISO 12167 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
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ISO 12167-1:2023(E)
Introduction
Hydrostatic bearings use external lubrication to support pressure on the bearings; thus, they are
less prone to wear and tear, run quietly, and have a wide useable speed, as well as high stiffness and
damping capacity. These properties also demonstrate the special importance of plain journal bearings
in different fields of application such as in machine tools.
Basic calculations described in this document may be applied to bearings with different numbers of
recesses and different width/diameter ratios for identical recess geometry.
Oil is fed to each bearing recess by means of a common pump with a constant pumping pressure (system
p = constant) and through preceding linear restrictors, e.g. capillaries.
en
In this document, only bearings with oil drainage grooves between the recesses are taken into account.
As compared to bearings without oil drainage grooves, this type needs higher power with the same
stiffness behaviour.
Equivalent calculation procedures that enable operating conditions to be estimated and checked
against acceptable conditions may also be used.
v
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INTERNATIONAL STANDARD ISO 12167-1:2023(E)
Plain bearings — Hydrostatic plain journal bearings with
drainage grooves under steady-state conditions —
Part 1:
Calculation of oil-lubricated plain journal bearings with
drainage grooves
1 Scope
This document specifies the calculation procedures to calculate and assess a given bearing design,
as well as to design a bearing as a function of some optional parameters. This document contains the
design of the required lubrication system including the calculation of the restrictor data.
This document applies to hydrostatic plain journal bearings under steady-state conditions.
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 12167-2:2021, Plain bearings — Hydrostatic plain journal bearings with drainage grooves under
steady-state conditions — Part 2: Characteristic values for the calculation of oil-lubricated plain journal
bearings with drainage grooves
3 Terms and definitions
No terms and definitions are listed in this document.
ISO and IEC maintain terminology databases for use in standardization at the following addresses:
— ISO Online browsing platform: available at https:// www .iso .org/ obp
— IEC Electropedia: available at https:// www .electropedia .org/
4 Symbols
Symbols and units are defined in Table 1.
Table 1 — Symbols
Symbol Term Unit
a Inertia factor 1
2
A Land area m
lan
*
Relative land area 1
A
lan
2
A Recess area m
p
b Width perpendicular to the direction of flow m
1
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ISO 12167-1:2023(E)
TTabablele 1 1 ((ccoonnttiinnueuedd))
Symbol Term Unit
π×D
 
b Width of axial outlet b = −+()lb m
ax ax cG
 
 Z 
b Width of circumferential outlet ()bB=−l m
c cax
b Width of drainage groove m
G
B Bearing width m
c Stiffness coefficient N/m
c Specific heat capacity of the lubricant (p is a constant) J/kg·K
p
C Radial clearance m
R
d Diameter of capillaries m
cp
D Bearing diameter (D : shaft; D : bearing; D ≈ D ≈ D ) m
J B J B
e Eccentricity (shaft displacement) m
f Relative film thickness [ f = h/C ] 1
R
′
f Relative film thickness at ϕϕ= 1
1,i
en,i
′
Relative film thickness at ϕϕ=
f 1
ex,i 2,i
F Load-carrying capacity (load) N
F Radial force in recess i N
i
F Horizontal component of the resultant force of recesses N
h
F Vertical component of the resultant force of recesses N
v
F* Characteristic value of load-carrying capacity 1
*
Characteristic value of effective load-carrying capacity 1
F
eff
*
Characteristic value of effective load-carrying capacity for N = 0 1
F
eff,0
h Local lubricant film thickness (clearance gap height) m
h Oil film thickness at entrance edge of recess m
en
h Oil film thickness at exit edge of recess m
ex
h Minimum lubricant film thickness (minimum clearance gap height) m
min
h Depth of recess m
p
i (index) Consecutive number 1
k Exponent 1
K Speed-dependent parameter 1
rot
K Normalized speed-dependent parameter 1
rot,nom
l Length in the direction of flow m
l Axial land length m
ax
l Circumferential land length m
c
l Length of capillaries m
cp
-1
N Rotational frequency (speed) s
opt
optimum 1
(index)
p Recess pressure, general Pa
p Specific bearing load [ p = F/(B × D)]
Pa
p Feed pressure (pumping pressure) Pa
en
p Pressure in recess i Pa
i
p * Pressure ratio in recess i 1
i
p Pressure in recess i, when ε = 0 Pa
i, 0
2
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ISO 12167-1:2023(E)
TTabablele 1 1 ((ccoonnttiinnueuedd))
Symbol Term Unit
p Maximum pressure in recess Pa
max
p Minimum pressure in recess Pa
min
P* Power ratio 1
P Frictional power W
f
P Frictional power in land area W
f,lan
P Frictional power in recess area W
f,p
*
Characteristic value for the frictional power 1
P
f
*
Characteristic value for the frictional power in land area 1
P
fl, an
*
Characteristic value for the frictional power in recess area 1
P
fp,
P Pumping power W
p
P Total power W
tot
*
Characteristic value of total power 1
P
tot
3
Q Lubricant flow rate (for complete bearing) m /s
Q* Lubricant flow rate parameter 1
3
Q Lubricant flow rate from recess in the axial direction m /s
ax
3
Q Lubricant flow rate from capillary into recess i m /s
cp, i
3
Q Inlet lubricant flow rate into recess in the circumferential direction m /s
en
3
Q Outlet lubricant flow rate from recess in the circumferential direction m /s
ex
3
R Flow resistance of capillaries Pa⋅s/m
cp
 
12××η l
ax
3
R Flow resistance of one axial land R =  Pa⋅s/m
lan, ax lan,ax
 3 
bC×
 ax R 
 
12××η l
c
3
R Flow resistance of one circumferential land R =  Pa⋅s/m
lan, c lan,c
 3 
bC×
 
cR
3
R Flow resistance of one recess, when ε = 0 Pa⋅s/m
P, 0
Re Reynolds number 1
Re Reynolds number in capillaries 1
cp
Re Reynolds number in recess 1
p
So Sommerfeld number 1
T Temperature °C
T Mean temperature in the bearings; see Formula (15) °C
B
T Temperature in capillaries °C
cp
T Inlet temperature °C
en
u Flow velocity m/s
U Circumferential speed m/s
w
Average velocity in restrictor m/s
Z Number of recesses 1
Position of first recess related to recess centre measured from load direction; see
α rad
Figure A.3
β Attitude angle of shaft °
γ Exponent in viscosity formula 1
Δp Pressure difference in land area Pa
lan
ΔT Temperature difference K
3
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ISO 12167-1:2023(E)
TTabablele 1 1 ((ccoonnttiinnueuedd))
Symbol Term Unit
ΔT Mean temperature difference in the bearings K
B
ΔT Temperature difference in capillaries K
cp
ε Relative eccentricity 1
η Dynamic viscosity Pa⋅s
η Dynamic viscosity for TT= Pa⋅s
B B
η Dynamic viscosity in capillaries Pa⋅s
cp
η Dynamic viscosity at 40 °C Pa⋅s
40
κ Resistance ratio 1
ξ Restrictor ratio 1
π Relative frictional pressure 1
f
π Optimum relative frictional pressure 1
f,opt
3
ρ Density kg/m
2
τ Shearing stress N/m
φ Angular coordinate measured from radius opposite to eccentricity, e; see Figure A.3 rad
φ Angle of resultant force of recesses rad
F
φ Angle between the centre of land and the centre of drainage groove rad
G
ψ
Relative bearing clearance 1
ψ
Optimum relative bearing clearance 1
opt
-1
ω Angular velocity (ω = 2 × π × Ν) s
5 Bases of calculation and boundary conditions
Calculation in accordance with this document is the mathematical determination of the operational
parameters of hydrostatic plain journal bearings as a function of operating conditions, bearing
geometry and lubrication data. This means the determination of eccentricities, load-carrying capacity,
stiffness, required feed pressure, oil flow rate, frictional and pumping power, and temperature rise.
Besides the hydrostatic pressure build up, the influence of hydrodynamic effects is also approximated.
Reynolds equation furnishes the theoretical basis for the calculation of hydrostatic bearings. In
most practical cases of application, it is, however, possible to arrive at sufficiently exact results by
approximation.
The approximation used in this document is based on two basic formulae intended to describe the
flow through the bearing lands, which can be derived from Reynolds equation when special boundary
conditions are observed. The Hagen-Poiseuille law describes the pressure flow in a parallel clearance
gap and the Couette formula the drag flow in the bearing clearance gap caused by shaft rotation. A
detailed presentation of the theoretical background of the calculation procedure is included in Annex A.
An example of the calculation procedure is given in Annex B.
The following important premises are applicable to the calculation procedures described in this
document:
a) all lubricant flows in the lubrication clearance gap are laminar;
b) the lubricant adheres completely to the sliding surfaces;
c) the lubricant is an incompressible Newtonian fluid;
d) in the whole lubrication clearance gap, as well as in the preceding restrictors, the lubricant is
partially isoviscous;
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ISO 12167-1:2023(E)
e) a lubrication clearance gap completely filled with lubricant is the basis of frictional behaviour;
f) fluctuations of pressure in the lubricant film normal to the sliding surfaces do not take place;
g) bearing and journal have completely rigid surfaces;
h) the radii of curvature of the surfaces in relative motion to each other are large in comparison to the
lubricant film thickness;
i) the clearance gap height in the axial direction is constant (axial parallel clearance gap);
j) the pressure over the recess area is constant;
k) there is no motion normal to the sliding surfaces.
The bearing consists of Z cylindrical segments and rectangular recess of the same size and is
supplied with oil through restrictors of the same flow characteristics. Each segment consists of a
circumferential part between two centre lines of axial drainage grooves. With the aid of the above-
mentioned approximation formulae, all parameters required for the design or calculation of bearings
can be determined. The application of the similarity principle results in dimensionless similarity values
for load-carrying capacity, stiffness, oil flow rate, friction, recess pressures, etc.
The results indicated in this document in the form of tables and diagrams are restricted to operating
ranges common in practice for hydrostatic bearings. Thus, the range of the bearing eccentricity
(displacement under load) is limited to ε = 0 to 0,5.
Limitation to this eccentricity range means a considerable simplification of the calculation procedure
as the load-carrying capacity is a nearly linear function of the eccentricity. However, the applicability
of this procedure is hardly restricted as in practice eccentricities ε > 0,5 are mostly undesirable for
reasons of operational safety. A further assumption for the calculations is the approximated optimum
[2]
restrictor ratio ξ = 1 for the stiffness behaviour.
As for the outside dimensions of the bearing, this document is restricted to the range bearing width/
bearing diameter B/D = 0,3 to 1, which is common in practical cases of application. The recess depth is
larger than the clearance gap height by a factor of 10 to 100. When calculating the friction losses, the
friction loss over the recess in relation to the friction over the bearing lands can generally be neglected
on account of the above premises. However, this does not apply when the bearing shall be optimized
with regard to its total power losses.
To take into account the load direction of a bearing, it is necessary to distinguish between the two
extreme cases, load in the direction of recess centre and load in the direction of land centre.
Apart from the aforementioned boundary conditions, some other requirements are to be mentioned for
the design of hydrostatic bearings in order to ensure their functioning under all operating conditions.
In general, a bearing shall be designed in such a manner that a clearance gap height of at least 50 % to
60 % of the initial clearance gap height is ensured when the maximum possible load is applied. With
this in mind, particular attention shall be paid to misalignments of the shaft in the bearing due to
shaft deflection which may result in contact between shaft and bearing edge and thus in damage of the
bearing. In addition, the parallel clearance gap required for the calculation is no longer present in such
a case.
In the case where the shaft is in contact with the bearing lands when the hydrostatic pressure is
switched off, it can be necessary to check the contact zones with regard to rising surface pressures.
It shall be ensured that the heat originating in the bearing does not lead to a non-permissible rise in the
temperature of the oil.
If necessary, a means of cooling the oil shall be provided. Furthermore, the oil shall be filtered in order
to avoid choking of the capillaries and damage to the sliding surfaces.
Low pressure in the relieved recess shall also be avoided, as this leads to air being drawn in from the
environment and this would lead to a decrease in stiffness (see 6.7).
5
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ISO 12167-1:2023(E)
6 Method of calculation
6.1 General
This document covers the calculation, as well as the design, of hydrostatic plain journal bearings. In
this case, calculation is understood to be the verification of the operational parameters of a hydrostatic
bearing with known geometrical and lubrication data. In the case of a design calculation, with the given
methods of calculation, it is possible to determine the missing data for the required bearing geometry,
the lubrication data and the operational parameters on the basis of a few initial data (e.g. required load-
carrying capacity, stiffness, rotational frequency).
In both cases, the calculations are carried out according to an approximation method based on the
Hagen-Poiseuille and the Couette formulae, mentioned in Clause 5, see also A.2.2 and A.2.3. The bearing
parameters calculated according to this method are given as relative values in the form of tables and
diagrams as a function of different parameters. The procedure for the calculation or design of bearings
is described in 6.2 to 6.7. This includes the determination of different bearing parameters with the
aid of the given calculation formulae or the tables and diagrams. The following calculation items are
explained in detail:
a) determination of load-carrying capacity with and without taking into account shaft rotation;
b) calculation of lubricant flow rate and pumping power;
c) determination of frictional power with and without consideration of losses in the bearing recesses;
d) procedure for bearing optimization with regard to minimum total power loss.
For all calculations, it is necessary to check whether the important premise of laminar flow in the
bearing clearance gap, in the bearing recess and in the capillary is met. This is checked by determining
the Reynolds numbers. Furthermore, the portion of the inertia factor in the pressure differences shall
be kept low at the capillary (see A.3.1).
If the boundary conditions defined in Clause 5 are observed, this method of calculation yields results
with deviations which can be neglected for the requirements of practice, in comparison with an exact
calculation by solving the Reynolds equation.
6.2 Load-carrying capacity
Unless indicated otherwise, it is assumed in the following that capillaries with a linear characteristic
are used as restrictors and that the restrictor ratio is ξ = 1. Furthermore, the difference is only made
between the two cases, “load in direction of recess centre” and “load in direction of land centre”. For this
reason, it is no longer mentioned in each individual case that the characteristic values are a function of
the three parameters, “restrictor type”, “restrictor ratio” and “load direction relative to the bearing”.
Even under the abovementioned premises, the characteristic value of load carrying capacity
[Formula (1)]
F p
*
F = = (1)
BD××pp
en en
still depends on the following parameters:
— number of recesses, Z;
— width/diameter ratio, B/D;
— relative axial land width, l /B;
ax
— relative land width in circumferential direction, l /D;
c
— relative groove width, b /D;
G
6
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ISO 12167-1:2023(E)
— relative journal eccentricity, ε;
— relative frictional pressure when the difference is only made between the two cases, “load on recess
centre” and “load on land centre”:
ηω×
B
π = (2)
f
2
p ×ψ
en
NOTE The Sommerfeld number, So, common with hydrodynamic plain journal bearings can be set up as
follows:
2 *
p×ψ
F
So= =
ηω× π
Bf
In ISO 12167-2:2021, Figures 1 and 2, the functions F*(ε, π ) and β(ε, π ) are represented for Z = 4, ξ = 1,
f f
B/D = 1, l /B = 0,1, l /D = 0,1, b /D = 0,05, i.e. restriction by means of capillaries, load in direction of
ax c G
centre of bearing recess.
ISO 12167-2:2021, Figures 1 and 2 show the influence of rotation on the characteristic value of load-
carrying capacity and the attitude angle.
For the calculation of a geometrically similar bearing, it is possible to determine the minimum lubricant
film thickness when values are given, e.g. for F, B, D, p , ω, ψ , and η (determination of η according to
en B B
6.6, if applicable).
All parameters are given for the determination of F* according to Formula (1) and π according to
f
Formula (2). For this geometry, the relevant values for ε and β can be taken from ISO 12167-2:2021,
Figures 1 and 2 and thus, h = C (1 − ε).
min R
According to the approximation method described in Annex A, it transpires that the characteristic
value of effective load-carrying capacity is no longer a function of the ratio B/D.
*
F F
*
F = = (3)
eff
b Zb×
bZ××bP×
cax
caxen
×
D B
If the resistance ratio
R
lb×
lan,ax
ax c
k== (4)
R lb×
lan,c cax
and the speed dependent parameter
ξπ××kl×
fc
K =
rot
D
K
rot
K = (5)
rot,nom
1+k
are introduced, there remains a dependence on the following parameters:
*
FZ(),,ϕεkK,,
effG rot
*
If, in addition, advantage is taken of the fact that the function F ()ε is nearly linear for ε ≤ 0,5, then it
eff
*
is practically sufficient to know that the function Ff()εϕ=04,,= ()Zk,,K for the calculation of
effG rot
the load carrying capacity.
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ISO 12167-1:2023(E)
For K = 0, i.e. for the stationary shaft, the characteristic value of effective load-carrying capacity for
rot
ε = 0,4 only depends on three parameters:
*
Ff()εϕ=04,,= ()Zk,
effG
*
Thus, in ISO 12167-2:2021, Figure 3, F ε =04, for Z = 4 and 6 can be given via κ for different φ
()
eff,0 G
values.
The influence of the rotational movement on the characteristic value of load-carrying capacity is taken
*
F
eff
into account by the ratio = fZ(),,ϕ kK, .
Grot
*
F
eff,0
**
For Z = 4, the ratio FF/ is shown in ISO 12167-2:2021, Figure 4. The hydrodynamically conditioned
effeff,0
increase of the load-carrying capacity can be easily recognized when presented in such a manner.
*
If, for example, Z and all parameters are given for the determination of F according to Formula (3), κ
eff
according to Formula (4) and K according to Formula (5), then the minimum lubricant film thickness
rot
developing during operation can be determined.
*
After having calculated φ , κ and K , the value for F ()ε =04, is taken from ISO 12167-2:2021,
G rot, nom eff,0
** *
Figure 3 and the value for FF/ ()ε =04, from ISO 12167-2:2021, Figure 4, F is calculated
effeff,0 eff
according to Formula (3) and then the relative eccentricity is obtained as follows:
*
04, × F
eff
ε =
*
F
eff *
()εε=04,,×=F ()04
eff,0
*
F
eff,0
and the minimum lubricant film thickness is h = C × (1 – ε).
min R
6.3 Lubricant flow rate and pumping power
The characteristic value for the lubricant flow rate is given by
Q×η
* B
Q = (6)
3
Cp×
Ren
It depends only slightly on the relative journal eccentricity ε, the load direction relative to the bearing
and the relative frictional pressure π , or the speed dependent parameter K .
f rot
By approximation, the lubricant flow rate can be calculated as follows (see also A.3.4):
l
ax
1−
ZB k+1
** B
QQ()εε≤0,5 ≈=()0 = ×× × (7)
l
61()+ξ D k
c
D
8
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ISO 12167-1:2023(E)
R
P 6××η l
1 cp
P Bax
==()ε 0 with ξ = and R =
P,0
3
1+ξ p R
en P,0 bC×+()1 k
ax R
The flow resistance of the capillaries according to A.3.2.2 is given by
128××η l
cp cp
R ×+1 a
()
cp=
4
π×d
cp
with the non-linear portion (inertia factor)
10, 8 4××Q ρ
a=×
32 η ××lZπ×
cp cp
By converting Formula (6), the lubricant flow rate can be calculated when the parameters η , C , p , ξ,
B R en
B/D, and l /B are given.
ax
For optimized bearings, Q* shall be taken from ISO 12167-2:2021, Table 1. The pumping power, without
considering the pump efficiency, is given by
23
pC×
* en R
PQ=×pQ=× (8)
pen
η
B
According to the approximation method, Q* is again determined according to Formula (7), thus it is the
characteristic value
...

DRAFT INTERNATIONAL STANDARD
ISO/DIS 12167-1
ISO/TC 123/SC 8 Secretariat: JISC
Voting begins on: Voting terminates on:
2022-09-14 2022-12-07
Plain bearings — Hydrostatic plain journal bearings with
drainage grooves under steady-state conditions —
Part 1:
Calculation of oil-lubricated plain journal bearings with
drainage grooves
Paliers lisses — Paliers lisses radiaux hydrostatiques avec rainures d'écoulement fonctionnant en régime
stationnaire —
Partie 1: Calcul pour la lubrification des paliers lisses radiaux avec rainures d'écoulement
ICS: 21.100.10
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ISO/DIS 12167-1:2022(E)
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ISO/DIS 12167-1:2022(E)
DRAFT INTERNATIONAL STANDARD
ISO/DIS 12167-1
ISO/TC 123/SC 8 Secretariat: JISC
Voting begins on: Voting terminates on:

Plain bearings — Hydrostatic plain journal bearings with
drainage grooves under steady-state conditions —
Part 1:
Calculation of oil-lubricated plain journal bearings with
drainage grooves
Paliers lisses — Paliers lisses radiaux hydrostatiques avec rainures d'écoulement fonctionnant en régime
stationnaire —
Partie 1: Calcul pour la lubrification des paliers lisses radiaux avec rainures d'écoulement
ICS: 21.100.10
COPYRIGHT PROTECTED DOCUMENT
THIS DOCUMENT IS A DRAFT CIRCULATED
FOR COMMENT AND APPROVAL. IT IS
© ISO 2022
THEREFORE SUBJECT TO CHANGE AND MAY
This document is circulated as received from the committee secretariat.
All rights reserved. Unless otherwise specified, or required in the context of its implementation, no part of this publication may
NOT BE REFERRED TO AS AN INTERNATIONAL
be reproduced or utilized otherwise in any form or by any means, electronic or mechanical, including photocopying, or posting on STANDARD UNTIL PUBLISHED AS SUCH.
the internet or an intranet, without prior written permission. Permission can be requested from either ISO at the address below
IN ADDITION TO THEIR EVALUATION AS
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TECHNOLOGICAL, COMMERCIAL AND
ISO copyright office
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CP 401 • Ch. de Blandonnet 8
STANDARDS MAY ON OCCASION HAVE TO
BE CONSIDERED IN THE LIGHT OF THEIR
CH-1214 Vernier, Geneva
POTENTIAL TO BECOME STANDARDS TO
Phone: [phone]
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NATIONAL REGULATIONS.
Website: www.iso.org ISO/DIS 12167-1:2022(E)
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ii
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PROVIDE SUPPORTING DOCUMENTATION. © ISO 2022

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ISO/DIS 12167-1:2022(E)
Contents Page
Foreword .iv
Introduction .v
1 Scope . 1
2 Normative references . 1
3 Terms and definitions . 1
4 Symbols, terms and units .1
5 Bases of calculation and boundary conditions . 4
6 Method of calculation .6
6.1 General . 6
6.2 Load-carrying capacity . 6
6.3 Lubricant flow rate and pumping power . 8
6.4 Frictional power . 9
6.5 Optimization . 10
6.6 Temperatures and viscosities . 11
6.7 Minimum pressure in recesses .12
Annex A (normative) Description of the approximation method for the calculation
of hydrostatic plain journal bearings .13
Annex B (informative) Example of calculation according to the method given in Annex A .23
Bibliography .31
iii
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ISO/DIS 12167-1:2022(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.
The committee responsible for this document is 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 12167-1:2016), which has been
technically revised.
The main changes are as follows:
— subclause titles have been added;
— symbols have been corrected and added in Table 1;
— calculation values in Annex B have been corrected;
— adjustments have been made to ISO/IEC Directives, Part 2:2021;
— typographical errors have been corrected.
A list of all parts in the ISO 12167 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
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ISO/DIS 12167-1:2022(E)
Introduction
Hydrostatic bearings use external lubrication to support pressure on the bearings; thus, are less prone
to wear and tear, run quietly, and have wide useable speed, as well as high stiffness and damping
capacity. These properties also demonstrate the special importance of plain journal bearings in
different fields of application such as in machine tools.
Basic calculations described in this document may be applied to bearings with different numbers of
recesses and different width/diameter ratios for identical recess geometry.
Oil is fed to each bearing recess by means of a common pump with constant pumping pressure (system
p = constant) and through preceding linear restrictors, e.g. capillaries.
en
The calculation procedures listed in this document enable the user to calculate and assess a given
bearing design, as well as to design a bearing as a function of some optional parameters. Furthermore,
this document contains the design of the required lubrication system including the calculation of the
restrictor data.
v
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DRAFT INTERNATIONAL STANDARD ISO/DIS 12167-1:2022(E)
Plain bearings — Hydrostatic plain journal bearings with
drainage grooves under steady-state conditions —
Part 1:
Calculation of oil-lubricated plain journal bearings with
drainage grooves
1 Scope
This document applies to hydrostatic plain journal bearings under steady-state conditions.
In this document, only bearings with oil drainage grooves between the recesses are taken into account.
As compared to bearings without oil drainage grooves, this type needs higher power with the same
stiffness behaviour.
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, in whole or in part, are normatively referenced in this document and are
indispensable for its application. For dated references, only the edition cited applies. For undated
references, the latest edition of the referenced document (including any amendments) applies.
ISO 3448, Industrial liquid lubricants — ISO viscosity classification
ISO 12167-2:2021, Plain bearings — Hydrostatic plain journal bearings with drainage grooves under
steady-state conditions — Part 2: Characteristic values for the calculation of oil-lubricated plain journal
bearings with drainage grooves
3 Terms and definitions
No terms and definitions are listed in this document.
ISO and IEC maintain terminology databases for use in standardization at the following addresses:
— ISO Online browsing platform: available at https:// www .iso .org/ obp
— IEC Electropedia: available at https:// www .electropedia .org/
4 Symbols, terms and units
Symbols and units are defined in Table 1.
Table 1 — Symbols, terms and units
Symbol Term Unit
a Inertia factor 1
2
A Land area m
lan
*
Relative land area 1
A
lan
1
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ISO/DIS 12167-1:2022(E)
Table 1 (continued)
Symbol Term Unit
2
A Recess area m
p
b Width perpendicular to the direction of flow m
π ×D
 
b Width of axial outlet b = −+lb m
()
ax ax cG
 
Z
 
b Width of circumferential outlet ()bB=−l m
c cax
b Width of drainage groove m
G
B Bearing width m
c Stiffness coefficient N/m
c Specific heat capacity of the lubricant (p = constant) J/kg·K
p
C Radial clearance m
R
d Diameter of capillaries m
cp
D Bearing diameter (D : shaft; D : bearing; D ≈ D ≈ D ) m
J B J B
e Eccentricity (shaft displacement) m
f Relative film thickness [ f = h/C ] 1
R
′
f Relative film thickness at ϕϕ= 1
en,i 1,i
Relative film thickness at ϕϕ= ′
f 1
2,i
ex,i
F Load-carrying capacity (load) N
F Radial force in recess i N
i
F Horizontal component of the resultant force of recesses N
h
F Vertical component of the resultant force of recesses N
v
F* Characteristic value of load-carrying capacity 1
*
Characteristic value of effective load-carrying capacity 1
F
eff
*
Characteristic value of effective load-carrying capacity for N = 0 1
F
eff,0
h Local lubricant film thickness (clearance gap height) m
h Oil film thickness at entrance edge of recess m
en
h Oil film thickness at exit edge of recess m
ex
h Minimum lubricant film thickness (minimum clearance gap height) m
min
h Depth of recess m
p
i (index) Consecutive number 1
k Exponent 1
K Speed-dependent parameter 1
rot
K Normalized speed-dependent parameter 1
rot,nom
l Length in the direction of flow m
l Axial land length m
ax
l Circumferential land length m
c
l Length of capillaries m
cp
-1
N Rotational frequency (speed) s
opt
optimum 1
(index)
p Recess pressure, general Pa
p Specific bearing load [ p = F/(B × D)]
Pa
p Feed pressure (pumping pressure) Pa
en
p Pressure in recess i Pa
i
2
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ISO/DIS 12167-1:2022(E)
Table 1 (continued)
Symbol Term Unit
P * Pressure ratio 1
i
p Pressure in recess i, when ε = 0 Pa
i, 0
P* Power ratio 1
P Frictional power W
f
p Maximum pressure in recess Pa
max
p Minimum pressure in recess Pa
min
P Frictional power in land area W
f,lan
P Frictional power in recess area W
f,p
*
Characteristic value for the frictional power 1
P
f
*
Characteristic value for the frictional power in land area 1
P
fl, an
*
P Characteristic value for the frictional power in recess area 1
fp,
P Pumping power W
p
P Total power W
tot
*
Characteristic value of total power 1
P
tot
3
Q Lubricant flow rate (for complete bearing) m /s
Q* Lubricant flow rate parameter 1
3
Q Lubricant flow rate from recess in the axial direction m /s
ax
3
Q Lubricant flow rate from capillary into recess i m /s
cp, i
3
Q Inlet lubricant flow rate into recess in the circumferential direction m /s
en
3
Q Outlet lubricant flow rate from recess in the circumferential direction m /s
ex
3
R Flow resistance of capillaries Pa⋅s/m
cp
 
12××η l
ax
3
R Flow resistance of one axial land R =  Pa⋅s/m
lan, ax lan,ax
 3 
bC×
 
ax R
 
12××η l
c
3
R Flow resistance of one circumferential land R = Pa⋅s/m
 
lan, c lan,c
 3 
bC×
 cR 
3
R Flow resistance of one recess, when ε = 0 Pa⋅s/m
P, 0
Re Reynolds number 1
Re Reynolds number in capillaries 1
cp
Re Reynolds number in recess 1
p
So Sommerfeld number 1
T Temperature °C
T Mean temperature in the bearings; see Formula (15) °C
B
T Temperature in capillaries °C
cp
T Inlet temperature °C
en
u Flow velocity m/s
U Circumferential speed m/s
w Average velocity in restrictor m/s
Z Number of recesses 1
Position of first recess related to recess centre measured from load direction;
α rad
see Figure A.3
β Attitude angle of shaft °
γ Exponent in viscosity formula 1
3
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ISO/DIS 12167-1:2022(E)
Table 1 (continued)
Symbol Term Unit
Δp Pressure difference in land area Pa
lan
ΔT Temperature difference K
ΔT Mean temperature difference in the bearings K
B
ΔT Temperature difference in capillaries K
cp
ε Relative eccentricity 1
η Dynamic viscosity Pa⋅s
η Dynamic viscosity for TT= Pa⋅s
B B
η Dynamic viscosity in capillaries Pa⋅s
cp
η Dynamic viscosity at 40 °C Pa⋅s
40
κ Resistance ratio 1
ξ Restrictor ratio 1
π Relative frictional pressure 1
f
π Optimum relative frictional pressure 1
f,opt
3
ρ Density kg/m
2
τ Shearing stress N/m
Angular coordinate measured from radius opposite to eccentricity, e;
φ rad
see Figure A.3
φ Angle of resultant force of recesses rad
F
φ Angle between the center of land and the center of drainage groove rad
G
ψ Relative bearing clearance 1
ψ Optimum relative bearing clearance 1
opt
-1
ω Angular velocity (ω = 2 × π × Ν) s
5 Bases of calculation and boundary conditions
Calculation in accordance with this document is the mathematical determination of the operational
parameters of hydrostatic plain journal bearings as a function of operating conditions, bearing
geometry and lubrication data. This means the determination of eccentricities, load-carrying capacity,
stiffness, required feed pressure, oil flow rate, frictional and pumping power, and temperature rise.
Besides the hydrostatic pressure build up, the influence of hydrodynamic effects is also approximated.
Reynolds equation furnishes the theoretical basis for the calculation of hydrostatic bearings. In
most practical cases of application, it is, however, possible to arrive at sufficiently exact results by
approximation.
The approximation used in this document is based on two basic formulae intended to describe the
flow through the bearing lands, which can be derived from Reynolds equation when special boundary
conditions are observed. The Hagen-Poiseuille law describes the pressure flow in a parallel clearance
gap and the Couette formula the drag flow in the bearing clearance gap caused by shaft rotation. A
detailed presentation of the theoretical background of the calculation procedure is included in Annex A.
The following important premises are applicable to the calculation procedures described in this
document:
a) all lubricant flows in the lubrication clearance gap are laminar;
b) the lubricant adheres completely to the sliding surfaces;
c) the lubricant is an incompressible Newtonian fluid;
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ISO/DIS 12167-1:2022(E)
d) in the whole lubrication clearance gap, as well as in the preceding restrictors, the lubricant is
partially isoviscous;
e) a lubrication clearance gap completely filled with lubricant is the basis of frictional behaviour;
f) fluctuations of pressure in the lubricant film normal to the sliding surfaces do not take place;
g) bearing and journal have completely rigid surfaces;
h) the radii of curvature of the surfaces in relative motion to each other are large in comparison to the
lubricant film thickness;
i) the clearance gap height in the axial direction is constant (axial parallel clearance gap);
j) the pressure over the recess area is constant;
k) there is no motion normal to the sliding surfaces.
The bearing consists of Z cylindrical segments and rectangular recess of the same size and is
supplied with oil through restrictors of the same flow characteristics. Each segment consists of a
circumferential part between two centre lines of axial drainage grooves. With the aid of the above-
mentioned approximation formulae, all parameters required for the design or calculation of bearings
can be determined. The application of the similarity principle results in dimensionless similarity values
for load-carrying capacity, stiffness, oil flow rate, friction, recess pressures, etc.
The results indicated in this document in the form of tables and diagrams are restricted to operating
ranges common in practice for hydrostatic bearings. Thus, the range of the bearing eccentricity
(displacement under load) is limited to ε = 0 to 0,5.
Limitation to this eccentricity range means a considerable simplification of the calculation procedure
as the load-carrying capacity is a nearly linear function of the eccentricity. However, the applicability
of this procedure is hardly restricted as in practice eccentricities ε > 0,5 are mostly undesirable for
reasons of operational safety. A further assumption for the calculations is the approximated optimum
[1]
restrictor ratio ξ = 1 for the stiffness behaviour.
As for the outside dimensions of the bearing, this document is restricted to the range bearing width/
bearing diameter B/D = 0,3 to 1, which is common in practical cases of application. The recess depth is
larger than the clearance gap height by a factor of 10 to 100. When calculating the friction losses, the
friction loss over the recess in relation to the friction over the bearing lands can generally be neglected
on account of the above premises. However, this does not apply when the bearing shall be optimized
with regard to its total power losses.
To take into account the load direction of a bearing, it is necessary to distinguish between the two
extreme cases, load in the direction of recess centre and load in the direction of land centre.
Apart from the aforementioned boundary conditions, some other requirements are to be mentioned for
the design of hydrostatic bearings in order to ensure their functioning under all operating conditions.
In general, a bearing shall be designed in such a manner that a clearance gap height of at least 50 % to
60 % of the initial clearance gap height is ensured when the maximum possible load is applied. With
this in mind, particular attention shall be paid to misalignments of the shaft in the bearing due to
shaft deflection which may result in contact between shaft and bearing edge and thus in damage of the
bearing. In addition, the parallel clearance gap required for the calculation is no longer present in such
a case.
In the case where the shaft is in contact with the bearing lands when the hydrostatic pressure is
switched off, it might be necessary to check the contact zones with regard to rising surface pressures.
It shall be ensured that the heat originating in the bearing does not lead to a non-permissible rise in the
temperature of the oil.
If necessary, a means of cooling the oil shall be provided. Furthermore, the oil shall be filtered in order
to avoid choking of the capillaries and damage to the sliding surfaces.
5
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ISO/DIS 12167-1:2022(E)
Low pressure in the relieved recess shall also be avoided, as this leads to air being drawn in from the
environment and this would lead to a decrease in stiffness (see 6.7).
6 Method of calculation
6.1 General
This document covers the calculation, as well as the design, of hydrostatic plain journal bearings. In
this case, calculation is understood to be the verification of the operational parameters of a hydrostatic
bearing with known geometrical and lubrication data. In the case of a design calculation, with the given
methods of calculation, it is possible to determine the missing data for the required bearing geometry,
the lubrication data and the operational parameters on the basis of a few initial data (e.g. required load-
carrying capacity, stiffness, rotational frequency).
In both cases, the calculations are carried out according to an approximation method based on the
Hagen-Poiseuille and the Couette formulae, mentioned in Clause 5, see also A.2.2 and A.2.3. The bearing
parameters calculated according to this method are given as relative values in the form of tables and
diagrams as a function of different parameters. The procedure for the calculation or design of bearings
is described in 6.2 to 6.7. This includes the determination of different bearing parameters with the
aid of the given calculation formulae or the tables and diagrams. The following calculation items are
explained in detail:
a) determination of load-carrying capacity with and without taking into account shaft rotation;
b) calculation of lubricant flow rate and pumping power;
c) determination of frictional power with and without consideration of losses in the bearing recesses;
d) procedure for bearing optimization with regard to minimum total power loss.
For all calculations, it is necessary to check whether the important premise of laminar flow in the
bearing clearance gap, in the bearing recess and in the capillary is met. This is checked by determining
the Reynolds numbers. Furthermore, the portion of the inertia factor in the pressure differences shall
be kept low at the capillary (see A.3.1).
If the boundary conditions defined in Clause 5 are observed, this method of calculation yields results
with deviations which can be neglected for the requirements of practice, in comparison with an exact
calculation by solving the Reynolds equation.
6.2 Load-carrying capacity
Unless indicated otherwise, it is assumed in the following that capillaries with a linear characteristic
are used as restrictors and that the restrictor ratio is ξ = 1. Furthermore, the difference is only made
between the two cases, “load in direction of recess centre” and “load in direction of land centre”. For this
reason, it is no longer mentioned in each individual case that the characteristic values are a function of
the three parameters, “restrictor type”, “restrictor ratio” and “load direction relative to the bearing”.
Even under the abovementioned premises, the characteristic value of load carrying capacity
[Formula (1)]
F p
*
F = = (1)
BD××pp
en en
still depends on the following parameters:
— number of recesses, Z;
— width/diameter ratio, B/D;
— relative axial land width, l /B;
ax
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ISO/DIS 12167-1:2022(E)
— relative land width in circumferential direction, l /D;
c
— relative groove width, b /D;
G
— relative journal eccentricity, ε;
— relative frictional pressure when the difference is only made between the two cases, “load on recess
centre” and “load on land centre”:
ηω×
B
π = (2)
f
2
p ×ψ
en
NOTE The Sommerfeld number, So, common with hydrodynamic plain journal bearings can be set up as
follows:
2 *
p×ψ F
So= =
ηω× π
Bf
In ISO 12167-2:2021, Figures 1 and 2, the functions F*(ε, π ) and β(ε, π ) are represented for Z = 4, ξ = 1,
f f
B/D = 1, l /B = 0,1, l /D = 0,1, b /D = 0,05, i.e. restriction by means of capillaries, load in direction of
ax c G
centre of bearing recess.
These figures show the influence of rotation on the characteristic value of load-carrying capacity and
the attitude angle.
For the calculation of a geometrically similar bearing, it is possible to determine the minimum lubricant
film thickness when values are given, e.g. for F, B, D, p , ω, ψ, and η (determination of η according to
en B B
6.6, if applicable).
All parameters are given for the determination of F* according to Formula (1) and π according to
f
Formula (2). For this geometry, the relevant values for ε and β can be taken from ISO 12167-2:2021,
Figures 1 and 2 and thus, h = C (1 − ε).
min R
According to the approximation method described in Annex A, it transpires that the characteristic
value of effective load-carrying capacity is no longer a function of the ratio B/D.
*
F F
*
F = = (3)
eff
b Zb×
bZ××bP×
cax
caxen
×
D B
If the resistance ratio
R
lb×
lan,ax
ax c
κ == (4)
R lb×
lan,c cax
and the speed dependent parameter
ξκ××π ×l
fc
K =
rot
D
K
rot
K = (5)
rot,nom
1+κ
7
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ISO/DIS 12167-1:2022(E)
are introduced, there remains a dependence on the following parameters:
*
FZ(),,ϕκ,,K ε
effG rot
*
If, in addition, advantage is taken of the fact that the function F ()ε is nearly linear for ε ≤ 0,5, then it
eff
*
is practically sufficient to know that the function Ffεϕ=04,,= ZK,,κ for the calculation of
() ()
effG rot
the load carrying capacity.
For K = 0, i.e. for the stationary shaft, the characteristic value of effective load-carrying capacity for
rot
ε = 0,4 only depends on three parameters:
*
Ff()εϕ=04,,= ()Z ,κ
effG
*
Thus, in ISO 12167-2:2021, Figure 3, F ε =04, for Z = 4 and 6 can be given via κ for different φ
()
G
eff,0
values.
The influence of the rotational movement on the characteristic value of load-carrying capacity is taken
*
F
eff
into account by the ratio = fZ(),,ϕκ,K .
Grot
*
F
eff,0
**
For Z = 4, the ratio FF/ is shown in ISO 12167-2:2021, Figure 4. The hydrodynamically conditioned
effeff,0
increase of the load-carrying capacity can be easily recognized when presented in such a manner.
*
If, e.g. Z and all parameters are given for the determination of F according to Formula
...