ISO 12168-1:2019
(Main)Plain bearings — Hydrostatic plain journal bearings without drainage grooves under steady-state conditions — Part 1: Calculation of oil-lubricated plain journal bearings without drainage grooves
Plain bearings — Hydrostatic plain journal bearings without drainage grooves under steady-state conditions — Part 1: Calculation of oil-lubricated plain journal bearings without drainage grooves
This document specifies a calculation method of oil-lubricated plain journal bearings without drainage grooves under steady-state conditions. It applies to hydrostatic plain journal bearings under steady-state conditions. In this document, only bearings without oil drainage grooves between the recesses are taken into account.
Paliers lisses — Paliers lisses radiaux hydrostatiques sans rainure d'écoulement fonctionnant en régime stationnaire — Partie 1: Calcul pour la lubrification des paliers lisses radiaux sans rainure d'écoulement
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Standards Content (Sample)
INTERNATIONAL ISO
STANDARD 12168-1
Second edition
2019-11
Plain bearings — Hydrostatic
plain journal bearings without
drainage grooves under steady-state
conditions —
Part 1:
Calculation of oil-lubricated plain
journal bearings without drainage
grooves
Paliers lisses — Paliers lisses radiaux hydrostatiques sans rainure
d'écoulement fonctionnant en régime stationnaire —
Partie 1: Calcul pour la lubrification des paliers lisses radiaux sans
rainure d'écoulement
Reference number
ISO 12168-1:2019(E)
©
ISO 2019
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ISO 12168-1:2019(E)
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ii © ISO 2019 – All rights reserved
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ISO 12168-1:2019(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 . 3
6 Method of calculation . 5
6.1 General . 5
6.2 Load-carrying capacity . 5
6.3 Lubricant flow rate and pumping power . 7
6.4 Frictional power. 9
6.5 Optimization . 9
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) Examples of calculation .23
Bibliography .31
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ISO 12168-1:2019(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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expressions related to conformity assessment, as well as information about ISO's adherence to the
World Trade Organization (WTO) principles in the Technical Barriers to Trade (TBT), see www .iso .org/
iso/ foreword .html.
This document was prepared by Technical Committee ISO/TC 123, Plain bearings, Subcommittee SC 8,
Calculation methods for plain bearings and their applications.
This second edition cancels and replaces the first edition (ISO 12168-1:2001), of which it constitutes a
minor revision.
The changes compared to the previous edition are as follows:
— adjustment to ISO/IEC Directives, Part 2:2018;
— correction of typographical errors.
A list of all parts in the ISO 12168 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 2019 – All rights reserved
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ISO 12168-1:2019(E)
Introduction
The functioning of hydrostatic bearings is characterized by the fact that the supporting pressure of the
bearing is generated by external lubrication. The special advantages of hydrostatic bearings are lack
of wear, quiet running, wide useable speed range as well as high stiffness and damping capacity. These
properties are also the reason for the special importance of hydrostatic bearing units in different fields
of application such as machine tools.
The bases of calculation described in this document apply to bearings with different numbers of
recesses and different width/diameter ratios for identical recess geometry. In this document, only
bearings without oil drainage grooves between the recesses are taken into account. As compared to
bearings with oil drainage grooves, this type needs less power with the same stiffness behaviour.
The oil is fed to each bearing recess by means of a common pump with constant pump pressure
(system p = constant) and via preceding linear restrictors (e.g. in the form of 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.
© ISO 2019 – All rights reserved v
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INTERNATIONAL STANDARD ISO 12168-1:2019(E)
Plain bearings — Hydrostatic plain journal bearings
without drainage grooves under steady-state conditions —
Part 1:
Calculation of oil-lubricated plain journal bearings without
drainage grooves
1 Scope
This document specifies a calculation method of oil-lubricated plain journal bearings without drainage
grooves under steady-state conditions.
It applies to hydrostatic plain journal bearings under steady-state conditions.
In this document, only bearings without oil drainage grooves between the recesses are taken into
account.
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 12168-2:2019, Plain bearings — Hydrostatic plain journal bearings without drainage grooves under
steady-state conditions — Part 2: Characteristic values for the calculation of oil-lubricated plain journal
bearings without drainage grooves
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, terms and units
Symbols, terms and units are shown in Table 1.
Table 1 — Symbols, terms and units
Symbol Term Unit
a Inertia factor 1
2
A Land area m
lan
A
lan
* *
1
A Relative land area A =
lan lan
π ××BD
2
A Recess area m
p
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ISO 12168-1:2019(E)
Table 1 (continued)
Symbol Term Unit
b Width perpendicular to the direction of flow m
π×D
b Width of axial outlet b = m
ax
ax
Z
b Width of circumferential outlet ()bB= −l m
c
cax
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 CD=−D / 2 m
()
R
RB J
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 Load-carrying capacity (load) N
* *
F Characteristic value of load-carrying capacity [F = F/(B × D × p )] 1
en
*
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 Minimum lubricant film thickness (minimum clearance gap height) m
min
h Depth of recess m
p
K Speed-dependent parameter 1
rot
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
p Recess pressure, general Pa
p []() Pa
Specific bearing load pF=×BD
p Feed pressure (pump pressure) Pa
en
p Pressure in recess i Pa
i
p Pressure in recess i, when ε = 0 Pa
i,0
* *
P Power ratio (P = P /P ) 1
f p
P Frictional power W
f
P Pumping power W
p
P Total power (P = P + P ) W
tot tot p f
*
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
R Flow resistance of capillaries Pa⋅s/m
cp
12××η l
ax
3
Flow resistance of one axial land R =
R Pa⋅s/m
lan,ax
lan,ax
3
b ×C
ax R
12××η l
c
3
R Flow resistance of one circumferential land R = Pa⋅s/m
lan,c
lan,c
3
b ×
C
R
c
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ISO 12168-1:2019(E)
Table 1 (continued)
Symbol Term Unit
3
R Flow resistance of one recess, when ε = 0, RR=05, Pa⋅s/m
()
P,0
P,0lan,ax
Re Reynolds number 1
So Sommerfeld number 1
T Temperature °C
ΔT Temperature difference K
u Flow velocity m/s
U Circumferential speed m/s
w Average velocity in restrictor m/s
Z Number of recesses 1
α Position of 1st recess related to recess centre rad
β Attitude angle of shaft °
γ Exponent in viscosity formula 1
ε Relative eccentricity (ε = e/C ) 1
R
η Dynamic viscosity Pa⋅s
R
lb×
lan,ax
ax c
κ Resistance ratio κ = = 1
R lb×
lan,c cax
R
cp
ξ Restrictor ratio ξ = 1
R
P,0
ηω×
B
π Relative frictional pressure π = 1
f
f
2
p ×
ψ
en
3
ρ Density kg/m
2
τ Shearing stress N/m
φ Angular coordinate rad
2×C
R
ψ 1
Relative bearing clearance ψ =
D
−1
ω Angular velocity ()ω =×2 π× N s
5 Bases of calculation and boundary conditions
Calculation within the meaning of 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.
The Reynolds equation provides the theoretical bases 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 for describing the flow via
the bearing lands, which can be derived from the Reynolds equation when special boundary conditions
are observed. The Hagen-Poiseuille law describes the pressure flow in a parallel clearance gap and
the Couette equation 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.
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ISO 12168-1:2019(E)
The following important premises apply 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;
e) a lubrication clearance gap completely filled with lubricant is the basis for the frictional behaviour;
f) fluctuations of pressure in the lubricant film normal to the sliding surfaces do not take place;
g) half 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.
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 as well as
recess pressures.
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 the factor 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, difference is made between the two extreme cases,
the load in the direction of the recess centre and the load in the direction of the 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 assured 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 can result in contact between the shaft and the 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.
As the shaft contacts 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.
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ISO 12168-1:2019(E)
It shall be assured 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).
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 and rotational frequency).
In both cases, the calculations are carried out according to an approximation method based on the
Hagen-Poiseuille and the Couette equations, mentioned in Clause 5. 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) the determination of load-carrying capacity with and without consideration of shaft rotation;
b) the calculation of lubricant flow rate and pumping power;
c) the determination of frictional power with and without consideration of losses in the bearing
recesses;
d) the procedure for bearing optimization with regard to minimum total power loss.
For all calculations, it shall be checked in addition 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.3.2.2).
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.
Examples of calculation are given in Annex B.
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, 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”.
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ISO 12168-1:2019(E)
Even under the above-mentioned premises, the characteristic value of the load-carrying capacity
F p
*
F = = (1)
BD×× p p
en en
still depends on the following parameters:
— the number of recesses Z;
— the width/diameter ratio B/D;
— the relative axial land width l /B;
ax
— the relative land width in circumferential direction l /B;
c
— the relative journal eccentricity ε;
— the relative frictional pressure:
η ×ω
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 × Fψ
So = =
× ωπ
η
B f
In ISO 12168-2:2019, 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 = 0,16, l /B = 0,26, i.e. restriction by means of capillaries and load in the direction of the
ax c
centre of bearing recess.
These figures represent a comparison between the approximation and the more precise solution by
means of the Reynolds equation. Further, the influence of rotation on the characteristic value of the
load-carrying capacity and on the attitude angle can be realized.
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 12168-2:2019,
Figures 1 and 2 and thus, h = C (1 − ε).
min R
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ISO 12168-1:2019(E)
According to the approximation method described in Annex A, this results in a dependence of the
characteristic value of the effective load-carrying capacity formed with the so-called “effective bearing
width” B − l
ax
F
*
F = (3)
eff
( Bl−× ) Dp ×
ax en
on lesser parameters. In the case of this definition, the width/diameter ratio B/D can be dropped as
parameter. Maintained are the number of recesses Z, the resistance ratio
l l
ax ax
× 1−−
2
R
lb×
B B
B Z
lan,ax
ax c
κ = = = ×× (4)
l
R lb× D π
c
lan,ac cax
D
the relative journal eccentricity ε and the speed dependent parameter determining the ratio of
hydrodynamic to hydrostatic pressure build-up:
l η ×ω l
cc
B
K =×πκ ×= ξ ××κξ (5)
rotf
2
D D
p ψ
en
*
If, in addition, advantage is taken of the fact that the function (ε ) is nearly linear for ε ≤ 0,5, then it
F
eff
*
is practically sufficient to know the function Ff(=εκ0,4)=(ZK,, ) for the calculation of the load-
effrot
carrying capacity.
**
In ISO 12168-2:2019, Figure 3, the function FF( εε= 0,4) = ( = 0,4); (=Kf 0)= (,Z )κ and in
eff,0eff rot
*
F
eff
Figure 4 the function =(fZ =4,,κ K ) are presented for the case “load in direction of recess
rot
*
F
eff,0
centre”. The hydrodynamically conditioned increase of the load-carrying capacity can be recognized
well when presented in such 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 κ and K , F ( ε = 0,4) is taken from ISO 12168-2:2019, Figure 3 and
rot
eff,0
* * *
(/FF ),ε =04 from ISO 12168-2:2019, Figure 4, F is calculated according to Formula (3)
()
effeff,0 eff
and with
*
0,4 ×
F
eff
ε =
* * *
FF ( εε== 0,4) ×F ( 0,4)
()
eff eff,0 eff,0
the minimum lubricant film thickness hC= (1− )ε is obtained.
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
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ISO 12168-1:2019(E)
By approximation, the lubricant flow rate can be calculated as follows (see also A.3.3):
11π
*
*
0Q ((εε≤≈,5) Q ==0) × × (7)
16 +ξ BD lB
()
ax
where
R
cp
2
ba−=4 c ;
R
P,0
6 ×× η l
Bax
R = .
P,0
3
bC×
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):
1,08 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 12168-2:2019, Table 1. The pumping power, without
considering the pump efficiency, is given by Formula (8):
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 of both the flow rate and the pumping power.
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ISO 12168-1:2019(E)
6.4 Frictional power
The characteristic value for the frictional power is given by
PC ×
fR
*
P = (9)
f
2
η ×× UB ×D
B
Friction is generated in the lands as well as in the recess area. The land area related to the total surface
of the bearing π × B × D is given by
l l l
Z
ax cax
*
A =×2 + ×× 1 −× 2
lan
B π D B
According to the approximation method, the characteristic value for the frictional power in the land
area is given by
π
**
PA= ×
f,lanlan
2
1 −ε
and in the recess area by
C
R
**
P =×π 4 ×× (1−A )
f,P lan
h
p
Thus, the characteristic value for the total amount of friction is given by
4 × C
1 1
R
**
PA=×π × + ×− 1 (10)
flan
*
2 h
A
1 − p
lan
ε
The actual frictional power is obtained by converting Formula (9) as follows
2
η ××UB × D
B
*
PP=×
ff
C
R
6.5 Optimization
When optimizing according to the power consumption, the total power loss, i.e. the sum of pumping
and frictional power, is minimized. According to 6.3 and 6.4, the total power is given by
23 2
pC× ××UB × D
η
en R B
* *
PP== + PQ × + P ×
totp f f
η C
B R
With Formulae (1) and (2), the following formula can be written:
*
P
Q
f
PF=× ω ××C ×+1 (11)
totR
*
P
4 × ()BD ××F π
p
f
[3]
Following a proposal of Vermeulen , the ratio of frictional to pumping power is introduced as an
* *
optional parameter P and designated with (P = P /P ). Thus, using Formula (11), the characteristic
f p
value for the total power loss is given by:
**
P
QP× (1 +)
tot
*
P = = (12)
tot
*
FC ××ω
4 × ()BD ××F π
R
f
Serial calculations have shown that the power minimum which can be obtained in the relatively wide
* *
range P = 1 to 3 depends only slightly on the chosen power ratio P . It is proposed to carry out an
*
approximated optimization with the mean value P = 2.
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ISO 12168-1:2019(E)
The relative frictional pressure in Formula (12) cannot be chosen freely as it is linked to the chosen
*
power ratio P :
*
**
P
B 1 PQ×
f
* 2
P =π ×× 4 ×=or π (13)
f f
*
B
D 2
Q
*
P ×
f
...
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