ISO 12168-1:2001
(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
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
Drsni ležaji - Hidrostatični radialni drsni ležaji brez odtočnih žlebov pri stacionarnih pogojih obratovanja - 1. del: Preračun oljnih hidrostatičnih radialnih drsnih ležajev brez drenažnih žlebov
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INTERNATIONAL ISO
STANDARD 12168-1
First edition
2001-12-15
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 12168-1:2001(E)
©
ISO 2001
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ISO 12168-1:2001(E)
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ii © ISO 2001 – All rights reserved
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ISO 12168-1:2001(E)
Contents Page
Foreword.iv
Introduction.v
1 Scope .1
2 Normative references.1
3 Bases of calculation and boundary conditions.1
4 Symbols, terms and units.3
5 Method of calculation.5
5.1 General.5
5.2 Load-carrying capacity .6
5.3 Lubricant flow rate and pumping power .7
5.4 Frictional power .8
5.5 Optimization .9
5.6 Temperatures and viscosities .10
5.7 Minimum pressure in recesses .11
Annex A (normative) Description of the approximation method for the calculation of hydrostatic plain
journal bearings.12
Annex B (normative) Examples of calculation.22
Bibliography.31
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ISO 12168-1:2001(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.
International Standards are drafted in accordance with the rules given in the ISO/IEC Directives, Part 3.
The main task of technical committees is to prepare International Standards. Draft International Standards adopted
by the technical committees are circulated to the member bodies for voting. Publication as an International
Standard requires approval by at least 75 % of the member bodies casting a vote.
Attention is drawn to the possibility that some of the elements of this part of ISO 12168 may be the subject of
patent rights. ISO shall not be held responsible for identifying any or all such patent rights.
ISO 12168-1 was prepared by Technical Committee ISO/TC 123, Plain bearings, Subcommittee SC 4, Methods of
calculation of plain bearings.
ISO 12168 consists of the following parts, under the general title 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
Part 2: Characteristic values for the calculation of oil-lubricated plain journal bearings without drainage grooves
Annexes A and B form a normative part of this part of ISO 12168.
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ISO 12168-1:2001(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 e.g. machine tools.
The bases of calculation described in this part of ISO 12168 apply to bearings with different numbers of recesses
and different width/diameter ratios for identical recess geometry. In this part of ISO 12168 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 part of ISO 12168 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 part of
ISO 12168 contains the design of the required lubrication system including the calculation of the restrictor data.
© ISO 2001 – All rights reserved v
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INTERNATIONAL STANDARD ISO 12168-1:2001(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 part of ISO 12168 applies to hydrostatic plain journal bearings under steady-state conditions.
In this part of ISO 12168 only bearings without oil drainage grooves between the recesses are taken into account.
2 Normative references
The following normative documents contain provisions which, through reference in this text, constitute provisions of
this part of ISO 12168. For dated references, subsequent amendments to, or revisions of, any of these publications
do not apply. However, parties to agreements based on this part of ISO 12168 are encouraged to investigate the
possibility of applying the most recent editions of the normative documents indicated below. For undated
references, the latest edition of the normative document referred to applies. Members of ISO and IEC maintain
registers of currently valid International Standards.
ISO 3448:1992, Industrial liquid lubricants — ISO viscosity classification
ISO 12168-2:2001, 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 Bases of calculation and boundary conditions
Calculation within the meaning of this part of ISO 12168 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' differential equation furnishes 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 part of ISO 12168 is based on two basic equations for describing the flow via the
bearing lands, which can be derived from Reynolds' differential 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 12168-1:2001(E)
The following important premises apply to the calculation procedures described in this part of ISO 12168:
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 equations, 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 part of ISO 12168 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
[1]
assumption for the calculations is the approximated optimum restrictor ratio ξ = 1 for the stiffness behaviour.
As for the outside dimensions of the bearing, this part of ISO 12168 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, load in the
direction of recess centre and load in the direction of land centre.
Apart from the afore-mentioned 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 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.
As 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.
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ISO 12168-1:2001(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 5.7).
4 Symbols, terms and units
See Table 1.
Table 1 — Symbols, terms and units
Symbol Term Unit
a Inertia factor 1
2
A
Land area m
lan
ʈA
lan
*
*
Relative land area =
A 1
A lan
lan Á˜
˯p¥ B ¥ D
2
A
Recess area m
p
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 = B –
( ) m
c bl
cax
B Bearing width m
c Stiffness coefficient N/m
.
c Specific heat capacity of the lubricant (p = constant)
p J/kg K
C Radial clearance =– / 2
CD()D m
R R BJ
d
Diameter of capillaries m
cp
Bearing diameter (D : shaft; D : bearing; D ≈ D ≈ D )
D m
J B J B
e Eccentricity (shaft displacement) m
F Load-carrying capacity (load) N
*
*
Characteristic value of load-carrying capacity [F = F/(B ¥ D ¥ p )]
F 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
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ISO 12168-1:2001(E)
Table 1 — (continued)
Symbol Term Unit
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 Specific bearing load È p=¥FB D ˘
( ) Pa
Î ˚
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
*
*
Power ratio (P = 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
R Flow resistance of one axial land =
Pa⋅s/m
lan,ax R
lan, ax
3
×
bC
ax
R
12××η l
c
3
R Flow resistance of one circumferential land =
Pa⋅s/m
lan,c R
lan,c
3
×
bC
c
R
3
R Flow resistance of one recess, when ε = 0, = 0,5
(RR ) Pa⋅s/m
P,0 P,0 lan,ax
Re Reynolds number 1
So Sommerfeld number 1
T Temperature °C
∆T 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
Relative eccentricity (ε = e/C )
ε 1
R
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ISO 12168-1:2001(E)
Table 1 — (continued)
Symbol Term Unit
Dynamic viscosity
η Pa⋅s
×
R
lan,ax lb
ax c
Resistance ratio κ = =
1
κ
×
R lan,c lb
cax
R
cp
Restrictor ratio ξ =
ξ 1
R
P,0
×ω
η
B
Relative frictional pressure = 1
π π
f f
2
p ×ψ
en
3
ρ Density kg/m
2
τ Shearing stress N/m
ϕ Angular coordinate rad
2×
C
R
Relative bearing clearence ψ =
ψ 1
D
-1
Angular velocity (2ω =×π×N)
ω s
5 Method of calculation
5.1 General
This part of ISO 12168 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 equations, mentioned in clause 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 5.2 to 5.7. 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 consideration of 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 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).
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ISO 12168-1:2001(E)
If the boundary conditions defined in clause 3 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 differential equation.
5.2 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”.
Even under the above mentioned premises, the characteristic value of load carrying capacity
F p
*
= = (1)
F
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;
×ω
η
B
the relative frictional pressure = (2)
π
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 = =
× ω
η
π
f
B
In Figures 1 and 2 of ISO 12168-2:2001, the functions F*(ε, p) and β (ε, p) 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, load in direction of centre of bearing
ax c
recess.
These figures represent a comparison between the approximation and the more precise solution by means of
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 5.6, if
en B B
applicable):
*
All parameters are given for the determination of F according to equation (1) and π according to equation (2). For
f
this geometry, the relevant values for ε and β can be taken from Figures 1 and 2 in ISO 12168-2:2001 and thus
h = C (1 - ε).
min R
According to the approximation method described in annex A, this results in a dependence of the characteristic
value of effective load-carrying capacity formed with the so-called “effective bearing width” B - l
ax
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ISO 12168-1:2001(E)
F
*
(3)
=
F
eff
( BD-¥ ) ¥ p
l
ax
en
on lesser parameters. In the case of this definition, espacially the width/diameter ratio B/D can be dropped as
parameter. Maintained are the number of recesses Z, the resistance ratio:
ʈ
ll
ax ax
¥-1
2
Á˜
˯
R ¥ ʈBZ B B
lan,ax lb
ax c
k = == ¥ ¥ (4)
Á˜
˯
¥pDlD
R lb
lan,c c ax
c
the relative journal eccentricity ε, and the speed dependent parameter determining the ratio of hydrodynamic to
hydrostatic pressure build-up:
h ¥w
ll
cc
B
=¥ kx ¥ = ¥kx ¥ (5)
K p
rot f
2
D D
p y
en
*
If, in addition, advantage is taken of the fact that the function ()e is nearly linear for ε u 0,5, then it is
F
eff
*
practically sufficent to know the function (ek = 0,4) = fZ ( , , ) for the calculation of the load carrying
FK
eff rot
capacity.
**
In Figure 3 of ISO 12168-2:2001, the function (ee = 0,4) = ( = 0,4); ( = 0) = fZ ( , k ) and in Figure 4
FF K
eff,0 eff rot
*
F
eff
the function = fZ ( = 4, k, ) are presented for the case “load in direction of recess centre”. The
K
rot
*
F
eff,0
hydrodynamically conditioned increase of the load carrying capacity can be recognized well when presented in
such manner.
*
If, e.g, Z and all parameters are given for the determination of according to equation (3), κ according to
F
eff
equation (4) and according to equation (5), then the minimum lubricant film thickness developing during
K
rot
operation can be determined.
*
After having calculated κ and K , ( e = 0,4) is taken from Figure 3 of ISO 12168-2:2001 and
F
eff,0
rot
** *
/ (e = 0,4) from Figure 4 of ISO 12168-2, is calculated according to equation 3 and with
FF
F
eff eff,0 eff
*
0,4 ¥
F
eff
e =
** *
FF¥¥ (ee = 0,4) F ( = 0,4)
eff
()eff,0 eff,0
the minimum lubricant film thickness = (1 - e ) is obtained.
hC
min R
5.3 Lubricant flow rate and pumping power
The characteristic value for the lubricant flow rate is given by
Q ¥h
* b
Q = (6)
3
¥ p
C
R en
It depends only slightly on the relative journal eccentricity ε, the load direction relative to the bearing and the
relative frictional pressure p or the speed dependent parameter K .
f rot
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ISO 12168-1:2001(E)
By approximation, the lubricant flow rate can be calculated as follows (see also A.3.3):
11p
* *
Q (e u 0,5) ª¥ Q (e = 0) = ¥ (7)
16 + x (B D l B
)
ax
6 ¥¥ l
R h
cp
ax
B
where x = and = .
R
P,0
3
R ¥
P,0 bC
ax
R
The flow resistance of the capillaries according to A.3.2.2 is given by
128 ¥¥
h l
cp
cp
= ¥ (1 + a)
R
cp
4
p¥
d
cp
with the non-linear portion (inertia factor):
1,08 4 ¥¥Q r
a = ¥
32 h ¥¥ Z
l
cp
cp
By converting equation (6), the lubricant flow rate can be calculated when the parameters h , C , p , ξ, B/D, and
B R en
l /B are given.
ax
*
For optimized bearings, Q shall be taken from Table 1 of ISO 12168-2:2001. The pumping power, without
considering the pump efficiency, is given by
2
3
p ¥
C
* en
R
= Q ¥¥ pQ = (8)
P
p
en
h
B
*
According to the approximation method, Q is again determined according to equation (7), thus it is the
characteristic value of both flow rate and pumping power.
5.4 Frictional power
The characteristic value for the frictional power is given by
¥
C
P
* f R
= (9)
P
f
2
¥¥ B ¥ D
h
U
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 p ¥ B ¥ D is given by
Z ʈ
ll l
ax c ax
*
=¥2 + ¥ ¥ 1 - 2 ¥
A
lan
Á˜
B pDB˯
According to the approximation method, the characteristic value for the frictional power in the land area is given by
p
**
=¥ ,
PA
f, lan lan
2
1 -
e
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ISO 12168-1:2001(E)
and in the recess area by
C
R
**
=p ¥4 ¥ ¥(1 - ).
PA
f,P lan
h
p
Thus the characteristic value for the total amount of friction is given by
È ˘
ʈ
114 ¥
C
R
**
=p ¥ ¥Í + ¥ - 1 ˙ (10)
PA
flan
Á˜
*
2
Í h ˙
˯A
1 - p lan
e
Î ˚
The actual frictional power is obtained by converting equation (9) as follows
2
¥¥UB ¥ D
h
B
*
=¥
PP
ff
C
R
5.5 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 5.3 and 5.4, the total power is given by
2 2
3
p ¥ h ¥¥UB ¥ D
C
* enRB*
== + ¥ + P ¥
Q
PP P
tot p f f
h
C
R
B
With equations (1) and (2) this can be written as follows
*
ʈ
Q
P
f
=¥FC w ¥ ¥ ¥ 1 + . (11)
P
tot Á˜
R
*
P
4 ¥¥(BD) F¥ ˯p
p
f
[2]
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 the characteristic value for the total power loss is given by
f p
*
*
Q ¥(1 + )
P P
tot
*
== (12)
P
tot
*
FC ¥¥w
4 ¥¥BD F¥p
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.
*
The relative frictional pressure in equation (12) cannot be chosen freely as it is linked to the chosen power ratio P :
*
**
B 1 ¥ Q
PP
* 2 f
= ¥¥4 ¥ or p = (13)
P p
f
f
*
D 2 B
*
Q
¥
P
f
D
*
When P , B/D, ε, h /C and ξ are given, the characteristic value of total power according to equation (12) becomes
p R
a function of Z, l /B, and l /B.
ax c
*
*
In Figures 5 and 6 of ISO 12168-2:2001, for P = 2, Z = 4, ξ = 1, B/D = 1, ε = 0,4 with or without friction in the
P
tot
recess (h /C = 40) is presented as a function of the geometrical parameters l /B and l /B.
p R ax c
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ISO 12168-1:2001(E)
*
*
In Figures 7 to 12 of ISO 12168-2:2001, for P = 2, ξ = 1, ε = 0,4, h = 40 C , is presented for different B/D and
P
tot p R
Z as a function of l /B and l /B, taking into account friction in the recesses. The land widths l /B and l /B, where
c
ax c ax
the total power is reduced t
...
SLOVENSKI STANDARD
SIST ISO 12168-1:2002
01-december-2002
'UVQLOHåDML+LGURVWDWLþQLUDGLDOQLGUVQLOHåDMLEUH]RGWRþQLKåOHERYSUL
VWDFLRQDUQLKSRJRMLKREUDWRYDQMDGHO 3UHUDþXQROMQLKKLGURVWDWLþQLKUDGLDOQLK
GUVQLKOHåDMHYEUH]GUHQDåQLKåOHERY
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
Ta slovenski standard je istoveten z: ISO 12168-1:2001
ICS:
21.100.10 Drsni ležaji Plain bearings
SIST ISO 12168-1:2002 en
2003-01.Slovenski inštitut za standardizacijo. Razmnoževanje celote ali delov tega standarda ni dovoljeno.
---------------------- Page: 1 ----------------------
SIST ISO 12168-1:2002
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SIST ISO 12168-1:2002
INTERNATIONAL ISO
STANDARD 12168-1
First edition
2001-12-15
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 12168-1:2001(E)
©
ISO 2001
---------------------- Page: 3 ----------------------
SIST ISO 12168-1:2002
ISO 12168-1:2001(E)
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ii © ISO 2001 – All rights reserved
---------------------- Page: 4 ----------------------
SIST ISO 12168-1:2002
ISO 12168-1:2001(E)
Contents Page
Foreword.iv
Introduction.v
1 Scope .1
2 Normative references.1
3 Bases of calculation and boundary conditions.1
4 Symbols, terms and units.3
5 Method of calculation.5
5.1 General.5
5.2 Load-carrying capacity .6
5.3 Lubricant flow rate and pumping power .7
5.4 Frictional power .8
5.5 Optimization .9
5.6 Temperatures and viscosities .10
5.7 Minimum pressure in recesses .11
Annex A (normative) Description of the approximation method for the calculation of hydrostatic plain
journal bearings.12
Annex B (normative) Examples of calculation.22
Bibliography.31
© ISO 2001 – All rights reserved iii
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SIST ISO 12168-1:2002
ISO 12168-1:2001(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.
International Standards are drafted in accordance with the rules given in the ISO/IEC Directives, Part 3.
The main task of technical committees is to prepare International Standards. Draft International Standards adopted
by the technical committees are circulated to the member bodies for voting. Publication as an International
Standard requires approval by at least 75 % of the member bodies casting a vote.
Attention is drawn to the possibility that some of the elements of this part of ISO 12168 may be the subject of
patent rights. ISO shall not be held responsible for identifying any or all such patent rights.
ISO 12168-1 was prepared by Technical Committee ISO/TC 123, Plain bearings, Subcommittee SC 4, Methods of
calculation of plain bearings.
ISO 12168 consists of the following parts, under the general title 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
Part 2: Characteristic values for the calculation of oil-lubricated plain journal bearings without drainage grooves
Annexes A and B form a normative part of this part of ISO 12168.
iv © ISO 2001 – All rights reserved
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SIST ISO 12168-1:2002
ISO 12168-1:2001(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 e.g. machine tools.
The bases of calculation described in this part of ISO 12168 apply to bearings with different numbers of recesses
and different width/diameter ratios for identical recess geometry. In this part of ISO 12168 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 part of ISO 12168 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 part of
ISO 12168 contains the design of the required lubrication system including the calculation of the restrictor data.
© ISO 2001 – All rights reserved v
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SIST ISO 12168-1:2002
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SIST ISO 12168-1:2002
INTERNATIONAL STANDARD ISO 12168-1:2001(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 part of ISO 12168 applies to hydrostatic plain journal bearings under steady-state conditions.
In this part of ISO 12168 only bearings without oil drainage grooves between the recesses are taken into account.
2 Normative references
The following normative documents contain provisions which, through reference in this text, constitute provisions of
this part of ISO 12168. For dated references, subsequent amendments to, or revisions of, any of these publications
do not apply. However, parties to agreements based on this part of ISO 12168 are encouraged to investigate the
possibility of applying the most recent editions of the normative documents indicated below. For undated
references, the latest edition of the normative document referred to applies. Members of ISO and IEC maintain
registers of currently valid International Standards.
ISO 3448:1992, Industrial liquid lubricants — ISO viscosity classification
ISO 12168-2:2001, 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 Bases of calculation and boundary conditions
Calculation within the meaning of this part of ISO 12168 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' differential equation furnishes 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 part of ISO 12168 is based on two basic equations for describing the flow via the
bearing lands, which can be derived from Reynolds' differential 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.
© ISO 2001 – All rights reserved 1
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SIST ISO 12168-1:2002
ISO 12168-1:2001(E)
The following important premises apply to the calculation procedures described in this part of ISO 12168:
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 equations, 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 part of ISO 12168 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
[1]
assumption for the calculations is the approximated optimum restrictor ratio ξ = 1 for the stiffness behaviour.
As for the outside dimensions of the bearing, this part of ISO 12168 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, load in the
direction of recess centre and load in the direction of land centre.
Apart from the afore-mentioned 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 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.
As 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.
2 © ISO 2001 – All rights reserved
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SIST ISO 12168-1:2002
ISO 12168-1:2001(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 5.7).
4 Symbols, terms and units
See Table 1.
Table 1 — Symbols, terms and units
Symbol Term Unit
a Inertia factor 1
2
A
Land area m
lan
ʈA
lan
*
*
Relative land area =
A 1
A lan
lan Á˜
˯p¥ B ¥ D
2
A
Recess area m
p
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 = B –
( ) m
c bl
cax
B Bearing width m
c Stiffness coefficient N/m
.
c Specific heat capacity of the lubricant (p = constant)
p J/kg K
C Radial clearance =– / 2
CD()D m
R R BJ
d
Diameter of capillaries m
cp
Bearing diameter (D : shaft; D : bearing; D ≈ D ≈ D )
D m
J B J B
e Eccentricity (shaft displacement) m
F Load-carrying capacity (load) N
*
*
Characteristic value of load-carrying capacity [F = F/(B ¥ D ¥ p )]
F 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
© ISO 2001 – All rights reserved 3
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SIST ISO 12168-1:2002
ISO 12168-1:2001(E)
Table 1 — (continued)
Symbol Term Unit
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 Specific bearing load È p=¥FB D ˘
( ) Pa
Î ˚
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
*
*
Power ratio (P = 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
R Flow resistance of one axial land =
Pa⋅s/m
lan,ax R
lan, ax
3
×
bC
ax
R
12××η l
c
3
R Flow resistance of one circumferential land =
Pa⋅s/m
lan,c R
lan,c
3
×
bC
c
R
3
R Flow resistance of one recess, when ε = 0, = 0,5
(RR ) Pa⋅s/m
P,0 P,0 lan,ax
Re Reynolds number 1
So Sommerfeld number 1
T Temperature °C
∆T 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
Relative eccentricity (ε = e/C )
ε 1
R
4 © ISO 2001 – All rights reserved
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SIST ISO 12168-1:2002
ISO 12168-1:2001(E)
Table 1 — (continued)
Symbol Term Unit
Dynamic viscosity
η Pa⋅s
×
R
lan,ax lb
ax c
Resistance ratio κ = =
1
κ
×
R lan,c lb
cax
R
cp
Restrictor ratio ξ =
ξ 1
R
P,0
×ω
η
B
Relative frictional pressure = 1
π π
f f
2
p ×ψ
en
3
ρ Density kg/m
2
τ Shearing stress N/m
ϕ Angular coordinate rad
2×
C
R
Relative bearing clearence ψ =
ψ 1
D
-1
Angular velocity (2ω =×π×N)
ω s
5 Method of calculation
5.1 General
This part of ISO 12168 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 equations, mentioned in clause 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 5.2 to 5.7. 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 consideration of 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 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).
© ISO 2001 – All rights reserved 5
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SIST ISO 12168-1:2002
ISO 12168-1:2001(E)
If the boundary conditions defined in clause 3 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 differential equation.
5.2 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”.
Even under the above mentioned premises, the characteristic value of load carrying capacity
F p
*
= = (1)
F
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;
×ω
η
B
the relative frictional pressure = (2)
π
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 = =
× ω
η
π
f
B
In Figures 1 and 2 of ISO 12168-2:2001, the functions F*(ε, p) and β (ε, p) 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, load in direction of centre of bearing
ax c
recess.
These figures represent a comparison between the approximation and the more precise solution by means of
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 5.6, if
en B B
applicable):
*
All parameters are given for the determination of F according to equation (1) and π according to equation (2). For
f
this geometry, the relevant values for ε and β can be taken from Figures 1 and 2 in ISO 12168-2:2001 and thus
h = C (1 - ε).
min R
According to the approximation method described in annex A, this results in a dependence of the characteristic
value of effective load-carrying capacity formed with the so-called “effective bearing width” B - l
ax
6 © ISO 2001 – All rights reserved
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SIST ISO 12168-1:2002
ISO 12168-1:2001(E)
F
*
(3)
=
F
eff
( BD-¥ ) ¥ p
l
ax
en
on lesser parameters. In the case of this definition, espacially the width/diameter ratio B/D can be dropped as
parameter. Maintained are the number of recesses Z, the resistance ratio:
ʈ
ll
ax ax
¥-1
2
Á˜
˯
R ¥ ʈBZ B B
lan,ax lb
ax c
k = == ¥ ¥ (4)
Á˜
˯
¥pDlD
R lb
lan,c c ax
c
the relative journal eccentricity ε, and the speed dependent parameter determining the ratio of hydrodynamic to
hydrostatic pressure build-up:
h ¥w
ll
cc
B
=¥ kx ¥ = ¥kx ¥ (5)
K p
rot f
2
D D
p y
en
*
If, in addition, advantage is taken of the fact that the function ()e is nearly linear for ε u 0,5, then it is
F
eff
*
practically sufficent to know the function (ek = 0,4) = fZ ( , , ) for the calculation of the load carrying
FK
eff rot
capacity.
**
In Figure 3 of ISO 12168-2:2001, the function (ee = 0,4) = ( = 0,4); ( = 0) = fZ ( , k ) and in Figure 4
FF K
eff,0 eff rot
*
F
eff
the function = fZ ( = 4, k, ) are presented for the case “load in direction of recess centre”. The
K
rot
*
F
eff,0
hydrodynamically conditioned increase of the load carrying capacity can be recognized well when presented in
such manner.
*
If, e.g, Z and all parameters are given for the determination of according to equation (3), κ according to
F
eff
equation (4) and according to equation (5), then the minimum lubricant film thickness developing during
K
rot
operation can be determined.
*
After having calculated κ and K , ( e = 0,4) is taken from Figure 3 of ISO 12168-2:2001 and
F
eff,0
rot
** *
/ (e = 0,4) from Figure 4 of ISO 12168-2, is calculated according to equation 3 and with
FF
F
eff eff,0 eff
*
0,4 ¥
F
eff
e =
** *
FF¥¥ (ee = 0,4) F ( = 0,4)
eff
()eff,0 eff,0
the minimum lubricant film thickness = (1 - e ) is obtained.
hC
min R
5.3 Lubricant flow rate and pumping power
The characteristic value for the lubricant flow rate is given by
Q ¥h
* b
Q = (6)
3
¥ p
C
R en
It depends only slightly on the relative journal eccentricity ε, the load direction relative to the bearing and the
relative frictional pressure p or the speed dependent parameter K .
f rot
© ISO 2001 – All rights reserved 7
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SIST ISO 12168-1:2002
ISO 12168-1:2001(E)
By approximation, the lubricant flow rate can be calculated as follows (see also A.3.3):
11p
* *
Q (e u 0,5) ª¥ Q (e = 0) = ¥ (7)
16 + x (B D l B
)
ax
6 ¥¥ l
R h
cp
ax
B
where x = and = .
R
P,0
3
R ¥
P,0 bC
ax
R
The flow resistance of the capillaries according to A.3.2.2 is given by
128 ¥¥
h l
cp
cp
= ¥ (1 + a)
R
cp
4
p¥
d
cp
with the non-linear portion (inertia factor):
1,08 4 ¥¥Q r
a = ¥
32 h ¥¥ Z
l
cp
cp
By converting equation (6), the lubricant flow rate can be calculated when the parameters h , C , p , ξ, B/D, and
B R en
l /B are given.
ax
*
For optimized bearings, Q shall be taken from Table 1 of ISO 12168-2:2001. The pumping power, without
considering the pump efficiency, is given by
2
3
p ¥
C
* en
R
= Q ¥¥ pQ = (8)
P
p
en
h
B
*
According to the approximation method, Q is again determined according to equation (7), thus it is the
characteristic value of both flow rate and pumping power.
5.4 Frictional power
The characteristic value for the frictional power is given by
¥
C
P
* f R
= (9)
P
f
2
¥¥ B ¥ D
h
U
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 p ¥ B ¥ D is given by
Z ʈ
ll l
ax c ax
*
=¥2 + ¥ ¥ 1 - 2 ¥
A
lan
Á˜
B pDB˯
According to the approximation method, the characteristic value for the frictional power in the land area is given by
p
**
=¥ ,
PA
f, lan lan
2
1 -
e
8 © ISO 2001 – All rights reserved
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SIST ISO 12168-1:2002
ISO 12168-1:2001(E)
and in the recess area by
C
R
**
=p ¥4 ¥ ¥(1 - ).
PA
f,P lan
h
p
Thus the characteristic value for the total amount of friction is given by
È ˘
ʈ
114 ¥
C
R
**
=p ¥ ¥Í + ¥ - 1 ˙ (10)
PA
flan
Á˜
*
2
Í h ˙
˯A
1 - p lan
e
Î ˚
The actual frictional power is obtained by converting equation (9) as follows
2
¥¥UB ¥ D
h
B
*
=¥
PP
ff
C
R
5.5 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 5.3 and 5.4, the total power is given by
2 2
3
p ¥ h ¥¥UB ¥ D
C
* enRB*
== + ¥ + P ¥
Q
PP P
tot p f f
h
C
R
B
With equations (1) and (2) this can be written as follows
*
ʈ
Q
P
f
=¥FC w ¥ ¥ ¥ 1 + . (11)
P
tot Á˜
R
*
P
4 ¥¥(BD) F¥ ˯p
p
f
[2]
Following a proposal of Vermeulen , the ratio of frictional to pumping power is introduc
...
NORME ISO
INTERNATIONALE 12168-1
Première édition
2001-12-15
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
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
Numéro de référence
ISO 12168-1:2001(F)
©
ISO 2001
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ISO 12168-1:2001(F)
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ISO 12168-1:2001(F)
Sommaire Page
Avant-propos .iv
Introduction.v
1 Domaine d'application .1
2 Références normatives .1
3 Bases de calcul et conditions aux limites.1
4 Symboles, termes et unités .3
5 Méthode de calcul.5
5.1 Généralités .5
5.2 Portance .6
5.3 Débit de lubrifiant et puissance de pompage.8
5.4 Puissance de frottement.9
5.5 Optimisation.9
5.6 Températures et viscosités .11
5.7 Pression minimale sur les alvéoles .12
Annexe A (normative) Description de la méthode d'approximation pour le calcul des paliers lisses
radiaux hydrostatiques .13
Annexe B (normative) Exemples de calcul.23
Bibliographie.32
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ISO 12168-1:2001(F)
Avant-propos
L'ISO (Organisation internationale de normalisation) est une fédération mondiale d'organismes nationaux de
normalisation (comités membres de l'ISO). L'élaboration des Normes internationales est en général confiée aux
comités techniques de l'ISO. Chaque comité membre intéressé par une étude a le droit de faire partie du comité
technique créé à cet effet. Les organisations internationales, gouvernementales et non gouvernementales, en
liaison avec l'ISO participent également aux travaux. L'ISO collabore étroitement avec la Commission
électrotechnique internationale (CEI) en ce qui concerne la normalisation électrotechnique.
Les Normes internationales sont rédigées conformément aux règles données dans les Directives ISO/CEI,
Partie 3.
La tâche principale des comités techniques est d'élaborer les Normes internationales. Les projets de Normes
internationales adoptés par les comités techniques sont soumis aux comités membres pour vote. Leur publication
comme Normes internationales requiert l'approbation de 75 % au moins des comités membres votants.
L'attention est appelée sur le fait que certains des éléments de la présente partie de l'ISO 12168 peuvent faire
l'objet de droits de propriété intellectuelle ou de droits analogues. L'ISO ne saurait être tenue pour responsable de
ne pas avoir identifié de tels droits de propriété et averti de leur existence.
L'ISO 12168-1 a été élaborée par le comité technique ISO/TC 123, Paliers lisses, sous-comité SC 4, Méthodes de
calcul des paliers lisses.
L'ISO 12168 comprend les parties suivantes, présentées sous le titre général 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
Partie 2: Caractéristiques du calcul pour la lubrification des paliers lisses radiaux sans rainure d'écoulement
Les annexes A et B constituent des éléments normatifs de la présente partie de l’ISO 12168.
iv © ISO 2001 – Tous droits réservés
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ISO 12168-1:2001(F)
Introduction
Le fonctionnement des paliers hydrostatiques est caractérisé par le fait que la pression d'appui du palier est
générée par une lubrification externe. L'absence d'usure, le fonctionnement silencieux, la grande plage de vitesses
utilisables ainsi que la forte rigidité et la capacité d'amortissement constituent les avantages particuliers des paliers
hydrostatiques. Ces propriétés expliquent également le bien-fondé de l'importance particulière des ensembles avec
paliers hydrostatiques dans différents domaines d'application tels que par exemple les machines-outils.
Les bases de calcul décrites dans la présente partie de l'ISO 12168 s'appliquent aux paliers ayant des nombres
d’alvéoles et des rapports largeur/diamètre différents pour une géométrie d’alvéole identique. Dans la présente
partie de l'ISO 12168, seuls les paliers non équipés de rainures de vidange de l'huile entre les alvéoles sont pris en
compte. Par comparaison avec les paliers avec rainures de vidange de l'huile, ce type de palier requiert une
puissance plus élevée avec le même comportement à la rigidité.
L'huile est injectée dans chaque alvéole de palier à l'aide d'une pompe commune à pression constante (circuit
p = constant) et par l'intermédiaire des restricteurs linéaires placés en amont, par exemple sous la forme de
en
capillaires.
Les méthodes de calcul énumérées dans la présente partie de l'ISO 12168 doivent permettre à l'utilisateur de
calculer et d'évaluer un type de palier donné ainsi que de concevoir un palier en fonction de certains paramètres
facultatifs. De plus, la présente partie de l'ISO 12168 contient le modèle type du circuit de lubrification requis y
compris le calcul des données propres aux restricteurs.
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NORME INTERNATIONALE ISO 12168-1:2001(F)
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
1 Domaine d'application
La présente partie de l'ISO 12168 s'applique aux paliers lisses radiaux hydrostatiques dans des conditions de
régime stationnaire.
Dans la présente partie de l'ISO 12168, seuls les paliers sans rainure d’écoulement de l'huile entre les alvéoles
sont pris en compte.
2 Références normatives
Les documents normatifs suivants contiennent des dispositions qui, par suite de la référence qui y est faite,
constituent des dispositions valables pour la présente partie de l'ISO 12168. Pour les références datées, les
amendements ultérieurs ou les révisions de ces publications ne s'appliquent pas. Toutefois, les parties prenantes
aux accords fondés sur la présente partie de l'ISO 12168 sont invitées à rechercher la possibilité d'appliquer les
éditions les plus récentes des documents normatifs indiqués ci-après. Pour les références non datées, la dernière
édition du document normatif en référence s'applique. Les membres de l'ISO et de la CEI possèdent le registre des
Normes internationales en vigueur.
ISO 3448:1992, Lubrifiants liquides industriels — Classification ISO selon la viscosité
ISO 12168-2:2001, Paliers lisses — Paliers lisses radiaux hydrostatiques sans rainures d’écoulement fonctionnant
en régime stationnaire — Partie 2: Caractéristiques du calcul pour la lubrification des paliers lisses radiaux sans
rainure d'écoulement
3 Bases de calcul et conditions aux limites
Le calcul relevant du domaine d'application de la présente partie de l'ISO 12168 est la détermination mathématique
des paramètres de fonctionnement des paliers radiaux lisses hydrostatiques en fonction des conditions de
fonctionnement, de la géométrie des paliers et des données de lubrification, c'est-à-dire la détermination des
excentricités, de la portance, de la rigidité, de la pression d'alimentation requise, du débit d'huile, de la puissance
de frottement et de pompage, et de l'augmentation de la température. Outre le développement de la pression
hydrostatique, l'influence de l'effet hydrodynamique est également estimée.
L'équation différentielle de Reynolds fournit les bases théoriques du calcul des paliers hydrostatiques. Dans la
plupart des cas pratiques d'application, il est toutefois possible d'obtenir des résultats suffisamment exacts par
approximation.
L'approximation utilisée dans la présente partie de l'ISO 12168 est fondée sur deux équations de base utilisées
pour décrire l'écoulement par l'intermédiaire des butées des paliers, que l'on peut dériver de l'équation différentielle
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ISO 12168-1:2001(F)
de Reynolds lorsqu'on observe des conditions aux limites particulières. La loi de Hagen-Poiseuille décrit le débit de
pression dans un jeu parallèle, et l'équation de Couette décrit l'écoulement dans le jeu des paliers dû à la rotation
des arbres. Une présentation détaillée du contexte théorique de la méthode de calcul est incluse dans l'annexe A.
Les hypothèses fondamentales suivantes s'appliquent aux méthodes de calcul décrites dans la présente partie de
l'ISO 12168:
a) tous les écoulements de lubrifiant dans le jeu de lubrification sont laminaires;
b) le lubrifiant adhère entièrement aux surfaces de glissement;
c) le lubrifiant est un fluide newtonien incompressible;
d) dans l'ensemble du jeu de lubrification ainsi que dans les restricteurs en amont, le lubrifiant est partiellement
isovisqueux;
e) un jeu de lubrification entièrement rempli de lubrifiant constitue la base du comportement au frottement;
f) absence de variations de pression du film d'huile perpendiculaire aux surfaces de glissement;
g) le demi-palier et le demi-tourillon ont des surfaces totalement rigides;
h) les rayons de courbure des surfaces, dont le mouvement des unes par rapport aux autres est relatif, sont
grands par comparaison à l'épaisseur du film d'huile;
i) la hauteur du jeu dans la direction des axes est constante (jeu parallèle axial);
j) la pression exercée sur l’alvéole est constante;
k) absence de mouvement perpendiculaire aux surfaces de glissement.
À l'aide des équations d'approximation mentionnées ci-dessus, tous les paramètres requis pour la conception ou le
calcul des paliers peuvent être déterminés. L'application du principe de similarité entraîne des valeurs de similarité
non dimensionnées de la portance, de la rigidité, du débit d'huile, du frottement, des pressions d’alvéole, etc.
Les résultats indiqués dans la présente partie de l'ISO 12168 sous la forme de tableaux et de diagrammes se
limitent aux plages de fonctionnement communes dans la pratique pour les paliers hydrostatiques. Ainsi, la plage
d'excentricité des paliers (déplacement sous charge) se limite à ε = 0 à 0,5.
La limitation de cette plage d'excentricité signifie une simplification considérable de la méthode de calcul dans la
mesure où la portance est une fonction quasi linéaire de l'excentricité. Toutefois, l'applicabilité de cette procédure
est rarement limitée, dans la mesure où dans la pratique les excentricités ε > 0,5 sont la plupart du temps non
souhaitées pour des raisons de sécurité de fonctionnement. Une autre hypothèse relative aux calculs est le rapport
[1]
restricteur optimal par approximation ξ = 1 pour le comportement à la rigidité.
Pour ce qui concerne les dimensions extérieures du palier, la présente partie de l'ISO 12168 se limite au domaine
largeur/diamètre des paliers B/D = 0,3 à 1 qui est commun dans les cas d'application pratiques. La profondeur
d’alvéole est plus importante que la hauteur du jeu par un facteur de 10 à 100. Pour le calcul des pertes par
frottement, la perte par frottement sur l’alvéole par rapport au frottement sur les butées des paliers peut
généralement être négligée compte tenu des hypothèses mentionnées ci-dessus. Cela ne s'applique toutefois pas
lorsque le palier doit être optimisé eu égard à ses pertes de puissance totales.
Afin de prendre en compte la direction de la charge d'un palier, on distingue les deux cas extrêmes que sont la
charge dans la direction de l'axe d’alvéole et la charge dans la direction de l'axe des butées.
Mis à part les conditions aux limites mentionnées, certaines autres exigences doivent être mentionnées pour la
conception des paliers hydrostatiques afin de s'assurer de leur fonctionnement dans toutes les conditions de
fonctionnement. En général, un palier doit être désigné de sorte qu'une hauteur de jeu minimale comprise entre
50 % et 60 % de la hauteur de jeu initiale soit assurée lorsque la charge maximale possible est appliquée. Dans ce
2 © ISO 2001 – Tous droits réservés
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ISO 12168-1:2001(F)
type de liaison, il faut accorder une attention toute particulière aux désalignements de l'arbre dans le palier dus aux
déformations de l'arbre, susceptibles d'apparaître dans un contact entre l'arbre et le bord du palier, endommageant
ainsi le palier. De plus, le jeu parallèle requis pour le calcul n'est plus nécessaire dans ce cas.
Dans la mesure où l'arbre est en contact avec les butées de paliers lorsque la pression hydrostatique est
interrompue, il pourrait s'avérer nécessaire de vérifier les zones de contact eu égard aux pressions de surface
naissantes.
On doit s'assurer que la chaleur du palier n'entraîne pas une augmentation inadmissible de la température de
l'huile.
Si nécessaire, le refroidissement de l'huile doit être prévu. De plus, l'huile doit être filtrée afin d'éviter
l'encrassement des capillaires et l'endommagement des surfaces de glissement.
Une faible pression de l’alvéole dégagée doit également être évitée, dans la mesure où, dans le cas contraire, l'air
provient du milieu ambiant, et entraînerait une diminution de la rigidité (voir 5.7).
4 Symboles, termes et unités
Voir Tableau 1.
Tableau 1 — Symboles, termes et unités
Symbole Terme Unité
a Facteur d'inertie 1
2
A
Surface des butées m
lan
ʈ
A
* lan
*
Surface relative des butées =
A
A lan 1
lan Á˜
˯p¥ B ¥ D
2
A
Surface d’alvéole m
p
b Largeur perpendiculaire au sens d'écoulement m
π ¥ D
È ˘
b Largeur de la sortie axiale b=
m
ax ax
Í ˙
Z
Î ˚
b
Largeur de la sortie périphérique ( = B – l ) m
c b
cax
B Largeur de palier m
c Coefficient de rigidité N/m
c
Chaleur massique du lubrifiant (p = constante) J/kg⋅K
p
C
Jeu radial =() – 2 m
R CD D
R BJ
d Diamètre des capillaires m
cp
D Diamètre des paliers (D : arbre; D : demi-palier; D ≈ D ≈ D ) m
J B J B
e Excentricité (déplacement de l'arbre) m
F Portance (charge) N
* *
Valeur caractéristique de la portance [F = F/(B ¥ D ¥ p )]
F 1
en
*
F Valeur caractéristique de la portance effective 1
eff
*
F Valeur caractéristique de la portance effective pour N = 0 1
eff,0
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ISO 12168-1:2001(F)
Tableau 1 — (suite)
Symbole Terme Unité
h
Épaisseur locale du film d'huile (hauteur de jeu) m
h
Épaisseur minimale du film d'huile (hauteur de jeu minimale) m
min
h
Profondeur d’alvéole m
p
K
Paramètre selon la vitesse 1
rot
l
Longueur dans le sens de l'écoulement m
l Longueur axiale des butées m
ax
l
Longueur de la circonférence des butées m
c
l
Longueur des capillaires m
cp
-1
N Fréquence de rotation (vitesse) s
p
Pression dans l’alvéole, en général Pa
p Charge spécifique des paliers [ p = F/(B × D)]
Pa
p
Pression d'alimentation (pression de refoulement de la pompe) Pa
en
p
Pression dans l’alvéole i Pa
i
p
Pression dans l’alvéole i, lorsque e = 0
Pa
i,0
Rapport de puissance (P* = P /P )
P* 1
f p
P
Puissance de frottement W
f
P Puissance de pompage W
p
P Puissance totale (P = P +P )
W
tot tot p f
*
P Valeur caractéristique de la puissance totale 1
tot
3
Q Débit de lubrifiant (pour un palier complet) m /s
*
Q Paramètre de débit de lubrifiant 1
3
R
Résistance à l'écoulement des capillaires Pa⋅s/m
cp
12××η l
ax
3
R Résistance à l'écoulement d'une butée axiale =
Pa⋅s/m
lan, ax R
lan, ax
3
×
bC
ax
R
ʈ
12¥¥h l
c
3
R Résistance à l'écoulement d'une butée circonférentielle R =
Pa⋅s/m
lan, c Á˜lan, c
3
˯bC¥
cR
3
R Résistance à l'écoulement d'une alvéole, lorsque e = 0, RR= 0,5
Pa⋅s/m
P, 0 ()P,0 lan, ax
Re Nombre de Reynolds 1
So Nombre de Sommerfeld 1
T Température °C
DT Différence de température K
u Vitesse d'écoulement m/s
U
Vitesse périphérique m/s
w
Vitesse moyenne du restricteur m/s
Z
Nombre d’alvéoles 1
4 © ISO 2001 – Tous droits réservés
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ISO 12168-1:2001(F)
Tableau 1 — (suite)
Symbole Terme Unité
a Position de la première alvéole par rapport à l'axe d’alvéole rad
b Angle de calage de l'arbre °
g Exposant de la formule de viscosité 1
Excentricité relative (e = e/C )
e 1
R
h Viscosité dynamique Pa⋅s
ʈ
R
lb¥
lan,ax
ax c
Rapport de résistance k==
k 1
Á˜
Rl ¥b
˯lan,c c ax
ʈ
R
cp
Rapport restricteur x =
ξ 1
Á˜
R
˯P,0
ʈ
hw¥
B
p Pression de frottement relative p = 1
Á˜
f f
2
p ¥ y
˯
en
3
r
Masse volumique kg/m
2
t Effort de cisaillement N/m
j Coordonnée angulaire rad
ʈ2 ¥ C
R
Jeu relatif des paliers y =
y 1
Á˜
˯
D
-1
w Vitesse angulaire (w = 2 ¥ p ¥ N) s
5 Méthode de calcul
5.1 Généralités
La présente partie de l'ISO 12168 couvre le calcul ainsi que la conception des paliers radiaux lisses
hydrostatiques. Dans ce cas, le terme calcul doit être perçu comme la vérification des paramètres de
fonctionnement d'un palier hydrostatique avec des données géométriques et de lubrification connues. Dans le cas
d'un calcul théorique, avec les méthodes de calcul données, il est possible de déterminer les données absentes
pour la géométrie de palier requise, les données de lubrification ainsi que les paramètres de fonctionnement sur la
base de quelques données initiales (par exemple portance requise, rigidité, fréquence de rotation).
Dans les deux cas, les calculs sont effectués selon une méthode d'approximation fondée sur les équations de
Hagen-Poiseuille et de Couette respectivement, mentionnées à l'article 3. Les paramètres relatifs aux paliers
calculés selon cette méthode sont donnés comme valeurs relatives sous la forme de tableaux et de diagrammes
en fonction des paramètres différents. La procédure de calcul ou de conception des paliers est décrite en 5.2 à 5.7.
Cela inclut la détermination des différents paramètres des paliers à l'aide des formules de calcul données ou des
tableaux et des diagrammes respectivement. Les éléments de calcul suivants sont expliqués de manière détaillée:
a) détermination de la portance en tenant ou en ne tenant pas compte de la rotation des arbres;
b) calcul du débit d'huile et de la puissance de pompage;
c) détermination de la puissance de frottement en tenant et en ne tenant pas compte des pertes des alvéoles
dans les paliers;
d) procédure d'optimisation des paliers eu égard à la perte minimale totale de puissance.
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ISO 12168-1:2001(F)
Pour tous les calculs, il doit en outre être vérifié si l'hypothèse fondamentale de l'écoulement laminaire dans le jeu
et l’alvéole des paliers, ainsi que dans le capillaire, est vraie. Cela peut être vérifié en déterminant les nombres de
Reynolds. De plus, la part du facteur d'inertie dans les différences de pression doit être faible au niveau du
capillaire (voir A.3.2.2).
Si les conditions aux limites définies à l'article 3 sont respectées, cette méthode de calcul donne des résultats avec
des écarts pouvant être négligés eu égard aux exigences de pratique, par comparaison avec un calcul exact par
résolution de l'équation différentielle de Reynolds.
5.2 Portance
Sauf indication contraire, on suppose que les capillaires présentant une caractéristique linéaire sont utilisés comme
restricteurs et que le rapport restricteur est ξ = 1. De plus, on distingue uniquement les deux cas «charge dans la
direction de l'axe d’alvéole» et «charge dans la direction de l'axe des butées». Pour cette raison, il n'est plus fait
mention, dans chaque cas individuel, du fait que les valeurs caractéristiques sont fonction des trois paramètres
«type de restricteur», «rapport restricteur» et «direction de la charge par rapport au palier».
Même dans les hypothèses mentionnées ci-dessus, la valeur caractéristique de la portance
F p
*
F== (1)
B¥¥Dp p
en en
dépend pourtant des paramètres suivants:
nombre d’alvéoles Z;
rapport largeur/diamètre B/D;
largeur axiale relative des butées l /B;
ax
largeur relative périphérique des butées l /B;
c
excentricité relative des tourillons e ;
pression de frottement relative
h ¥ w
B
p = (2)
f
2
p ¥y
en
NOTE Le nombre de Sommerfeld So couramment utilisé avec les paliers radiaux lisses hydrodynamiques peut être défini
de la manière suivante:
2*
pF¥y
So==
h ¥wp
Bf
*
Dans les Figures 1 et 2 de l'ISO 12168-2:2001, les fonctions F (e, p ) et b (e, p ) sont représentées pour Z = 4,
f f
ξ = 1, B/D = 1, l /B = 0,16, l /B = 0,26, c’est-à-dire la restriction au moyen des capillaires, la charge dans la
ax c
direction de l'axe de l’alvéole des paliers.
Ces figures illustrent la comparaison entre le calcul approché et la solution plus précise à l'aide de l'équation de
Reynolds. De plus, l'influence de la rotation sur la valeur caractéristique de la portance et l'angle de calage peut
être explicitée.
Pour le calcul d'un palier géométriquement similaire, il est possible de déterminer l'épaisseur minimale du film
d'huile lorsque les valeurs sont données, par exemple pour F, B, D, p , ω, ψ et η (détermination de η
en B B
conformément à 5.6, le cas échéant).
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ISO 12168-1:2001(F)
Tous les paramètres sont donnés pour la détermination de F* selon l'équation (1) et de p selon l'équation (2). Pour
f
ce type de géométrie, les valeurs correspondantes pour e et b peuvent être obtenues à partir des Figures 1 et 2 de
l'ISO 12168-2:2001, et l'on obtient h = C ¥ (1 - e).
min R
Selon la méthode d'approximation décrite dans l'annexe A, cela entraîne une dépendance de la valeur
caractéristique de la portance utile, constituée de ce qu'on appelle la «largeur effective des paliers» B - l ,
ax
F
*
F = (3)
eff
B-¥lD¥p
()
ax en
vis-à-vis d'un nombre réduit de paramètres. Dans le cas de cette définition, le rapport largeur/diamètre B/D peut
plus particulièrement être réduit en tant que paramètre. On maintient toutefois le nombre d’alvéoles Z, le rapport de
résistance
ʈ
ll
ax ax
¥- 1
2
Á˜
R
lan,ax ¥ BZBB˯
lb ʈ
ax c
k = == ¥ ¥ (4)
Á˜
˯
RD ¥ πlD
lb
lan,c cax c
l'excentricité relative du tourillon e et le paramètre selon la vitesse, qui détermine le rapport montée en pression
hydrodynamique/hydrostatique:
h ¥ w
ll
ccB
=¥pk ¥ x = ¥ k ¥ x (5)
K
rot f
2
D D
p y
en
*
Si, de plus, l'on tire avantage du fait que la fonction F (e) est quasi linéaire pour e u 0,5, alors il suffit, dans la
eff
pratique, de connaître la fonction
*
(0ek==,4) fZ ( , ,K )
F
eff rot
pour calculer la portance.
Dans la Figure 3 de l'ISO 12168-2:2001, la fonction
**
(ee==0,4) (= 0,4); ( = 0)=fZ ( , k )
FF K
eff,0 eff rot
et, dans la Figure 4, la fonction
*
F
eff
==fZ (4, k ,K )
rot
*
F
eff,0
sont illustrées pour le cas de la «charge appliquée dans la direction de l'axe d’alvéole». L'augmentation sous
condition hydrodynamique de la portance peut être parfaitement reconnue lorsqu'elle est illustrée de cette manière.
*
Si, par exemple, Z et tous les paramètres sont donnés pour la détermination de F selon l'équation (3), de k selon
eff
l'équation (4), et de K selon l'équation (5), l'épaisseur minimale du film d'huile qui se développe en cours de
rot
fonctionnement peut être déterminée.
*
Après calcul de k et K , la valeur F (e = 0,4) est donnée à la Figure 3 de l'ISO 12168-2:2001 et la valeur
rot eff,0
* * *
F / F (e = 0,4) est donnée à la Figure 4 de l'ISO 12168-2:2001; F étant calculée selon l'équation (3) et
eff eff, 0 eff
avec
© ISO 2001 – Tous droits réservés 7
---------------------- Page: 12 ----------------------
ISO 12168-1:2001(F)
*
0,4 ¥F
eff
e =
*
**
¥= ( ee0,4) ¥ F (= 0,4)
FF
()eff eff,0 eff,0
on obtient l'épaisseur minimale du film d'huile suivante:
h = C ¥ (1 - e)
min R
5.3 Débit de lubrifiant et puissance de pompage
La valeur caractéristique du débit de lubrifiant est la suivante:
Q ¥ h
* B
Q = (6)
3
Cp¥
Ren
Elle dépend uniquement sensiblement de l'excentricité relative du tourillon e, de la direction de la charge par
rapport au palier et de la pression de frottement relative p ou du paramètre selon la vitesse K .
f rot
Par approximation, le débit de lubrifiant peut être calculé de la manière suivante (voir également A.3.3):
11π
**
(7)
QQ ( 0eeu ,5) ª¥ ( = 0)= ¥
16 + x B D l B
( )
ax
6 ¥¥h l
R
cp ax
B
où x = et = .
R
P,0
3
¥
R bC
P,0
ax
R
La résistance à l'écoulement des capillaires selon l'équation A.3.2.2 est la suivante:
128¥¥h l
cp cp
R = ¥+1 a
( )
cp
4
π ¥ d
cp
avec la part non linéaire (facteur d'inertie)
1, 08 4¥¥Q r
a=¥
32 h ¥¥lZ
cp cp
En convertissant l'équation (6) avec ce dernier facteur, on peut calculer le débit de lubrifiant lorsque les paramètres
η , C , p , ξ, B/D et l /B sont connus.
B R en ax
*
Pour des paliers optimisés, la valeur de Q doit être celle donnée dans le Tableau 1 de l'ISO 12168-2:2001. La
puissance de pompage est la suivante (sans tenir compte du rendement de la pompe):
23
p ¥ C
* en R
PQ=¥p =Q¥ (8)
pen
h
B
* *
Selon la méthode d'approximation, Q est déterminé de nouveau selon l'équation (7). Ainsi, Q est la valeur
caractéristique du débit et de la puissance de pompage.
8 © ISO 2001 – Tous droits réservés
---------------------- Page: 13 ----------------------
ISO 12168-1:2001(F)
5.4 Puissance de frottement
La valeur caractéristique de la puissance de frottement est la suivante:
PC¥
*
fR
P = (9)
f
2
h¥¥UB¥D
B
Le frottement se produit dans les butées ainsi que dans la surface d’alvéole. La surface d'appui par rapport à la
surface totale du palier p ¥ B ¥ D est la suivante:
llZ ʈl
* ax c ax
A =¥21+ ¥ ¥ -2¥
lan
Á˜
B πDB˯
Selon la méthode d'approximation, la val
...
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