ISO 31657-1:2025

International
Standard
ISO 31657-1
First edition
Plain bearings — Hydrodynamic
2025-10
plain journal bearings under
steady-state conditions —
Part 1:
Calculation of multi-lobed and
tilting pad journal bearings
Paliers lisses — Paliers lisses hydrodynamiques radiaux
fonctionnant en régime stabilisé —
Partie 1: Calcul pour les paliers radiaux multilobés et à patins
oscillants
Reference number
© ISO 2025
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Published in Switzerland
ii
Contents Page
Foreword .iv
Introduction .v
1 Scope . 1
2 Normative references . 1
3 Terms and definitions . 1
4 Symbols and units. 1
5 General principles, assumptions and preconditions . 7
6 Calculation method .12
6.1 General . 12
6.2 Load-carrying capacity .14
6.3 Frictional power .14
6.4 Lubricant flow rate .14
6.5 Heat balance . . 15
6.6 Maximum lubricant film temperature .17
6.7 Maximum lubricant film pressure .18
6.8 Operating states .18
6.9 Further influencing parameters .18
6.10 Stiffness and damping coefficients .19
Annex A (informative) Calculation examples .24
Bibliography .38

iii
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
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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).
ISO draws attention to the possibility that the implementation of this document may involve the use of (a)
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This document was prepared by Technical Committee ISO/TC 123, Plain bearings, Subcommittee SC 8,
Calculation methods for plain bearings and their applications.
This first edition of ISO 31657-1 cancels and replaces ISO/TS 31657-1:2020, which has been technically
revised.
The main changes are as follows:
— Clause 7 has been deleted;
— correction of typographical errors.
A list of all parts in the ISO 31657 series can be found on the ISO website.
Any feedback or questions on this document should be directed to the user’s national standards body. A
complete listing of these bodies can be found at www.iso.org/members.html.

iv
Introduction
The aim of this document is the operationally-safe design of plain journal bearings for medium or high
journal circumferential velocities, U , up to approximately 90 m/s by applying a calculation method for oil-
J
lubricated hydrodynamic plain bearings with complete separation of journal and bearing sliding surfaces by
a lubricating film.
For low circumferential velocities up to approximately 30 m/s, circular cylindrical bearings are normally
applied. For these bearings, a similar calculation method is given in ISO 7902-1, ISO 7902-2 and ISO 7902-3.
Based on practical experience, the calculation procedure is usable for application cases where specific
bearing load times circumferential speed, pU⋅ , does not exceed approximately 200 MPa·m/s.
J
This document discusses multi-lobed journal bearings with two, three and four equal, symmetrical sliding
surfaces, which are separated by laterally-closed lubrication pockets, and symmetrically-loaded tilting-pad
journal bearings with four and five pads. Here, the curvature radii, R , of the sliding surfaces are usually
B
chosen larger than half the bearing diameter, D, so that an increased bearing clearance results at the pad ends.
The calculation method described here can also be used for other gap forms, for example asymmetrical
multi-lobed journal bearings like offset-halves bearings, pressure-dam bearings or other tilting-pad journal
bearing designs, if the numerical solutions of the basic formulae are available for these designs.
For a reliable evaluation of the results of this calculation method, it is indispensable to consider the
physical implications of these assumptions as well as practical experiences for instance from temperature
measurements carried out on real machinery under typical operating conditions. Applied in this sense, this
document presents a simple way to predict the approximate performance of plain journal bearings for those
unable to access more complex and accurate calculation techniques.
Unsteady operating states are not recorded. The stiffness and damping coefficients of the plain journal
bearings required for the linear vibration and stability investigations are indicated in ISO 31657-2 and
ISO 31657-3.
NOTE Equivalent calculation procedures exist that enable operating conditions to be estimated and checked
against acceptable conditions. Another calculation procedure is equally admissible.

v
International Standard ISO 31657-1:2025(en)
Plain bearings — Hydrodynamic plain journal bearings under
steady-state conditions —
Part 1:
Calculation of multi-lobed and tilting pad journal bearings
1 Scope
This document specifies the general principles, assumptions and preconditions for the calculation of multi-
lobed and tilting-pad journal bearings by means of an easy-to-use calculation procedure based on numerous
simplifying assumptions.
The calculation method applies to the design and optimisation of plain bearings, for example in turbines,
compressors, generators, electric motors, gears and pumps. It is restricted to steady-state operation, i.e. in
continuous operating states the load according to size and direction and the angular velocity of the rotor are
constant.
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 31657-2, Plain bearings — Hydrodynamic plain journal bearings under steady-state conditions — Part 2:
Characteristic values for calculation of multi-lobed journal bearings
ISO 31657-3, Plain bearings — Hydrodynamic plain journal bearings under steady-state conditions — Part 3:
Characteristic values for calculation of tilting pad journal bearings
ISO 31657-4, Plain bearings — Hydrodynamic plain journal bearings under steady-state conditions — Part 4:
Permissible operational parameters for calculation of multi-lobed and tilting pad journal bearings
3 Terms and definitions
No terms and definitions are listed in this document.
ISO and IEC maintain terminology databases for use in standardization at the following addresses:
— ISO Online browsing platform: available at https:// www .iso .org/ obp
— IEC Electropedia: available at https:// www .electropedia .org/
4 Symbols and units
Table 1 contains the symbols used in the ISO 31657 series.

Table 1 — Symbols and units
Symbol Description Unit
B Bearing width m
B
*
*
Relative bearing width, width ratio as given by: B =
B
D
b Width of lubricant pocket m
P
b
* P
*
b Relative width of lubricant pocket, as given by: b =
P P
B
Bearing radial clearance, as given by: CR=−R
C
m
RJ
R
C
Effective radial bearing clearance m
R,eff
c Stiffness coefficient of lubricant film (i,k = 1,2) N/m
ik
Non-dimensional stiffness coefficient of lubricant film, as given by:
*
ψ
c
* eff
ik
c = ⋅=ci(),,k 12
ik ik
2⋅⋅B ηω⋅
eff
c
Specific heat capacity (p = constant) J/(kg K)
p
D Nominal bearing diameter (inside diameter of journal bearing) m
D Maximum value of D m
max
D Minimum value of D m
min
D
Journal diameter (diameter of the shaft section located inside of a journal bearing) m
J
D Maximum value of D
m
Jm, ax J
D Minimum value of D
m
Jm, in J
d Damping coefficient of lubricant film (i,k = 1,2) N s/m
ik
Non-dimensional damping coefficient of lubricant film, as given by:
*
ψ 1
d
* eff
ik
d = ⋅⋅ω di(),,k =12
ik ik
2⋅⋅B ηω⋅
eff
e Eccentricity (distance between journal and bearing axis) m
Eccentricity of the bearing sliding surfaces (pads) of a multi-lobed or tilting-pad journal bear-
e m
B
ing
F Bearing force, bearing load, nominal bearing load, load-carrying capacity N
F Friction force, as given by: Ff=⋅F N
f f
f
*
*
Friction force parameter, as given by: F =⋅So 1
F
f
f
ψ
eff
F
Bearing force at transition to mixed friction N
tr
f Coefficient of friction 1
f
Journal deflection m
J
h(φ) Local lubricant film thickness m
h()ϕ
*
*
Relative local lubricant film thickness, as given by: h ()ϕ = 1
h ()ϕ
C
R
h Minimum admissible lubricant film thickness at transition to mixed friction m
lim,tr
Minimum admissible relative lubricant film thickness at transition to mixed friction, as given
* h
lim,tr
h * 1
lim,tr
by: h =
lim,tr
C
R,eff
TTabablele 1 1 ((ccoonnttiinnueuedd))
Symbol Description Unit
h Minimum lubricant film thickness, minimum gap m
min
h
* min
Minimum relative lubricant film thickness, minimum relative gap, as given by: h =
min
*
C
h
R,eff
min
h
Minimum lubricant film thickness at transition to mixed friction m
min,tr
Minimum relative lubricant film thickness at transition to mixed friction, as given by:
* h
min,tr 1
h *
min,tr
h =
min,tr
C
R,eff
Local gap at ε = 0 , gap function
h ()ϕ m
h ()ϕ
* 0
Relative local gap at ε = 0 , profile function, as given by: h ()ϕ =
* 0
h ()ϕ C
R
h
Maximum gap at ε = 0 m
0,max
h
0,max
*
Maximum relative gap at ε = 0 , gap ratio, as given by: h =
*
0,max
h
C
0,max
R
Profile factor (relative difference between lobe or pad bore radius and journal radius), as given
ΔR 1
K B 1
P
by: K ==
P
Cm1−
R
K
Effective profile factor 1
P,eff
K
Profile factor at 20 °C 1
P,20
M Mixing factor 1
m Preload factor, preload of bearing or pad sliding surface 1
−1
N Rotational speed (rotational frequency) of the rotor (revolutions per time unit) s
−1
N Critical speed (critical rotational frequency) s
cr
Rotational speed (rotational frequency) at the stability speed limit of the rotor supported by
−1
N s
lim
plain bearings
−1
N Resonance speed (resonance rotational frequency) of the rotor supported by plain bearings s
rsn
Rotational speed (rotational frequency) at transition to mixed friction, transition rotational
−1
N
s
tr
speed, transition rotational frequency
O Centreline of plain bearing 1
B
O
Centreline of sliding surface No. i 1
i
O
Centreline of journal 1
J
Frictional power, as given by: PF=⋅U
P W
ff J
f
P
Heat flow via the lubricant W
th,L
p Lubricant film pressure, local lubricant film pressure Pa
F
p Pa
Specific bearing load, as given by: p =
BD⋅
p Lubricant supply pressure Pa
en
p ⋅ψ
* * en eff
Lubricant supply pressure parameter, as given by: p =
p
en
en
ηω⋅
eff
p Maximum admissible lubricant film pressure Pa
lim
TTabablele 1 1 ((ccoonnttiinnueuedd))
Symbol Description Unit
p
Maximum admissible specific bearing load at transition to mixed friction Pa
lim,tr
p Maximum lubricant film pressure Pa
max
p
* max
*
Maximum lubricant film pressure parameter, as given by: p = 1
p
max
max
p
F
tr
p Pa
Specific bearing load at transition to mixed friction, as given by: p =
tr
tr
BD⋅
Lubricant flow rate, as given by: QQ=+ Q
Q m /s
3p
Q Minimum admissible lubricant flow rate m /s
lim
Q
Lubricant flow rate due to supply pressure m /s
p
Q
p
*
*
Lubricant flow rate parameter due to supply pressure, as given by: Q =
Q 1
p
p
*
pQ⋅
en 0
3 3
Q m /s
Reference value of Q, as given by: QR=⋅ωψ⋅
0 eff
Q Lubricant flow rate at the entrance into the lubrication gap (circumferential direction) m /s
Lubricant flow rate at the exit of the lubrication gap (circumferential direction), as given by:
Q
m /s
QQ=− Q
21 3
Lubricant flow rate parameter at the exit of the lubrication gap (circumferential direction), as
* Q
* 2 1
Q
given by: Q =
Q
Q Lubricant flow rate due to hydrodynamic pressure build-up (side flow rate) m /s
Lubricant flow rate parameter due to hydrodynamic pressure build-up (side flow parameter),
*
Q
* 3
Q
as given by: Q =
Q
D
R m
Journal bearing inside radius, as given by: R=
R
Lobe or pad bore radius of a multi-lobed or tilting-pad journal bearing m
B
Journal radius (radius of the shaft section located inside of a journal bearing), as given by:
R D
m
J J
R =
J
Rz Surface finish ten-point average of bearing sliding surface m
B
Rz
Surface finish ten-point average of journal sliding surface m
J
ρω⋅⋅RC⋅
R,eff
Re Reynolds number, as given by: Re= 1
η
eff
Re Critical Reynolds number 1
cr
F⋅ψ
eff
So Sommerfeld number, as given by: So= 1
BD⋅⋅ηω⋅
eff
So Sommerfeld number at transition to mixed friction 1
tr
S Displacement amplitude of the rotor (mechanical oscillation) m
T Temperature °C
t Time s
T Bearing temperature °C
B
T Effective temperature of lubricant film °C
eff
T Lubricant temperature at the bearing entrance °C
en
TTabablele 1 1 ((ccoonnttiinnueuedd))
Symbol Description Unit
T Lubricant temperature at the bearing exit °C
ex
T
Journal temperature °C
J
T
Maximum admissible bearing temperature °C
lim
T Maximum temperature of lubricant film °C
max
T Lubricant temperature at the entrance into the lubrication gap (circumferential direction) °C
T Lubricant temperature at pressure profile trailing edge (circumferential direction) °C
Circumferential speed of the journal, sliding velocity
U
m/s
J
UR=⋅ω
JJ
U
Minimum admissible circumferential speed at transition to mixed friction m/s
lim,tr
U
Circumferential speed at transition to mixed friction m/s
tr
u Velocity component in the φ-direction m/s
u
Average velocity component in the φ-direction m/s
w Velocity component in the z- direction m/s
w Average velocity component in the z-direction m/s
x Coordinate of journal radial motion, normal to direction of load m
Relative coordinate of journal radial motion, normal to direction of load, as given by:
*
x
*
x
x =
C
R
Coordinate normal to sliding surface (across the lubricant film, in the radial direction); coordi-
y m
nate of journal radial motion, in direction of load
y
*
*
Relative coordinate of journal radial motion, in direction of load, as given by: y =
y
C
R
y Coordinate normal to sliding surface (across the lubricant film) m
h
Z Number of sliding surfaces (pads), number of pockets per bearing 1
Coordinate parallel to the sliding surface, normal to direction of motion (normal to circumfer-
z m
ential direction, in the axial direction)
−1
α
Linear thermal expansion coefficient of bearing material K
lB,
−1
α
Linear thermal expansion coefficient of journal material K
lJ,
β Attitude angle (angular position of journal eccentricity related to the direction of load) °
β
Angle between direction of load and position of minimum lubricant film thickness °
hm, in
ΔF Component of additional dynamic force in x-direction N
x
ΔF
Component of additional dynamic force in y-direction N
y
Component of additional dynamic force parameter in x-direction, as given by:
*
ΔψF ⋅ 1
ΔF
* xeff
x
ΔF =
x
BD⋅⋅ηω⋅
eff
Component of additional dynamic force parameter in y-direction, as given by:
*
ΔψF ⋅ 1
ΔF
yeff
y
*
ΔF =
y
BD⋅⋅ηω⋅
eff
Difference between lobe or pad bore radius and journal radius, as given by: ΔRR=− R
ΔR m
BB J
B
ΔT
Heating of lubricant between bearing entrance and exit, as given by: ΔTT=−T K
ex en
ΔT Maximum admissible heating of lubricant between bearing entrance and exit K
lim
TTabablele 1 1 ((ccoonnttiinnueuedd))
Symbol Description Unit
Difference between maximum temperature of lubricant film and lubricant temperature in the
ΔT
K
max
lubricant pocket, as given by: ΔTT=−T
maxmax 1
Non-dimensional difference between maximum temperature of lubricant film and lubricant
*
ρψ⋅⋅c
peff 1
ΔT *
max
temperature in the lubricant pocket, as given by: ΔΔT = ⋅ T
max max
pf⋅
Difference between lubricant temperature at the entrance into the lubrication gap and
ΔT
K
lubricant temperature at the bearing entrance, as given by: ΔTT=−T
11 en
Difference between lubricant temperature at pressure profile trailing edge and lubricant
ΔT
K
temperature at the entrance into the lubrication gap, as given by: ΔTT=−T
22 1
δ
Journal misalignment angle (angular deviation of journal) °
J
e
Relative eccentricity: ε =
ε 1
C
Re, ff
η Dynamic viscosity of the lubricant Pa∙s
η Effective dynamic viscosity in the lubricant film Pa∙s
eff
ρ Density of the lubricant kg/m
φ Angular coordinate in circumferential direction °
ϕ Angular coordinate of pivot position of pad (tilting-pad bearing) °
F
ϕ Angular coordinate of lubricant pocket centreline °
P
Angular coordinate of bearing sliding surface (segment or pad) centreline at multi-lobed or
ϕ °
tilting-pad journal bearings (with non-tilted pads), see Figure 1, a)
ϕ Angular coordinate at the entrance into the gap °
ϕ Angular coordinate at the end of the hydrodynamic pressure build-up °
ϕ Angular coordinate at the exit of the gap °
C
R
ψ ‰
Relative bearing clearance, as given by: ψ =
R
Tolerance of ψ, as given by: Δψψ=−ψ
Δψ ‰
maxmin
ψ Effective relative bearing clearance ‰
eff
ψ Maximum value of ψ ‰
max
ψ Minimum value of ψ ‰
min
Δψ Thermal change of ψ ‰
th
ψ
Relative bearing clearance at 20 °C ‰
Ω Angular span of bearing sliding surface (segment or pad), as given by: Ω =−ϕϕ °
Angular distance between leading edge and pivot position of pad (tilting-pad bearing), as given
Ω °
F
by: Ω =−ϕϕ
FF 1
Relative angular distance between leading edge and pivot position of pad (tilting-pad bearing),
*
Ω
*
F
as given by: ΩΩ= /Ω
FF
360°
Ω °
Angular span of lubricant pocket, as given by: ΩΩ= −
P
P
Z
−1
ω Angular speed of the rotor, as given by: ωπ=⋅2 ⋅ N s
−1
ω Angular speed at transition to mixed friction s
tr
5 General principles, assumptions and preconditions
The bearing bore form of multi-lobed journal bearings [see Figure 1, a)] and tilting-pad journal bearings
h ϕ
()
* 0
[with non-tilted pads according to Figure 1, b)] is described by the profile function h ()ϕ = in the
C
R
e
case of a centric journal position ε ==0 . The angle φ is counted, starting from the load direction, in the
C
R
journal rotational direction.
Formula (1) applies to the shell segment or pad i with the angular length Ω =−ϕϕ :
ii31,,i
ΔΔR R
 
* B B
h ()ϕϕ=+ −11⋅−cos()ϕ ,,i = .,Z (1)
00,ii  ,
C C
 
R R
with the profile factor
RR−
ΔR e
BJ
B B
K == =+1 ,
P
C C C
R R R
minimum clearance
DD−
J
CR=−R =
RJ
and the lubricant film thickness ratio as given by Formula (2):
h
* * 0,max
hhϕ == (2)
()
00Pm,,i ax
C
R
Here, the position of the sliding surface (segment or pad) axis (curvature centre "point") of the shell segment
or pad i is uniquely described by the sliding surface eccentricity e and the associated angle coordinate ϕ .
B 0,i
*
In the case of cylindrical bearings, K = 1 and h ϕ =1 .
()
P 0
NOTE Instead of the profile factor, K the "preload factor", m, is frequently used internationally; the following
P,
relation exists between both variables:
K =
P
1−m
In the case of an eccentric position of the journal (ε, β), Formula (3) applies to the lubricant film thickness,
h(φ), of the multi-lobed journal bearings [(see Figure 1, c)]:
**

hC()ϕϕ=⋅hC()=⋅ h ()ϕε−⋅cos(]ϕβ− (3)
)
RR 0

In the case of tilting-pad journal bearings [see Figure 1, d)], the individual pads automatically adjust
themselves (optimally) so that the lubricant film force F passes through the supporting pad pivot,
i
[9]
respectively. For a more precise calculation of tilting-pad journal bearings, the elasticities in the pad
support and the elastic and thermal deformations of the pads shall be considered.
The pressure formation in the lubrication gaps is basically calculated with the numerical solutions of the
Reynolds differential equation for a finite bearing width:
1 ∂ ∂p ∂ ∂p ∂h
 
33 
⋅ h ⋅ + h ⋅ =⋅6 ηω⋅⋅ (4)
 
 
∂ϕϕ∂ ∂z  ∂z  ∂ϕ
 
R
J
with ωπ=⋅2 ⋅ N angular speed of the rotor.

For derivation of the Reynolds differential equation, reference is made to Reference [5], for the numerical
solution to Reference [6].
When solving Formula (4), the following idealising assumptions and preconditions are made, whose
[7]
permissibility shall be estimated according to Clause 6, if necessary.
a) The lubricant corresponds to a Newtonian fluid.
b) All flow processes of the lubricant are laminar.
c) The lubricant adheres fully to the sliding surfaces.
d) The lubricant is incompressible.
e) At the leading edge of the segment or pad, the lubrication gap is completely filled with lubricant.
f) Inertia effects, gravitation and magnetic forces of the lubricant are negligible.
g) The components forming the lubrication gap are rigid or their deformation is negligible; the surfaces of
the journal and bearing bore are ideal circular cylinders or cylindrical segments.
h) The curvature radii of the surfaces moving relative to one another are large in comparison to the
lubricant film thicknesses.
i) The lubricant film thickness in an axial direction (z coordinate) is constant.
j) Pressure changes in the lubricant film normal to the sliding surfaces (in the lubricant film thickness
direction) are negligible.
k) A movement normal to the sliding surfaces (in the lubricant film thickness direction) is not considered
here, in contrast to 6.10.
l) The lubricant film is isoviscous in the entire lubrication gap.
m) The lubricant is supplied at the leading edge of the segments or pads respectively; the level of the supply
pressure is negligible compared to the lubricant film pressures themselves.
The boundary conditions for the lubricant film pressure build-up satisfy the continuity condition.
The following applies respectively to the individual segments or pads (see Figures 2 and 3):
— at the lateral bearing edge pzϕ,/=±B 20= ;
()
— in the lubrication pocket and on the sealing land pz()ϕ, =0 ;
∂p
   
— at the pressure profile trailing edge pzϕ (),,z = ϕ ()zz =0 ;
   
∂ϕ
   
∂p ∂p
— at the beginning of cavitation area pzϕ (),,z = ϕϕ()zz = ()zz, =0 ;
[] [] []
∂ϕ ∂z
— at the end of cavitation area pzϕ ,0z = .
[]()
[12][13]
The cavitation theory according to Jakobsson, Floberg and Olsson is used in the cavitation area and on
its edge for fulfilment of the continuity condition.
The numerical integration of the Reynolds differential equation is done using the transformation of the
pressure proposed in Reference [6] by conversion into a difference formula, which is applied to a grid of
nodal points and which leads to a system of linear formulas.
After specifying the boundary conditions, the integration yields the pressure profile in the circumferential
and axial direction.
The maximum lubricant film temperature is calculated using the numerical solution of the energy equation
averaged by integration with respect to the lubricant film thickness, h
 
h
u ∂T ∂T η 1  ∂u   ∂w 
⋅ +⋅w = ⋅⋅  + ⋅⋅dy (5)
    h
∫
R ∂ϕ ∂zcρ⋅ h ∂y ∂y
 
Jp  hh  
 
[6] [10] [11]
for the two-dimensional temperature distribution T(φ, z).
This includes
hh
u =⋅ ud⋅=yw, ⋅⋅wdy
hh
∫∫
h h
the flow rates averaged over the lubricant film thickness h in the circumferential and axial direction.
When deriving the energy equation, see Formula (5), it is also assumed besides the above preconditions that
no heat is dissipated from the lubrication gap by thermal conduction (adiabatic calculation).
When solving Formula (5) the following boundary conditions apply (see Figure 3):
— at the entrance gap Tzϕ , =T ;
()
∂T
— in the axial bearing centre ()ϕ,0z= =0 .
∂z
The numerical integration of Formula (5) is carried out similar to the solution of the Reynolds differential
equation, see Formula (4), using a suitable difference formula and yields for the specified boundary
conditions the temperature distribution in the circumferential and axial direction.
The application of the similarity principle in the hydrodynamic plain bearing theory leads to dimensionless
similarity variables for the interesting characteristic values (such as load-carrying capacity, frictional
power, lubricant flow rate and relative bearing width). Use of the similarity variables reduces the number of
necessary numerical solutions of the Reynolds differential equation, Formula (4), and the energy equation,
Formula (5), which are summarised in ISO 31657-2 and ISO 31657-3.
As a rule, other solutions can also be used, insofar as they satisfy the conditions indicated in this document
and a corresponding numerical accuracy.
ISO 31657-4 contains operational guide values for checking the calculation results, in order to ensure the
functionality of the plain bearings.
In special use cases, operational guide values different from ISO 31657-4 can be agreed.

a) Multi-lobed journal bearing c) Lubrication gap geometry of the multi-lobed
journal bearing in eccentric position of the jour-
nal
b) Tilting-pad journal bearing (with non-tilting d) Lubrication gap geometry of the tilting-pad
pads) journal bearing (with non-tilting pads) in eccen-
tric position of the journal
*
DD− C C
j RR hC()ϕϕ=×h ()
00R
ψ ==
C = =−RR
R j
D
R
*
ΔRR=− R RR−
BB j
Bj hC=×h
00,,maxmR ax
K =
p
C
R
*
D e
hCϕϕ=×h
() ()
R
eR=− =−RR ε =
BB B
2 C
R
Figure 1 — Bearing bore shape of multi-lobed journal bearing, tilting-pad journal bearing (with non-
tilting pads) and lubrication gap geometry of the same bearings in eccentric position of the journal

a) Segment centre b) Lubrication pocket centre
Figure 2 — Pressure distribution p(φ, z) in the lubricating gap of a multi-lobed journal bearing
loaded on the segment centre and the lubrication pocket centre

Key
1 segment or pad
2 journal
3 pocket
4 in
5 out
Figure 3 — Distribution of lubricant film temperature, lubricant film pressure and lubricant film
thickness in bearing centre (z = 0) as well as lubricant flow rate and heat balance in the lubrication
pocket and lubrication gap (schematic diagram)
6 Calculation method
6.1 General
Calculation refers to the mathematical determination of the functional capability based on operational
characteristic values (see Figure 4), which are to be compared with permissible operational parameter values.
The operational characteristic values determined in different operating states shall be permissible with their
permissible operational parameter values. All continuous operating states shall be examined for this.
The safety against wear is ensured when a complete separation of the sliding partners is attained by the
lubricant. Continuous operation in the mixed friction area leads to premature functional incapability. Short-
term operation in the mixed friction area, for example when starting up and running down of machines
with slide bearings, is unavoidable and does not usually lead to bearing damage. At high loads, a hydrostatic
jacking can be required during slow start-up or run-down. Running-in and adapting wear for compensating
the surface form deviations from the ideal form are permissible as long as these occur with local and time
restrictions and without signs of overload.

The limits of the mechanical load are given by the strength of the bearing material. Minor plastic
deformations are permissible as long as they do not impair the functional capability of the plain bearing.
The limits of the thermal load result from the high-temperature strength of the bearing material, but also
from the viscosity temperature dependence and the tendency of the lubricant to age.
The calculation of the functional capability of plain bearings presupposes that the operating conditions
are known for all continuous operating states. In practice, however, additional disturbing influences
frequently occur, which are still unknown during the design and which are also not always accessible to a
mathematical approach. It is therefore recommended to work with a corresponding safety interval between
the operational characteristic values and the permissible operating parameter values. Disturbing influences
are, for example:
— disturbing forces (e.g. imbalances, vibrations);
— form deviations from the ideal geometry (e.g. operational deformations, production tolerances, assembly
deviations);
— lubricant impurities due to solid, liquid and gaseous foreign bodies;
— corrosion, electro-erosion.
Information on some further influencing variables is given in 6.9.
The applicability of this document, in which a laminar flow in the lubrication gap is presupposed, shall be
[8][9]
checked by the Reynolds number:
ρω⋅⋅RC⋅
41,3
Re, ff
Re= ≤≈Re (6)
cr
η
K ⋅ψ
eff
Peff
C
Re, ff
with ψ = effective relative bearing clearance.
eff
R
In the case of plain bearings with Re > Re (e.g. due to high circumferential velocities), higher power losses
cr
[7] [8] [9]
shall be expected. The load-carrying capacity can rise.
Bearings with turbulent flow can only be calculated approximately according to this document.
The plain bearing calculation grasps the following, based on the known bearing dimensions and operating data:
— the relation between bearing load-carrying capacity and lubricant film thickness;
— the frictional power;
— the lubricant flow rate;
— the heat balance;
— the maximum lubricant film temperature and the maximum lubricant film pressure;
all these interacting with one another. The solution happens in an iterative process whose sequence is
summarised in the calculation flow chart according to Figure 4.
A parameter variation can be performed for the optimisation of individual parameters. It is possible to
modify the calculation procedure

6.2 Load-carrying capacity
Characteristic for the load-carrying capacity is the (non-dimensional) Sommerfeld number:
F⋅ψ
eff
So= (7)
BD⋅⋅ηω⋅
eff
e B
*
whose dependence on the relative eccentricity ε = , the relative bearing width B = and the profile
C D
Re, ff
*
function h ()ϕ is indicated in ISO 31657-2 and ISO 31657-3. The state variables η , ψ consider thermal
0 eff eff
influences (see 6.5 and 6.9).
*
From this, with the attitude angle β [So, B*, h ()ϕ ] according to ISO 31657-2 and ISO 31657-3, the components
of the static bearing flexibility ε cos β, ε sin β depending on the static load parameter So can be determined.
In the case of multi-lobed and tilting-pad journal bearings, the static displacement e and minimum lubricant
film thickness h add up vectorially to the radial journal mobility (see Figure 5). The dependence indicated
min
h
** * min
 
in ISO 31657-2 and ISO 31657-3hSoB,,h ϕ = yields through comparison with the permissible
()
min 0
 
C
R,eff
h
lim,tr
*
operational parameter valueh = the load-carrying capacity at the transition to mixed friction.
lim,tr
C
R,eff
6.3 Frictional power
The losses due to frictional power in a hydrodynamic plain bearing shall be determined by the dimensionless
*
friction force, F , (or the friction coefficient, f ):
f
f
*
F =⋅So (8)
f
ψ
eff
*
whose dependence on So, B* and h ϕ shall be as indicated in ISO 31657-2 and ISO 31657-3. Here, it is
()
assumed that the lubricant supply pressure, p , remains very low and, in cavitation areas, the friction force
en
[12] [13]
has a linear dependence on calculated degree of filling.
The friction power in the bearing or the heat flow caused by it is
PF=⋅U (9)
ff J
with Ff=⋅F friction force and UR=⋅ω circumferential speed of the journal.
f JJ
NOTE Particularly in the lubricant pockets of multi-lobed journal bearings and between the pads of tilting-pad
journal bearings already at moderate circumferential speeds, turbulent flow can occur, leading to increasing power
losses in these areas not taken into account in the calculation procedure described in this document. In addition,
depending on the geometrical design of the lubricant supply and lubricant scraper elements (if applicable), churning
losses can arise in the cavities between the tilting-pads. The extent of these power losses not considered in this
calculation method depend also on the filling degree of the cavities (see 6.5).
6.4 Lubricant flow rate
The lubricant supplied to the bearing via lubrication pockets forms a load-carrying lubricant film for
separating the sliding surfaces. The pressure formation in the lubricant film forces the lubricant out from
the sides of the bearing.
[6] [10]
This is the fraction, Q , of lubricant flow rate due to inherent pressure formation:
*
QQ=⋅Q (10)
30 3
D
with QU=⋅C ⋅=R ⋅⋅ωψ reference lubricant flow rate
0 JR,eff eff
* *
and Q = f [So, B*, h ()ϕ ] according to ISO 31657-2 and ISO 31657-3.
3 0
The lubricant supply pressure, p , at the inlet into the bearing also forces lubricant out from the sides of
en
[10]
the plain bearing. This is the fraction, Q , of lubricant flow rate due to supply pressure:
p
**
QQ=⋅pQ⋅ (11)
pe0 np
* *
with Q = f [So, B*, h ()ϕ ] according to ISO 31657-2
p 0
p ⋅ψ
* en eff
and p = as lubricant supply pressure parameter.
en
ηω⋅
eff
The lubricant supply pressure, p is normally between 0,05 and 0,2 MPa (above the ambient pressure).
en,
In the case of tilting-pad journal bearings, the lubricant flow rate Q is normally set by throttling the supply
p
or discharge flow, or via corresponding nozzles (in case of injection lubrication).
The total lubricant flow rate is:
QQ=+ Q (12)
3p
For the (later) calculation of the lubricant temperature in the pockets, the lubricant flow rate, Q , which
enters in the circumference direction through the narrowest lubrication gap into the divergent gap, is also
required:
*
QQ=−QQ=⋅Q (13)
21 30 2
* *
with Q = f [So, B*, h ()ϕ ] according to ISO 31657-2 and ISO 31657-3.
2 0
Lubrication pockets as defined in this document are design elements for the distribution of the lubricant
over the bearing width. The recesses machined into the sliding surfaces of the bearing extend in the axial
direction and should be as short as possible in the circumferential direction.
*
Relative pocket widths should be bb=≤/,B 08 .
PP
Although greater values increase the oil flow rate, the oil escaping at the narrow throttling lands at the sides
does not take part in the heat dissipation. This applies more so if the side lands have axial grooves.
*
For the calculation of the flow rate fraction Q a relative pocket width of b = 0,8 is presupposed in this
p P
document. The effect of the lubricant inertia forces is not considered here.
The depth of the lubrication pockets is significantly greater than the bearing clearance.
6.5 Heat balance
The thermal state of the plain bearing results from the heat balance.
The heat flow resulting from the frictional power, P , in the bearing is dissipated to the surroundings via the
f
bearing housing and via the lubricant escaping from the bearing.

Pressure-lubricated multi-lobed and tilting-pad journal bearings (forced lubrication) primarily dissipate the
heat via the lubricant (recooling):
P = P (14)
f th,L
By neglecting the convective heat dissipation via the bearing housing, an additional safety results with the
design. Formula (15) applies for the heat dissipation by the lubricant:
Pc=⋅ρρ⋅⋅QT()−Tc=⋅ ⋅⋅QTΔ (15)
th,L pexenp
In the case of mineral lubricants, the volume-specific heat capacity is:
ρ⋅=c ()17,,…18 ⋅⋅10 J/ mK
()
p
In practice, with regard to the lubricant service life and/or the available cooling capacity of the lubricating
system, the heating of lubricant, ΔT, frequently has to be limited to a certain extent, ΔT , (e.g. 20 … 25 K) by
lim
increasing the total lubricant flow rate, Q, appropriately as shown in Formula (16):
P
f
Q = (16)
lim
ρ⋅⋅cTΔ
plim
Mixing processes in the lubrication pockets:
As a multi-lobed and tilting-pad journal bearing comprises several pads, it is necessary to consider not only
the lubricant flow rate of an individual pad, but that of the complete bearing and hence also the reciprocal
effect of the individual lubricant flow rate fractions. The lubricant escaping at the end of the segments or
pads is mixed with freshly supplied lubricant in the following oil pocket. This means that the lubricant
temperature, T , at the entrance of the lubrication gap is higher than that of lubricant freshly supplied with
the temperature, T (see Figure 3).
en
To simplify, the same temperature, T , and, when calculating it, for all segments or pads, an averaged oil
heating to the temperature, T , is presupposed for all oil pockets. When determining the temperature
difference as shown in Formula (17):
ΔTT=−T (17)
11 en
an empirical factor must be introduced, as a purely theoretical treatment of this mixing problem has not yet
led to satisfactory results.
For adaptation to the experience gained up to the present (see Reference [11]), it is possible via a heat balance
at the lubrication pockets (see Figure 3) to introduce a mixing factor, M, as follows:
Q
ΔΔT = ⋅ T (18)
1 2
MQ⋅+()1−MQ⋅
The limiting values are considered for explanation of the mixing factor. A mixing factor M = 0 means no
mixing in the lubrication pockets, i.e. the lubricant flow rate leaving the lubrication gaps Q fully enters into
the following lubrication gaps. As a result, a high lubricant flow rate, Q, would be ineffective, as the majority
of this freshly supplied lub
...