MITcalc - Hydrodynamic plain journal bearings

The program is designed for design and control of hydrodynamic radial plain bearing. In the program it is possible to solve other tasks such as lubricant viscosity, shaft and bearing fit tolerances, parameters of friction materials, calculation of bearing clearance, etc.

1. Design of radial plain bearing dimensions based on load and operating conditions
2. Control calculation of various types of bearings
3. Bearing clearance design and calculation of bearing and shaft tolerances
4. Proposal of suitable lubricant, oil database, viscosity calculations
5. Calculation of operating temperatures, coefficient of friction, lubricant flow and other parameters
6. Database of sliding materials and their parameters

The calculation is based on data, procedures and algorithms from specialized literature, standards and company catalogues.

[1] Strojně technická příručka (Svatopluk Černoch)
[2] Shigley’s Mechanical Engineering Design
[3] Strojírenská příručka (1992)
[4] Machinery’s Handbook (26th Edition)
[5] Části a mechanismy strojů II
[6] Tribologie (Jan Bečka)

Standards: ISO 7902, DIN 31652, ISO 2909, ISO 3448, ISO 3547, ISO 12129, ISO 4381, ISO 4382 , ISO 2909, ISO 3448, DIN 504

Company catalogues: SKF, FAG, INA, Timken, RBC Bearings Incorporated, Tecnamic GmbH...

Control, structure and syntax of calculations

Information on the project

Information on the purpose, use and control of the paragraph "Information on the project" can be found in the document "Information on the project".

Theory

The function of the plain bearing is based on the use of fluid friction between the shaft and bearing (or friction segment and pressure plate in the case of thrust bearings). When the shaft and bearing move together, the lubricant is drawn into the narrow gap between the bearing and shaft. This creates a thin film of lubricant that separates the bearing and shaft and causes fluid friction. The creation and maintenance of fluid friction is dependent on numerous parameters. Starting with dimensions, choice of materials, surface quality, choice of lubricant, etc. See figure (f...coefficient of friction, v...sliding speed).

1. Steady state / Dry friction
2. Low speed / Mixed friction
3. High speed / Liquid friction

The calculation is used to control (suggest) whether the bearing, under steady operating conditions, operates in the fluid friction range.

Due to the importance of plain bearings, this issue is addressed in a number of professional publications and standards. The calculation results are comparable for normal use. Therefore, we use the procedures defined by ISO 7902 - journal bearings, ISO 12131, ISO 12130 - plain thrust pad bearings (and related ISO standards) in our calculations. The ISO standard is comparable to DIN 31652, DIN 31654..... The calculations are extended with some additional parameters and recommendations based on professional and company literature.

EN	Table - Symbols and their designations
Symbol	Designation	Unit
	ISO 7902 - 1, 2
A	Area of heat-emitting surface (bearing housing)	m²
bG	Width of lubrication groove	m
bP	Width of lubrication pocket	m
B	Nominal bearing width	m
BH	Length of the axial housing	m
cp	Specific heat capacity of the lubricant	J/kg/°K
C	Nominal bearing clearance	m
CR,eff	Effective bearing radial clearance	m
dL	Lubrication hole diameter	m
D	Nominal bearing diameter (inside diameter)	m
DH	Outside diameter of the housing	m
DJ	Nominal shaft diameter	m
DJmax	Maximum value of DJ	m
DJmin	Minimum value of DJ	m
Dmax	Maximum value of D	m
Dmin	Minimum value of D	m
e	Eccentricity between the axis of the shaft and the bearing axis	m
f	Coefficient of friction in the loaded area of the lubricant film (f= Ff/F)	[~]
f'	Coefficient of friction in both the loaded and unloaded area of the lubricant film	[~]
F	Bearing force (nominal load)	N
Ff	Friction force in the loaded area of the lubricant film	N
Ff'	Frictional force in both the loaded and the unloaded area of the lubricant film	N
h	Local lubricant film thickness	m
heff	Effective lubricant film thickness	m
hG	Depth of lubrication groove	m
hlim	Minimum permissible lubricant film thickness	m
hmin	Minimum lubricant film thickness	m
hp	Depth of lubrication pocket	m
H	Length of the total height of the pedestal bearing	m
kA	Outer heat transmission coefficient	W/m²/°K
NB	Rotational frequency of the bearing	/s
NJ	Rotational frequency of the shaft	/s
NF	Rotational frequency of the force F	/s
p	Local lubricant film pressure	Pa
p'	Specific bearing load	Pa
pen	Lubricant feed pressure	Pa
plim	Maximum permissible lubricant film pressure	Pa
plim'	Maximum permissible specific bearing load	Pa
Pf	Frictional power	W
Pf'	Frictional power in both the loaded and the unloaded area of the lubricant film	W
Pth	Heat flow rate	W
Pth,amb	Heat flow rate to the ambient	W
Pth,f	Heat flow rate due to frictional power	W
Pth,L	Heat flow rate in the lubricant	W
qL	Coefficient related to lubricant flow rate due to feed pressure	[~]
qP	Coefficient related to lubricant flow rate from pocket	[~]
Q	Lubricant flow rate	m³/s
Q3	Lubricant flow rate due to hydrodynamic pressure	m³/s
Q3*	Lubricant flow rate parameter due to hydrodynamic pressure	[~]
Qp	Lubricant flow rate due to feed pressure	m³/s
Qp*	Lubricant flow rate parameter due to feed pressure	[~]
RzB	Average peak-to-valley height of bearing sliding surface	m
RzJ	Average peak-to-valley height of shaft mating surface	m
Re	Reynolds number	[~]
So	Sommerfeld number	[~]
Sou	Transition Sommerfeld number	[~]
Tamb	Ambient temperature	°C
TB	Bearing temperature	°C
TB,0	Assumed initial bearing temperature	°C
TB,1	Calculated bearing temperature resulting from iteration procedure	°C
Teff	Effective lubricant temperature	°C
Ten	Lubricant temperature at bearing entrance	°C
Tex	Lubricant temperature at bearing exit	°C
Tex,0	Assumed initial lubricant temperature at bearing exit	°C
Tex,1	Calculated lubricant temperature at bearing exit	°C
TJ	Shaft temperature	°C
Tlim	Maximum permissible bearing temperature	°C
TL'	Mean lubricant temperature	°C
UB	Linear velocity (peripheral speed) of bearing	m/s
UJ	Linear velocity (peripheral speed) of shaft	m/s
Va	Air ventilating velocity	m/s
x	Coordinate parallel to the sliding surface in the circumferential direction	m
y	Coordinate perpendicular to the sliding surface	m
z	Coordinate parallel to the sliding surface in the axial direction	m
αLB	Linear heat expansion coefficient of the bearing	/°K
αLJ	Linear heat expansion coefficient of the shaft	/°K
β	Attitude angle (angular position of the shaft eccentricity related to the direction of load)	°
ε	Relative eccentricity [ε = 2e/(D — DJ)]	[~]
εu	Transition eccentricity	[~]
η	Dynamic viscosity of the lubricant	Pa.s
ηeff	Effective dynamic viscosity of the lubricant	Pa.s
ν	Kinematic viscosity of the lubricant	m²/s
ξ	Coefficient of resistance to rotation in the loaded area of the lubricant film	[~]
ξ'	Coefficient of resistance to rotation in both the loaded and the unloaded area of the lubricant film	[~]
ξG	Coefficient of resistance to rotation in the area of circumferential groove	[~]
ξP	Coefficient of resistance to rotation in the area of the pocket	[~]
Rho	Density of lubricant	kg/m³
Rho20	Density of lubricant at 20°C	kg/m³
φ	Angular coordinate in the circumferential direction	rad
φ1	Angular coordinate of pressure leading edge	rad
φ2	Angular coordinate of pressure trailing edge	rad
ψ	Relative bearing clearance	[~]
ψ'	Mean relative bearing clearance	[~]
ψeff	Effective relative bearing clearance	[~]
ψmax	Maximum relative bearing clearance	[~]
ψmin	Minimum relative bearing clearance	[~]
ωB	Angular velocity of bearing	/s
ωF	Angular velocity of rotating force	/s
ωH	Hydrodynamic angular velocity	/s
ωJ	Angular velocity of shaft	/s
Ω	Angular span of bearing segment	°
ΩG	Angular span of lubrication groove	°
ΩP	Angular span of lubrication pocket	°
	ISO 7902 - 3
E'	Dimensionless modulus of elasticity	[~]
G'	Dimensionless shear modulus	[~]
hwav	Waviness of sliding surface	m
hwav,eff	Effective waviness of sliding surface	m
hwav,eff,lim	Maximum permissible effective waviness	m
hwav,lim	Maximum permissible waviness	m
i	Number of waviness	[~]
IG	Length of lubrication groove	m
IP	Length of lubricant pocket	m
LH	Length of bearing housing	m
NF	Rotational frequency of the bearing force	/s
Q1	Lubricant flow rate at the inlet to clearance gap	m³/s
Q2	Lubricant flow rate at the outlet to clearance gap	m³/s
γ	Angle of misalignment of the shaft	rad
φwav	Period of waviness	°
	MITCalc
BD	Bearing width ratio, relative bearing width B/D	[~]
D1	Bearing outside diameter	m
D2	Housing outside diameter	m
E	Modulus of elasticity in tension (Young modulus)	Mpa
EI	Lower deviation	μm
ES	Upper deviation	μm
Fmax	Maximum bearing force	N
fo	Minimum particle size that will pass through the filter	μm
m	Bearing weight	kg/m³
Mf	Frictional moment	Nm
nB	Rotational speed of the bearing	/min
nF	Rotational speed of the force vector	/min
nJ	Rotational speed of the shaft	/min
nm	Limit speed - Mixed-film lubrication	m/s
nt	Limit speed - Formation of turbulence	m/s
p1	Contact pressure during bearing insertion in the bearing housing	Mpa
Tx	Temperature (for which the parameter is calculated)	°C
ya	Amount of misalignment of the shaft	m
yd	Amount of deflection on bearing width	m
βL	Thermal expansion coefficient	1e-6/K
ΔDo	Mean value of the interference	m
Δψ(P)	Pres fitt change of the relative bearing clearence	[~]
Δψ(T)	Thermal change of the relative bearing clearence	[~]
ν	Poisson number	[~]
φA,φE	Angles of leading edge and trailing edge	°

ISO 7902-1,2,3 (DIN 31652-1,2,3)

A number of assumptions and simplifications that have been experimentally validated and are listed in the ISO (DIN) standard are used in the calculations. The formulas, relationships and explanations of the calculation deviations from the ISO standard are given here.

Assumptions and preconditions

The following idealizing assumptions and preconditions are made, the permissibility of which has been sufficiently confirmed both experimentally and in practice.
a) The lubricant corresponds to a Newtonian fluid.
b) All lubricant flows are laminar.
c) The lubricant adheres completely to the sliding surfaces.
d) The lubricant is incompressible.
e) The lubricant clearance gap in the loaded area is completely filled with lubricant. Filling up of the unloaded area depends on the way the lubricant is supplied to the bearing.
f) Inertia effects, gravitational and magnetic forces of the lubricant are negligible.
g) The components forming the lubrication clearance gap are rigid or their deformation is negligible; their surfaces are ideal circular cylinders.
h) The radii of curvature of the surfaces in relative motion are large in comparison with the lubricant film thicknesses.
I) The lubricant film thickness in the axial direction (z-coordinate) is constant.
j) Fluctuations in pressure within the lubricant film normal to the bearing surfaces (y-coordinate) are negligible.
k) There is no motion normal to the bearing surfaces (y-coordinate).
l) The lubricant is isoviscous over the entire lubrication clearance gap.
m) The lubricant is fed in at the start of the bearing liner or where the lubrication clearance gap is widest; the magnitude of the lubricant feed pressure is negligible in comparison with the lubricant film pressures.

The calculation solves two cases, namely:
1. Bearing cooling by convection
2. Bearing cooling with pressure oil

Since the problem is not directly solvable, the following successive iteration was used:
1. The working temperature of the bearing is estimated (or the oil outlet temperature for cooling with pressure oil)
2. The lubricant parameters are calculated from the temperature
3. Other parameters characterising the bearing function are calculated
4. The power loss (friction) is calculated and thus the working temperature of the bearing (or the oil outlet temperature for cooling with pressure oil)
5. Based on the temperatures from points 1 and 4, the new working temperature for point 1 is approximated and the calculation is repeated.

Used formulas

Nominal bearing diameter proposal

A. Diameter providing liquid friction
D,1 = (400 * F / (n * η * 3.14159 * BD * QualityCoeff))^(1/3)

Where QualityCoeff is in the range of 0.8-2.0 and determines the quality, accuracy and stiffness of the design.

B. Diameter for maximum permissible specific bearing load plim
D,2 = (F / plim)^0.5 / BD^0.5

Reynolds number

Re = PI() * D * NJ * (ψeff * D / 2)) / (η / Rho) <= 41.3*(D / (ψeff * D))^0.5

Density of lubricant

He doesn't specify. Requires direct input of the value Rho * cp for the operating temperature of the bearing.

Specific heat capacity of the lubricant

He doesn't specify. Requires direct input of the value Rho * cp for the operating temperature of the bearing.

Viscosity

Oil is defined by temperature T1, T2, kinematic viscosity v1, v2, density and by thermal expansion coefficient.

The viscosity is calculated with use of the formulas in the standard (ASTMD341):
log(log(v + 0.7)) = A - B * log(T) ... [T...Kelvin, v...cSt, A,B ... constants]
B = {log(log(v2 + 0.7)) - log(log(v1+0.7))} / (log(T1)-log(T2))
A = log(log(v2 + 0.7)) + B * log(T2)
v = 10^(10^(A - B * log(T))) - 0.7 ... [v...Kinematic viscosity]
η = Rho * v ... [Rho...Density]

Relative bearing clearance design ψ'

A. ISO 7902-3, Table 5 - Empirical permissible values for the mean relative bearing clearance

B. Modiffied basic formula with diameter and pressure influence
ψ' = (0.8 * v^0.25 + 4.5 / D^0.25 - 1.2) * (1.25 / p^0.15) * 0.001 in range <0.0008 - 0.01>
p ... Specific bearing load [MPa]
v ... Sliding speed [m/s]
D ... Nominal bearing diameter [mm]

Effective relative bearing clearance

ψmax = (Dmax - DJmin) / D
ψmin = (Dmin - DJmax) / D
ψ' = 0.5 * (ψmax + ψmin)

Thermal change of the relative bearing clearence

For coefficients of linear expansion which differ for shaft and bearing, the thermal change of the relative bearing clearance is given by:

Changing the relative bearing clearance by pressing the bushing into the bearing housing:

Sommerfeld number

Parameters interpolated from ISO 7902-2 tables

ISO 7902-2 provides tabulated values for ε, β, f'/fi, f/fi and Q3' for the Ω angle range 90° - 360° and for the B/D ratio in the range 0.25 - 1.5.
Interpolation is performed using Bezier interpolation.

Based on the calculated value of So, the values obtained by interpolation are as follows:

ε (epsilon) ... Relative eccentricity [ε = 2e/(D — DJ)]
β (beta) ... Attitude angle (angular position of the shaft eccentricity related to the direction of load)
f'/ψeff ... The specific coefficient of friction, taking account of the unloaded area of lubricant film
f/ψeff ... The specific coefficient of friction in the loaded area of the lubricant film
Q3' ... Lubricant flow rate parameter due to hydrodynamic pressure

Minimum lubricant film thickness

Coefficient of friction

Hydrodynamic angular velocity

Heat flow rate due to frictional

Lubricant flow rate due to hydrodynamic pressure

Lubricant flow rate due to feed pressure

Heat flow rate to the ambient

Heat dissipation by convection. Heat dissipation by convection takes place by thermal conduction in the bearing housing and radiation and convection from the surface of the housing to the environment.

Should the area of the heat-emitting surface, A, of the bearing housing not be known exactly, it is possible to use approximate estimates depending on the bearing type mounting.

Calculated bearing temperature resulting from iteration procedure

Calculated lubricant temperature at bearing exit

ISO 7902-3

Minimum permissible lubricating film thickness

The objective of maintaining the minimum permissible hlim thickness of the lubricating film is to maintain full lubrication of the plain bearing to achieve the lowest possible wear and low susceptibility to failure.

The lubricant should be free of contaminating particles, otherwise increased wear, seizure and local overheating may occur, thereby impairing the proper function of the plain bearing. If necessary, suitable lubricant filtration must be provided.

The minimum permissible hlim thickness of the lubricating film as a characteristic parameter for the transition to mixed friction (see ISO 7902-1:2020, 7.6) can be determined from the following formula as shown in the figure.

RzB, RzJ ... Average peak-to-valley height of bearing / shaft sliding surface
f ... Mesh size of oil filter
B ... Nominal bearing width
γ ... Angle of misalignment of the shaft
yd ... Amount of deflection
hwav,eff ... Effective waviness of sliding surface

Process of calculation

A. Indicative calculation.

Quick indicative determination of bearing size and decision between convection cooling and pressurised oil cooling.

1. Enter the basic input data [2.1-2.7]
2. Select the bearing quality level [2.8]
3. Select/set the parameters of the shaft, bearing and bearing housing materials [2.9-2.23]
4. Press the button for Automatic design [2.25]
5. The Automatic design will attempt to estimate the remaining input parameters based on the general recommendations
6. On lines [2.85-2.116] the functional parameters can be checked immediately

B. Design calculation.

1. Make an indicative calculation (A)
2. Go through the input parameters [2.26-2.78] and tune the parameters you know
3. Start the temperature iteration with the ‘Iteration’ button [2.109]
4. If any of the functional parameters [2.85-2.116] do not fit, use the Optimisation at the end of the paragraph for a quick change and check

Useful hint: You can use an analysis that shows the dependence of the functional parameters on the relative bearing clearance in a graph.

C. Control calculation.

1. Enter the basic input data [2.1-2.7]
2. Go through the other input parameters [2.10-2.78] and define them
3. Start the temperature iteration with the ‘Iteration’ button [2.109]

Useful hint: Try studying the examples at the end of the help.

Calculation units, units conversion [1]

1.1 Calculation units

Select the desired system of calculation units in the list box. After switching over the units, all values will be changed immediately.

1.2 Units conversion

Type the value you want to convert into the input box on the left. Select source units and target units on the right.

Ra and Rz roughness

However, based on statistical evaluation, it can be said that with a very high probability the roughness Ra(Rz) derived from the roughness Rz(Ra) will lie within the range of values in parentheses. The biggest influence is the chosen machining technology.

Rz => Ra
Ramin = 0.03 * Rz^1.3
Ramax = 0.24 * Rz^1.06

Ra => Rz
Rzmin = 3.8 * Ra^0.95
Rzmax = 14.5 * Ra^0.75

Radial plain bearing design / check (ISO 7902, DIN 31562...) [2]

2.1 Basic input data

2.3 Rotational speed of the shaft, Angular velocity

The hydrodynamic speed, which determines the bearing function, is defined as follows:
n=abs (nJ+nB-2*nF)

Note: All rotational motions and angular directions are positive relative to the direction of shaft rotation.

B. The load vector rotates at the same speed as the shaft (centrifugal force)
nB=0, nF=nJ ....... n=nJ

C. Shaft and bearing rotate with the same speed
nB=nJ, nF=0 ....... n = 2 * nJ (bearing load capacity is half)

2.7 Bearing width ratio, relative bearing width

The load capacity of long bearings (except sphere plain bearings) is reduced by marginal overload (shaft deflection). The bearing has the highest load capacity at B/D~0.4
Therefore, B/D=0.3-1.0 is generally chosen (exceptionally B/D=1.5)
Smaller values are chosen for pressure lubrication where higher oil flow does not matter.

2.8 Level of accuracy, manufacturing, stiffness…

A number of input parameters allow selection within a certain range. Most of the time, this choice relates to the level of manufacturing, precision, rigidity of construction, etc.
The selection of one of the three levels from the list is taken into account in the proposed values (green cells) and in the ‘Automatic design’.

Level:
Low ... Conservative parameters, safer result.
Medium
Heigh ... Precision manufacturing, quality assembly, quality geometry of sliding surfaces. The parameters must be maintained even during operation.

2.9 Material selection (shaft, bearing, housing)

When calculating the bearing functionality, it is necessary to know especially the properties of the bearing friction layer and the permissible maximum bearing pressure.
For other materials used for the design, the expansion and Poisson’s constant must be defined. These are necessary to calculate the change in dimensions with temperature change.

If the bearing sleeve is pressed into the bearing housing [2.42], it is necessary to select the bearing housing material.
After unchecking the button on the right, the own values can be entered.

2.14 Material of the bearing sliding layer

Material groups are mentioned in the list with empirical plim values in parentheses. These values are based on ISO 7902-3,Tab3 / ISO 12130-3,Tab3, respecting the requirement for minimum distortion of the sliding surfaces that does not result in deterioration of the bearing operation. The selected value is used on the next line. After unchecking the button, one can enter the own value.

Materials marked with ‘*’ – ISO recommendation
Unmarked materials – professional literature

2.15 Maximum permissible specific bearing load

The green cell contains two values.
- The first value reflects the sliding velocity ‘v’ from the empirical value in the list on the previous line (larger plim is possible for small ‘v’ values).
- The second value in brackets indicates the maximum plim value that is achievable in exceptional cases (very low sliding speeds, precision, structure rigidity, etc.)

2.24 Automatic design

During the automatic design, other calculation parameters are estimated and set based on the load, speed, B/D ratio and settings [2.8] and the final iteration is started to solve the bearing operating temperature.

2.25 Start the "Automatic design"

Caution: After starting the design, most parameters will be reset to the estimated default values!!!

2.26 Lubricant selection (definition)

A) The proposed bearing uses a common oil management within the machine. The oil is defined.
B) The lubricant parameters can be modified to best suit the operating requirements of the bearing.

In general, the higher the RPM, the lower the lubricant VG, and the higher the temperature, the higher the lubricant VG.

Select the appropriate oil and its viscosity index from the list. The parameters are defined by the following 3 lines. If you know the exact parameters of the oil, tick the button on the right and define them. Oils are usually defined by their dynamic viscosity for two temperatures and density at 20°C.

A schematic diagram of standard oils according to ISO 3448 in the range VG10-VG460 is shown on the right and the curve of the defined oil is shown in blue.

Useful hint: Detailed definitions, a table of industrial oils and their comparison are given in paragraph [3.0].

2.27 ISO VG (Viscosity grade)

2.32 Thermal expansion coefficient

The expansion coefficient affects the density for temperatures other than 20°C and is usually ~0.75 for mineral oils.

In the green cell is the value estimated based on the density according to the table:
Rho [kg/m³] BetaL
800 ………...….. 0.77 - 0.85
850 ………...….. 0.72 - 0.84
900 ………...….. 0.63 - 0.81
950 ………...….. 0.50 - 0.77

2.34 Design of dimensions

Define the dimensions of the bearing and bearing housing. For orientation you can use the information table of dimensions on the right.

2.35 Nominal bearing diameter - proposal

- bearing load capacity above the limit speed
- permissible specific pressure in the bearing
- shaft bending check

From these checks a design diameter is determined, which, after rounding to a table value, is shown on the following line.

2.36 Nominal bearing diameter (inside diameter)

After unchecking the button on the right, the own value can be entered. Dimensions of bearing bushings according to ISO3547-1 are mentioned in the table on the right. The corresponding dimensions for the bearing diameter D are selected on the following lines.

Note: The ISO 3547-1 standard defines a range of diameter D=<2,300> and width B=<3,100>[mm]. Suppliers may also offer other dimensions (e.g. in the form of bearing strips).

2.41 Bearing weight

2.42 Bearing seating in the bearing housing by press fitting

If the bearing is pressed into the bearing housing (and is not further machined after pressing), the inner diameter of the bearing D is changed.

Caution: The overlap value is a table value. The actual effective overlap during pressing will be further influenced by the pressing method used, the roughness of the surfaces and possibly the temperature. A calculation can be used to determine this precisely: "Force Couplings of Shafts with Hubs"

2.43 Using a press-fit coupling for a bearing

If a pressed connection is used, the change in diameter D is included in the calculation of the relative bearing clearance [2.97].

Note: When using a press-fit connection and assuming that the bearing thickness is considerably less than the thickness of the bearing housing, select the bearing housing material [2.16] as the bearing material [2.20].

2.44 Selection of fit

Bearings from a selection list are used according to ISO (ANSI) for pressing the bearing housings. The recommended storage is marked with an asterisk.

Useful hint: Detailed information can be found in the program: "Tolerances"

2.45 Mean value of the interference

The overlap between the outer diameter of the bearing and the inner diameter of the bearing housing. The save option is on the previous line.

Useful hint: You can also use paragraph [5.0] or calculation: "Force Couplings of Shafts with Hubs"

2.47 Relative bearing clearance ψ' and tolerances selection

The design of the relative bearing clearance ψ' is used to calculate the hydrodynamically effective relative bearing clearance ψeff. This is an important parameter that significantly affects the bearing properties. When designing you can use 2 procedures that design ψ' based on the circumferential speed ‘v’ and the bearing diameter ‘D’ and allow you to select the appropriate tolerances for the shaft and bearing.

2.48 What procedure is used to determine ψ' ?

A. The ψ' proposal is based on the recommendations of ISO7902-3. The ψ' value is changed in steps (0.56; 0.8; 1.12; 1.32; 1.6; 1.9; 2.24; 3.15 [‰]). This is followed by ISO12129 standard, which specifies a bearing system applicable to metal plain bearings used in general mechanical engineering for middle relative bearing clearances (0.56< ψ'm <3.15 [‰] a 25< D <1250 [mm]). The corresponding tolerances are thus automatically defined for the proposed ψ'.

B. To calculate the design ψ' a formula is used that respects the speed, bearing diameter and specific pressure. For the design of the fit, it is possible to use the ISO286 standard (3 < D < 3150 [mm]) or to define the own tolerances. This procedure is more general.

Note: The unselected variant is displayed in grey.

2.50, 2.53 Relative bearing clearance

After unchecking the button on the right, the own value can be entered.
For the determination of the relative bearing clearance ψ', it is possible to use the suggested value or to follow, for example, the following recommendations.

Recommended values according to friction material:
Composition ... 0.0005 - 0.001
Bronze ... 0.0008 - 0.002
Aluminum Alloys ... 0.0012 - 0.0025
Cast iron, graphite ... 0.002 - 0.003
Plastics ... 0.0015 - 0.010

Recommended values according to the application (e.g. internal combustion engines):
Piston pin bearing ... 0.0006 - 0.001
Crankshaft bearing ... 0.008 - 0.0012
Main bearing ... 0.001 - 0.0014

The following recommendations are generally used:
- Small values are suitable for bearings with high specific pressure operating at low sliding speeds.
- As ψ' increases, the bearing load capacity decreases and there is a risk of shaft vibration and bearing cavitation.

2.51 Tolerance zone Bearing / Shaft

For a given D and ψ' the value is determined according to ISO12129.

The dimensions can be checked in the picture on the right.
The zero axis defines the nominal dimension D from which the tolerances are determined.
Blue... Tolerance field of shaft and bearing.
Red... Nominal bearing clearance
Dimensions in [μm] or [10ˉ³in]

2.54 Tolerance grade Bearing / Shaft

By selecting the degree of accuracy, you choose the range of minimum and maximum bearing clearance and thus the complexity of production. When the button is checked, the degree of accuracy is selected automatically depending on the diameter D and ψ'. Generally, the accuracy for the bearing is chosen to be one degree lower than for the shaft.
For example: Bearing...7, Shaft...6

The dimensions can be checked in the picture on the right.
The zero axis defines the nominal dimension D from which the tolerances are determined.
Blue... Tolerance field of shaft and bearing.
Red... Nominal bearing clearance
Dimensions in [μm] or [10ˉ³in]

2.55 Selection of fit

The drop-down list contains a list of ISO 286 fits. The list is sorted by ψ’ value. You can thus simply choose the fit that is closest to the desired ψ’ [2.53].
After selecting the fit, the corresponding bearing and shaft dimensions and bearing clearance values are recalculated below.

2.56 Tolerance zone Bearing / Shaft

If the button on the right is checked, the tolerance fields are displayed according to the selected accuracy level and the ISO 286 D diameter (see [2.54, 2.55]).
After unchecking the button, one can select the own values.

2.57 Bore and shaft dimensions

2.58 Bore of the bearing Dmin, Dmax

If it is needed to enter the values of bearing diameter Dmin, Dmax and shaft diameter DJmin, Djmax directly press the ‘►D.. ’ button.
Switch to paragraph [8.0] where you can enter each exact value.

By unchecking the button on the right the Dmin value can be changed; the other values of Dmax, DJmin and DJmax will be calculated based on the previous values, namely:
Dmax = Dmin + (ES-EI)
DJmin = (Dmin + Dmax) / 2 - C - (es-ei)
DJmax = DJmin + (es-ei)

2.64 Bearing mounting and bearing geometry selection

2.65 Bearing mounting method

When calculating the heating of a bearing that is cooled by convection, it is necessary to determine the area ‘A’, which is radiating heat.
If the surface area is not known, select the method of the bearing fit from the list. The next line gives an estimate of the area of "A".

2.66 Area of heat-emitting surface (bearing housing)

Depending on the dimensions of the bearing and the previous selection, the size of the cooling surface is estimated. After unchecking the button, it is possible to enter a custom value.

2.67 Outer heat transmission coeff, Air velocity

To calculate the bearing heating it is necessary to estimate the heat transfer coefficient.
For still air kA = 12 [W/m²/°K]
Air swirled by rotating shaft only kA = 15-20
For higher air speeds (e.g. fan on shaft), select from the air speed table.
After unchecking the button, one can enter the own value.

2.68 Angular span of bearing segment

You can select Ω=360° for the standard bearings.
For bearing housings with a wiper ring or split bearings, select the appropriate value.

2.69 Include friction in the unloaded part of the bearing?

In order to determine the coefficient of friction (and thus the bearing heating), it is necessary to specify whether friction in the unloaded part of the bearing will also be taken into account.

2.70 Lubricant hole type, dimensions and location

The selection of the type and dimensions of the bore or lubrication groove affects the lubricant flow parameter ‘Q3*’ due to the hydrodynamic pressure ‘pen’ (and thus the temperature of the outlet oil).
Select the type from the picture, define the dimensions below.

2.71 Lubrication hole diameter

The green box shows the recommended minimum value (ISO3547-3). After unchecking the button, one can enter the own value.
If the bearing contains lubrication grooves or lubrication pockets, enter their dimensions according to the picture on the right.

2.73 Lubricant feed pressure

The lubricant supply pressure ‘pen’ should be significantly lower than the specific load of the bearing to avoid additional hydrostatic load. It usually ranges between 0.05 MPa and 0.2 MPa. The depth of the lubrication grooves and lubrication pockets is significantly greater than the bearing clearance.

2.75 Maximum permissible lubricant film pressure

2.76 Minimum permissible lubricant film thickness

1. Table: In ISO 7902-3 (DIN 31652-3), empirical permissible hlim values are given for which Rz<4 [μm] for the shaft, small geometric errors of the sliding surfaces, careful assembly and adequate lubricant filtration are assumed.
2. Calculation (Calc): hlim is defined in detail in the auxiliary calculations [7.0] and the hlim value from this calculation is used.
3. User: Choosing your own value.

2.77 Maximum permissible bearing temperature

The maximum permissible bearing temperature depends on the bearing material and lubricant. As the temperature increases, the hardness and strength of the bearing material (lead and tin alloys due to their low melting points) decreases and the viscosity of the lubricant decreases. In addition, mineral oil-based lubricants age at temperatures above 80°C.
For these general empirical values, it is taken into account that the maximum value of the temperature field is higher than the calculated bearing temperature or the calculated lubricant outlet temperature.
For pressure lubrication, if the ratio of total lubricant volume to lubricant volume per minute (lubricant flow) is higher than 5, the temperature may be 10°C higher.
Values in brackets may be allowed in exceptional cases under special operating conditions.

2.80 Sliding speed

Small and medium sliding speeds are v = 1 - 15 [m/s] (3 - 50 [ft/s]).
Commonly, speeds up to 60 - 80 [m/s] (200-260 [ft/s]) are achieved. Special bearings up to 100 [m/s] (330 [ft/s]).
Friction materials based on rubber, hardened fabric or plastic may have a limited maximum speed.

2.82 Limit speed - Mixed-film lubrication

Note: With very precise manufacturing, assembly, sliding surface quality, etc., the speed can be half that.

2.83 Limit speed - formation of turbulence

An approximate value of high speeds at which turbulent lubricant flow can occur in the bearing, increasing losses and heat development.

2.84 Maximum bearing force

Approximate value of the maximum load at which the liquid friction becomes the limited friction.

2.85 Calculation of bearing thermal and functional characteristics

There are 2 calculations for when the bearing is:
- Cooled by convection
- Pressure oil cooled

In both cases, functional parameters (effective bearing clearance, oil film thickness, friction, heat output) are calculated based on the initial temperature and bearing input parameters (load, dimensions, operating conditions....). When calculating, it is necessary to use an iteration where the estimated initial temperature is gradually refined by the result of the calculation.

Parameters that are outside the recommended values (green cell) are marked with red text.
The influence of input parameters on the functional parameters of the bearing is given in the help.

Useful hint: You can quickly change the main input parameters in the area on the right, labelled ‘'Optimization’.

2.86 Bearing clearance used for calculation

During production, variations in diameters occur within specified tolerances. Thus, during assembly, a shaft with the maximum permissible diameter can be found with a bearing with the minimum permissible diameter Cmin and (or vice versa Cmax). It is therefore advisable to check these extreme options in the calculation.

2.87 Bearing cooling method

2.92 Reynolds number

The Reynolds number is used to check whether the flow is laminar. If Re is greater than the value in the green cell, the calculation values are not valid and the bearing parameters must be changed.

Effect of input parameters
▲Viscosity ..... => ▼Re, ▼So, ▼ε, ▲▼hmin, ▲T
▲Clearence ... => ▲Re, ▲So, ▲ε, ▲▼hmin, ▼T
▲Diameter .... => ▲Re, ▲So, ▼▲ε, ▲hmin, ▲T
▲Width ......... => ▼▲Re, ▼So, ▼ε, ▲hmin, ▼T

2.94 Dynamic viscosity of the lubricant

The recommended minimum viscosity is indicated in the green cell.
The optimal viscosity is then 2-3 times higher.
For slow-running, heavily or shock-loaded bearings, the recommended value is 5-6 times higher.

2.95 Thermal change of the relative bearing clearance

If the degrees of expansion of the shaft and bearing material are different, the bearing clearance changes with the temperature change.

2.96 Pres fitt change of the relative bearing clearance

If the bearing is pressed into the bearing housing (and is not further machined after pressing), the inner diameter of the bearing changes (see lines [2.42-2.46]).

2.98 Summerfield number

Characteristic parameter for load capacity.
1 < So < 15 ... the bearing operates in the liquid friction range, the running is stable in standard design and under normal operating conditions.
So < 1 ... due to low specific pressure at high sliding speed there is a risk of uneven running. It is necessary to change the dynamic viscosity or the B/D ratio.
So > 15 ... danger of contact between sliding surfaces.

2.99 Relative eccentricity [ε = 2e/(D — DJ)]

ε < 0.70 ... leads to unsteady operation due to self-excited oscillations
ε > 0.96 ... leads to boundary friction between the peaks of surface roughness
Larger ε values are found in larger bearing diameters with good surface quality.

2.100 Minimum lubricant film thickness

If the minimum oil film ‘hmin’ thickness is less than the required ‘hlim’ value (green cell)

- You can check the required hlim using the calculation [7.0]
- Adjust bearing parameters
- Reduce bearing temperature by better cooling (area, ventilation)

2.102 Specific coefficient of friction

The green cell indicates whether the following is used:
L + U - specific friction coefficient with respect to the unloaded area of the lubricating film ‘f '/ψeff’
L Only - specific friction coefficient in the loaded area of the lubricating film ‘f/ψeff’

2.106 Calculated bearing temperature

In addition to the functional parameters of the bearing, the calculation also results in the amount of thermal energy (friction) and, due to the ambient temperature and cooling surface, the end temperature of the bearing TB,1. If it is different from the initial estimate of TB,0, a new estimate of TB,2 must be made and added to TB,0. This iteration must be repeated until the difference between TB,0 and TB,1 is minimum (<1°C).

For multiple steps in a row, use the button "30x TB,2 ►TB,0 + 30x Tex,2 ►Tex,0".

If the temperature TB,1 exceeds the allowed Tlim temperature (green cell) you can:
- increase the heat dissipating surface
- increase the air flow rate (ventilation)
- use pressure lubrication.

Note: In exceptional cases and under special operating conditions, Tlim may be higher (110°C, 230°F).

2.115 Calculated lubricant temperature at bearing exit

In addition to the functional parameters of the bearing, the calculation also results in the amount of thermal energy (friction). The lubricant outlet temperature Tex,1 is determined from the lubricant inlet temperature Ten and the lubricant flow rate Q.
If it is different from the initial estimate of Tex,0, a new estimate of Tex,2 must be made and added to Tex,0. This iteration must be repeated until the difference between Tex,0 and Tex,1 is minimum (<1°C).

For one step of this iteration, use the button on the following line "▲Tex,0".

For multiple steps in a row, use the button "30x TB,2 ►TB,0 + 30x Tex,2 ►Tex,0".

If the temperature Tex,1 exceeds the allowed Tlim temperature (green cell) you can:
- increase lubricant flow by reducing viscosity
- increase lubricant flow by increasing bearing clearance

Note: In exceptional cases and under special operating conditions, Tlim may be higher (125°C, 257°F).

2.120 Dynamic viscosity graphs

2.121 User values

After entering the temperature Tx, you will find the lubricant parameters for the specified temperature.

2.122 Table of diameters and widths

mm (SI) …...…. ISO 3547-1, Table 2
in (Imperial) ... Compilation of data from major manufacturers

2.123 Parameter analysis

For easier optimization or search for a functional solution, it is possible to create a graph of the monitored values depending on the specific bearing clearance ψ'.

When the button is pressed, the monitored parameters for the range ψ'<0.0003 - 0.0095> are calculated sequentially.

2.126 Optimization

For easy optimization, you can change the most important bearing input parameters at the click of a button. Immediately after the change, the iteration is executed and you can see the change in results on the left.

Use the ▼▲ buttons to change the parameters and the ► button to restart the Automatic design.

Lubricant selection, comparison and specification [3]

For any mechanism where friction occurs, one of the most important issues is the choice of the appropriate lubricant. In this paragraph you can:

A...Select an oil from a table of commonly used oils
B...Select an oil according to ISO3448
C...Define the oil properties

At the same time, you can easily compare the properties of different oils for various temperatures.

3.1 Lubricant selection from lubricants table

Select the desired oil from the table. It is shown in purple in the graph.
After pressing the button on the right, the definition values of the selected oil are moved to C.

3.3 Lubricant selection from ISO 3448 table

Here you can select the oil based on ISO3448. It is shown in green in the graph. Select gradually:

- Viscosity grade ISO VG 2 - ISO VG 3200
Defines the kinematic viscosity for 40°C v [mm²/s]

- Viscosity index VI=0, VI=50, VI=95
Defines the rate of change in oil viscosity due to temperature change. This means that a lubricant with a higher viscosity index is desirable because it provides a more stable lubricating film over a wider temperature range.

- Position in range 0-1
The viscosity grade allows a certain range of kinematic viscosity. For example, ISOVG100 can have a range between 90-110 [mm²/s]. The VR coefficient defines the position in this range. The value of 0.5 is therefore in the middle.

After pressing the button on the right, the definition values of the selected oil are moved to C.

Note: The faint lines in the graph on the right show the dynamic viscosity for the range VG2-VG3200. The specified VI and VR are used in their determination.

3.7 Lubricant definition

If the supplier defines the oil parameters directly, enter them here.
- Enter the oil density and the temperature for which it is defined.
- Enter the coefficient of thermal expansion.
- Enter a known kinematic or dynamic viscosity for two temperatures T1, T2.

Kinematic or dynamic viscosity inputs are switched via the switches on the right.

3.10 Thermal expansion coefficient

The expansion coefficient affects the density for temperatures other than 20°C and is usually ~0.75 for mineral oils.

In the green cell is the value estimated based on the density according to the table:
Rho [kg/m³] βL
800 ………...….. 0.77 - 0.85
850 ………...….. 0.72 - 0.84
900 ………...….. 0.63 - 0.81
950 ………...….. 0.50 - 0.77

3.17 Viscosity index

The viscosity index is determined according to ISO2909. It is determined from the kinematic viscosity for 40°C and 100°C.

3.18 Transfer definition into paragraph [2.0]

If you need to transfer the oil parameters to the calculation in paragraph [2.0], press the button.

3.20 Viscosity, density and thermal capacity table

3.21 User values

After entering the temperature Tx, you will find the lubricant parameters for the specified temperature.

Roughness tables [4]

System of limits and fits ISO 286 (ANSI B4.1) and ISO 12129 [5]

This paragraph can be used to choose a fit and determine tolerances and deviations of machine parts according to the standard ISO 286 (ANSI B4.1) and ISO 12129.

5.3 A. Calculation of tolerance sizes ISO 286 (3 < D < 3150 mm)

ISO 286: is identical with the European standard EN 20286:1993 and defines an internationally recognized system of tolerances, deviations and fits. The standard ISO 286 is used as an international standard for linear dimension tolerances and has been accepted in most industrially developed countries in identical or modified wording as a national standard (JIS B 0401, DIN ISO 286, BS EN 20286, CSN EN 20286, etc.).

Note: Details in the calculation "Tolerances and fits" and in the calculation "Force couplings of shafts with hubs".

5.4 Hole tolerance zones

Select the tolerance range of the hole (marked A-ZC, see picture for position) and the accuracy level (numbers 1-18). The deviations with respect to the nominal dimension ‘D’ are immediately calculated and displayed in a graph on the right.

5.8 Shaft tolerance zones

Select the tolerance range of the shaft (marked a-zc, see picture for position) and the accuracy level (numbers 1-18). The deviations with respect to the nominal dimension ‘D’ are immediately calculated and displayed in a graph on the right.

5.12 Minimum / Maximum clearance (+) / Minimum / Maximum interference (-)

After selecting the tolerance, the values used in the bearing design are calculated.

5.17 B. Calculation of tolerance sizes ISO 12129 (25 < D < 1250 mm)

ISO 12129: standard has been established because it is not possible to use the ISO deviations given in ISO 286-1 and ISO 286-2 to develop clearance fits which correspond to the requirements of plain bearing engineering for approximately uniform mean relative bearing clearances for all nominal size ranges.

Caution: Validity for 25<D<1250 [mm]

The ISO 12129 standard defines bore and shaft deviations for the selected ‘Relative bearing clearance’ ψ' and ‘Nominal diameter’ D.

After selecting the relative bearing clearance from the list, the relevant tolerances and corresponding dimensions are immediately calculated, see the picture and graph on the right.

5.31 System of fit, Type of fit

Although there can be generally coupled parts without any tolerance zones, only two methods of coupling of holes and shafts are recommended due to constructional, technological and economic reasons.

A) Hole basis system
The desired clearances and interferences in the fit are achieved by combinations of various shaft tolerance zones with the hole tolerance zone "H". In this system of tolerances and fits, the lower deviation of the hole is always equal to zero.
B) Shaft basis system
The desired clearances and interferences in the fit are achieved by combinations of various hole tolerance zones with the shaft tolerance zone "h". In this system of tolerances and fits, the upper deviation of the hole is always equal to zero.

Depending on the mutual position of tolerance zones of the coupled parts, 3 types of fit can be distinguished:

A) Clearance fit
It is a fit that always enables a clearance between the hole and shaft in the coupling. The lower limit size of the hole is greater or at least equal to the upper limit size of the shaft.
B) Transition fit
It is a fit where (depending on the actual sizes of the hole and shaft) both clearance and interference may occur in the coupling. Tolerance zones of the hole and shaft partly or completely interfere.
C) Interference fit
It is a fit always ensuring some interference between the hole and shaft in the coupling. The upper limit size of the hole is smaller or at least equal to the lower limit size of the shaft.

On the next line is a list of fits, the preferred fits are marked with an asterisk.

Note: Details in the calculation "Tolerances and fits".

A) Hole basis system (ISO 286)::

B) Shaft basis system (ISO 286):

A) Hole basis system (ANSI B4.1) :

B) Shaft basis system (ANSI B4.1) :

Maximum permissible specific bearing load - graphs [6]

6.1 Selection of sliding layer material

Select the bearing friction layer material in the selection list. The graph on the right shows the plim versus ‘v’.

6.5 Sliding speed

The value v from paragraph [2.0] is set. After unchecking the button, one can enter the own v value.

Minimum permissible lubricant film thickness hlim ISO 7902-3 (DIN 31652-3) [7]

7.1 Nominal bearing diameter (inside diameter)

7.3 Average peak-to-valley height of bearing sliding surface

In cases where there may be limiting friction (frequent running, high pressures and low speeds, rocking motion, insufficient lubrication), the roughness should be kept to a minimum.

Roughness is generally chosen
Bearing: Ra 0.4-1.6 (Rz 1.6-6.3) [μm]
Shaft: Ra 0.2-0.4 (Rz 0.8-1.6) [μm]

The green boxes contain recommended and estimated values based on:
- diameter D
- width B
- level of accuracy, production, rigidity [2.8]

7.5 Minimum particle size that will pass through the filter

The quality of oil filtration is also included in the hlim calculation (in addition to the ISO and DIN standards). Commonly manufactured industrial filters achieve filtration fineness of up to 1 [μm]. Automotive filters achieve 5 [μm].

7.6 Angle of misalignment of the shaft

How the rotation "γ" or the deflection of the shaft in the bearing "yd" can be determined quite accurately in the shaft calculation.

Calculation of bearing clearance from min/max diameters [8]

Calculation of the relative bearing clearance and tolerance fields from precisely specified min/max bearing and shaft diameters.

8.1 Nominal bearing diameter (inside diameter)

The nominal diameter D [2.36] from paragraph [2.0] is preset.
After unchecking the button, one can enter the own value.
The diameter D is necessary to calculate the relative bearing clearance ψ'.

8.2 Bore of the bearing Dmin, Dmax

8.9 Transfer definition into paragraph [2.0]

in paragraph [2.0], so that the entered averages Dmin, Dmax, Djmin, Djmax would correspond.

Caution: If the button [8.1] is unchecked, the nominal diameter D [2.36] is also set.

Example 1 – Design calculation, optimization

A basic task for which we know the load, material parameters and we want to find a functional (optimal bearing).

Nominal load F=1500 N
Rotational speed of the shaft nJ=1450 /min
Bearing width ratio, relative bearing width B/D ~ 0.75
Shaft and bearing material=Steel (αLJ = 11.7 e-6/°K)
Maximum permissible specific bearing load plim=7MPa

Fill in requested input parameters, select the "Level of accuracy, manufacturing, stiffness…" as "High" and press the "Automatic design" button.

In the results we can see that the hmin value is less than the recommended value.

So it is necessary to adjust either the working conditions or the bearing dimensions.

1. Lubricant viscosity change.

Increase viscosity to VG150 (button "▲"). The hmin value increases and the bearing is within the permissible operating parameters.
In many cases, however, the viscosity of the lubricant is determined by the other elements of the machine (common oil management) and a different method must be chosen for optimisation.

2. Bearing clearance change.

The graphs show that the bearing clearance ψ' is already set so that hmin cannot be increased significantly in this case.

3. Bearing diameter change

Press the button gradually to increase the diameter. The diameter is changed according to the standardized series and the bearing width is traced and changed to maintain the required B/D ratio.
As the bearing gets larger, the lubricant pressure decreases and the minimum oil film thickness hmin increases.

Caution: It is necessary to keep in mind that the bearing dimensions change stepwise according to the standardised series, so there is a step-change of the functional parameters of the bearing.

4. Bearing width change

The same applies for changing the bearing width as for changing the diameter. Bearing increase => Lubricant pressure decrease => hmin increase.

Caution: Increasing the bearing width B increases the B/D ratio, which can be undesirable (materials, design).

5. Cooling parameter changes.

The operating temperature of the bearing significantly affects its functional parameters.

Operating temperature is affected by:
1. Cooling surface (increased A=> decreased temperature => increased hmin).
2. Air flow rate. (increased air velocity => decreased temperature => increased hmin).
3. Ambient temperature.

The ambient temperature (3.) is usually difficult to change. The cooling surface A (1.) is estimated according to the size of the bearing body and should be checked on the basis of the specific design.
The simplest way to do this is to increase the air flow rate, for example by mounting a fan on the shaft (2.).

Increasing the velocity from 1 m/s (used for natural flow resulting from shaft rotation) to 5 m/s will reduce the bearing temperature by 10°C and increase the hmin.
To recalculate the model, press the ‘Iteration’ button.

B. Pressure oil cooling.

With pressure oil cooling, the bearing temperature can be controlled by the temperature of the lubricant at the bearing inlet. For example, by increasing the radiator size or the volume of lubricant in the system.

You can try lowering the temperature to 40°C.
To recalculate the model, press the ‘Iteration’ button.

Caution: When using Auto design, all parameters are set to their default values. Therefore, we recommend reviewing all input parameters [2.26-2.78].

Example 2 – Determining tolerances

Bearing design is usually based on the relative bearing clearance ‘ψ'’, which determines the nominal bearing clearance ‘C’ (ψ' = C / D). Since it is not possible to manufacture a bearing or shaft with zero tolerance, it is necessary to set the appropriate tolerances for the nominal dimensions. Automatic design sets the selection preferences as follows.

Based on the degree of accuracy (2) and the nominal diameter D, it sets the tolerance field (4). It also sets Dmin=D (5) and thus Dmax=Dmin+(ES-EI).
Then DJmin and DJmax are calculated based on the relative bearing clearance (1).

This will ensure the bearing tolerance Hx (where Dmin=D [2.36] and x is the degree of precision ITB [2.54]).

1. Relative bearing clearance ψ'

Based on the friction speed, nominal diameter and load, the bearing clearance ψ' (green cell) is set and used for the calculation.
After unchecking the button, it is possible to enter a custom value (optimization, materials used,...).

2. Degree of precision bearing/shaft

It is set according to the recommendations of ISO12129-1, you can of course choose your own value.

3. Selection of tolerances

In the selection list (3), the bearing/shaft tolerance combinations according to ISO 286 are sorted by size. Select the one that is the closest to the desired value from line [2.53].

The dimensions Dmin/Dmax and DJmin and DJmax are recalculated to meet the D5\p4 fitting acc. to ISO286 and thus the relative bearing clearance ψ'=0.00186.

Useful hint: If you know directly the min/max dimensions of the bearing and shaft, the parameters ψ', ES-EI, es-ei can be easily calculated in paragraph [8.0].

Example 3 – Checking for min/max bearing clearance

Dimensions: see Example 1 + Optimization by changing oil viscosity (VG100 => VG150)

Both the shaft and the bearing are manufactured with a given dimensional tolerance. If no selective assembly is used, it is advisable to check the bearing function also for min/max bearing clearance [2.86].

For the nominal value of bearing clearance, the monitored parameters are within the range of required values (green highlighting).

However, if we set the bearing clearance for the combination Dmax (bearing) + DJmin (shaft) => Cmax, the bearing clearance changes and after recalculating the bearing model it is seen that hmin has dropped below the required value.

It would probably be necessary to try further optimization (e.g. by changing the oil from VG150 to VG220).

Example 4 – Control calculation (Example from ISO 7902 or DIN 31562)

Calculate the heat balance and determine if pressure lubrication with external oil cooling is necessary.

Basic input data

Bearing force (nominal load) F=36000 N ..... [2.2]
Rotational speed of the shaft nJ = 2000 /min ..... [2.3]
Bearing width ratio, relative bearing width B/D=0.5 ..... [2.7]
Shaft linear heat expansion coefficient (steel) αLJ=11 e^-6/K ..... [2.11]
Bearing linear heat expansion coefficient (aluminium alloy) αLJ=23 e^-6/K ..... [2.17]
Maximum permissible specific bearing load plim'=5 MPa ..... [2.15]

Detail specification

Lubricant selection = ISO VG 100, VI95 ..... [2.27]
Nominal bearing diameter (inside diameter) D=120 mm ..... [2.36]
Mean relative bearing clearance ψ'=0.001 ..... [2.53]
Area of heat-emitting surface (bearing housing) A=0.3 m² ..... [2.66]
Outer heat transmission coeff kA=20 W/m²/K ..... [2.67]
Angular span of bearing segment Ω=360 ° ..... [2.68]
Lubricant hole diameter dL=5 mm ..... [2.71]
Lubricant feed pressure pen=0.5 Mpa ..... [2.73]
Ambient temperature Tamb=40 °C ..... [2.88]
Lubricant temperature at bearing entrance Ten=58 °C ..... [2.87]

Limit operating conditions

Maximum permissible specific bearing load plim'=5 MPa
Maximum permissible bearing temperature Tlim=70 °C
Minimum permissible lubricant film thickness hlim=0.009 mm < hmin

Solution procedure:

1) Fill in the "Basic input data" ... [2.2 - 2.23]
2) Press the "Automatic design" button.
3) Enter the values in detail ... [2.27 - 2.78]
4) Enter Tamb=40°C and Ten=58°C ... [2.90, 2.88]
5) Enter 60°C to estimate the temperature TB,0 and Tex,0 ... [2.91, 2.89]
6) Press the "Iteration" button.

Both problems are solved at the same time:
A - Convection cooling
B - Oil pressure cooling

A. Convection cooling

After completion of the iteration (Table1-right), it is clear that the cooling is insufficient and the bearing temperature (TB,0=137.5°C) exceeds the required value of 70°C. There are other values (So, ɛ, hmin) outside the recommendation, so that operation of the bearing would lead to damage.

Table 2 gives an example from the ISO standard. The small differences in the results are due to differences in the interpolation of the table values.

Table 2 - Results of iterative calculations for heat dissipation by convection (ISO example)

B. Pressure oil cooling

In the case of using pressure oil, the operating temperature of the Teff bearing is lower than the required temperature and the parameters So, ɛ, hmin are within the recommended values.

Table 3 – Results of iterative calculation for heat dissipation by lubricant in full bearing

Table 4 - Results of iterative calculation for heat dissipation via the lubricant in full bearing (ISO example)

Example 5 – Example of a possible optimization

The calculation allows to easily change important input parameters and thus quickly optimize the bearing function.

Use the data from Example 3 and try to reduce the operating temperature of the bearing Teff or the friction dissipated power Pth,f.

▲Viscosity ..... => ▼Re, ▼So, ▼ε, ▲▼hmin, ▲T
▲Clearence ... => ▲Re, ▲So, ▲ε, ▲▼hmin, ▼T
▲Diameter .... => ▲Re, ▲So, ▼▲ε, ▲hmin, ▲T
▲Width ......... => ▼▲Re, ▼So, ▼ε, ▲hmin, ▼T

which tell us that as the viscosity of the oil decreases with the temperature. Press the highlighted button three times to get a new result immediately.

By changing the viscosity of the oil, the losses were reduced by a third and the temperature was reduced by 5°C.

Example 6 – Comparison tables of calculation results (1728 bearings)

For comparative purposes, we have used the ‘Automatic design’ to design bearings for the combination of the parameters listed below and collated them into 288 clear tables.

A rough, approximate estimate of the bearing size and cooling method can thus be obtained from these tables.

Addressed ranges

Bearing force (nominal load) F=10N - 500000N [2.2]
Rotational speed of the shaft n=10 - 10000 /min. [2.3]
Bearing width ratio, relative bearing width B/D = 0.4; 0.8; 1.2 [2.7]
Maximum permissible specific bearing load plim=5MPa; 10MPa [2.15]
Level of accuracy, manufacturing, stiffness… Low; High [2.8]

The name of the sheet is made from: L ... Low, BD0.8 ... B/D=0.8, P5 ... plim=5MPa

Example of solved tables for (B/D=0.8; plim=10MPa; High level)

Different tables for cooling type (A ... Convection cooling, B … Oil pressure cooling)

Setting calculations, change the language

Workbook (calculation) modifications

List of Standards, literature list

ISO 7902:2020
Hydrodynamic plain journal bearings under steady-state conditions Circular cylindrical bearings
Part1, Part2, Part3

DIN 31652:2017
Gleitlager - Hydrodynamische Radial-Gleitlager im stationären Betrieb
Teil1, Teil2, Teil3

ISO 2909
Petroleum products — Calculation of viscosity index from kinematic viscosity

Bearing / Oil Temperature
	<50°C	60°C	75°C	90°C
n [/min]	<120°F	140°F	167°F	194°F
< 300	68	100	220	---
300-1500	46	68	100-150	150
1500-3000	32	32-46	68-100	100
3000-6000	32	32	46-68	68-100
6000-10000	32	32	32	32-46

Clearence fit		Transition fit	Interference fit
H5/g4	H8/h8	H5/js4	H6/n5
H5/h4	H8/h9	H5/k4	H6/p5
H6/f5	H9/c9	H5/m4	H6/r5
H6/g5	H9/d9 *	H6/js5	H6/s5
H6/h5	H9/e8	H6/j5	H6/t5
H7/c8	H9/e9 *	H6/k5	H6/u5
H7/d8	H9/f8	H6/m5	H7/p6 *
H7/e7	H9/f9	H7/js6	H7/r6 *
H7/e8	H9/h8	H7/j6 *	H7/s6 *
H7/f7 *	H9/h9	H7/k6 *	H7/s7
H7/g6 *	H10/d10	H7/m6	H7/t6
H7/h6 *	H10/h9	H7/n6 *	H7/u6
H8/c8	H10/h10	H8/js7	H7/u7
H8/d8	H11/a11	H8/j7	H8/s7
H8/d9	H11/b11	H8/k7	H8/t7
H8/e8 *	H11/c11 *	H8/m7	H8/u7
H8/e9	H11/d9	H8/n7	H8/u8
H8/f7 *	H11/d11	H8/p7	H8/x8
H8/f8	H11/h9	H8/r7	H8/z8
H8/f9	H11/h11
H8/g7	H12/b12
H8/h7	H12/h12

Clearence fit		Transition fit	Interference fit
G5/h4	H8/h8	JS5/h4	N6/h5
H5/h4	H9/h8	K5/h4	P7/h6 *
F7/h5	D9/h9	M5/h4	R7/h6
G6/h5	D10/h9 *	JS6/h5	S7/h6 *
H6/h5	E9/h9 *	J6/h5	T7/h6
D8/h6	F8/h9	K6/h5	U7/h6
E8/h6	F9/h9	M6/h5	U8/h7
F7/h6	H8/h9	JS7/h6
F8/h6 *	H9/h9	J7/h6
G7/h6 *	H10/h9	K7/h6 *
H7/h6 *	H11/h9	M7/h6
D8/h7	D10/h10	N7/h6 *
E8/h7	H10/h10	JS8/h7
F8/h7	A11/h11	J8/h7
H8/h7	B11/h11	K8/h7
D8/h8	C11/h11 *	M8/h7
D9/h8	D11/h11	N8/h7
E8/h8	H11/h11
E9/h8	B12/h12
F8/h8	H12/h12
F9/h8

Hydrodynamic plain journal bearings

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