Plain bearings · DIN 31652

Hydrodynamic journal bearing to DIN 31652: film thickness, friction, temperature

The calculator runs the complete steady-state design of a fully enclosing journal bearing to DIN 31652-1: from load, speed, geometry, clearance and oil viscosity it determines the Sommerfeld number, relative eccentricity, attitude angle, minimum film thickness, friction power and lubricant flow. The bearing temperature follows from the heat balance – for convection cooling through the housing or for pressure-fed lubrication with oil cooling – and is iterated until it agrees with the temperature-dependent viscosity.

Ω = 360°, 0.125 ≤ B/D ≤ 1, laminar flow, steady load. The characteristic functions come from DIN 31652-2 (eq. 1–4, 6–15), the guide values from DIN 31652-3.

DIN31652

Bearing and load

Bearing load F [N]Shaft speed nJ [min⁻¹]Bearing speed nB [min⁻¹]Rotational frequency of the load direction nF [min⁻¹]Nominal bearing diameter D [mm]Load-carrying bearing width B [mm]Relative assembly clearance ψ₂₀ [‰]Thermal expansion coefficient of the bearing αlB [10⁻⁶/K]Thermal expansion coefficient of the shaft αlJ [10⁻⁶/K]

Lubricant

Dynamic viscosity η₁ [mPa·s]at temperature T₁ [°C]Dynamic viscosity η₂ [mPa·s]at temperature T₂ [°C]Oil density ρ [g/cm³]

Data sheets often give kinematic viscosities – convert them with the calculator Dynamic/kinematic viscosity conversion or interpolate with Lubricant viscosity vs temperature.

Heat dissipationType of heat dissipation

Oil inlet temperature Ten [°C]Feed pressure pen [MPa]Lubricant feed elementFeed bore diameter dH [mm]Effective lubricant film temperature

Guide values to DIN 31652-3Bearing material group

Custom minimum permissible film thickness hmin,lim [µm]Custom permissible bearing temperature TB,lim [°C]Total oil volume to flow per minute above 5

So = F·ψ²/(D·B·η·ω), ε from DIN 31652-2 eq. (1), hmin = 0.5·D·ψ·(1 − ε), Pf = F·(μ/ψ)·ψ·(D/2)·ω, heat balance per DIN 31652-1 eq. (26)/(27).

Operating state at thermal equilibrium

Define bearing, lubricant and heat dissipation. The defaults are example A.1 of DIN 31652-1.

Inputs and method

Designing a journal bearing to DIN 31652

The calculator determines the thermal-equilibrium operating state of a hydrodynamic plain journal bearing under steady load: Sommerfeld number, relative eccentricity, attitude angle, minimum film thickness, friction power, oil flow and bearing temperature – and checks the results against the DIN 31652-3 guide values.

Inputs

Bearing load F, speeds of shaft, bearing and load direction, nominal diameter D, load-carrying width B, relative assembly clearance ψ₂₀ with the expansion coefficients of bearing and shaft, the dynamic viscosity of the oil at two temperatures, and the type of heat dissipation: housing surface, heat transfer coefficient and ambient temperature for convection cooling, or oil inlet temperature, feed pressure and feed element for pressure-fed lubrication. For the checks you select the bearing material group or custom limits.

Calculation

From a starting temperature follow the viscosity (Vogel equation, DIN 31652-1 eq. 34) and the hot clearance (eq. 35, 37), and from these the Sommerfeld number So = F·ψ²/(D·B·η·ω) (eq. 7). The relative eccentricity ε is found by inverting the characteristic function DIN 31652-2 eq. (1), the attitude angle β per eq. (2), friction μ/ψ per eq. (3), flow Q₁ per eq. (4) and Qp per eq. (6) to (15). The friction power Pf = F·μ·U (eq. 11) yields a new temperature through the heat balance (eq. 26 or 27); the calculator averages and repeats until convergence. Then hmin = 0.5·D·ψ·(1 − ε) (eq. 9), the Reynolds number (eq. 6) and the DIN 31652-3 checks follow.

Example

Example A.1 of DIN 31652-1: D = 120 mm, B = 60 mm, F = 36 kN, nJ = 2,000 min⁻¹, ψ₂₀ = 1 ‰, aluminium shell (23·10⁻⁶/K) on a steel shaft (11·10⁻⁶/K), oil with 98 mPa·s at 40 °C and 9.7 mPa·s at 100 °C. With convection cooling (A = 0.3 m², k = 20 W/(m²·K), Tamb = 40 °C) the iteration gives TB ≈ 137 °C – far above the required 80 °C. With pressure-fed lubrication (Ten = 58 °C, pen = 0.5 MPa, bore dH = 5 mm opposite the load) the calculator converges at Teff = 70.9 °C, ψeff = 1.61 ‰, So = 2.56, ε = 0.839, β = 27.9°, hmin = 15.6 µm, Pf = 1.78 kW and Q = 4.66 l/min; the standard quotes 71.0 °C, ε = 0.839, hmin = 15.57 µm, Pf = 1,777 W and Q = 4.67 l/min for its computer program.

Sources and limits: DIN 31652-1:2017-01 (clause 4.1 eq. 2, 4, 6; 4.3 eq. 7, 9; 4.4 eq. 10, 11; 4.5 eq. 12–15; 4.6 eq. 16–18, 22–27, 31, 34; 4.7 eq. 35–37; Annex A.1 with Tables A.1 and A.2 as reference test), DIN 31652-2:2017-01 with Corrigendum 1:2019-04 (eq. 1–4 characteristic functions, eq. 6–15 feed elements, Tables 1 and 2 as cross-check), DIN 31652-3:2017-01 (Table 1 hmin,lim, Table 2 p̄lim, Table 4 TB,lim). Limits: fully enclosing circular bearing Ω = 360°, 0.125 ≤ B/D ≤ 1, laminar flow, steady load, rigid and aligned sliding surfaces, Newtonian mineral oil; the closed-form eq. (1) deviates from the tabulated Reynolds solutions by up to about 6 %. Half bearings (Ω = 180°), misalignment, elastic deformation, pressure dependence of viscosity and dynamic loads are not covered.

Technical article

Designing a journal bearing to DIN 31652 in detail

What does the DIN 31652 bearing calculation deliver?

DIN 31652 is the calculation standard for hydrodynamic plain journal bearings under steady-state conditions. Such a bearing carries the shaft not by solid contact but by the pressure that builds up in the wedge-shaped lubricant gap between journal and shell once the shaft rotates. Whether that film carries the load is decided by four interlinked quantities: the minimum film thickness hmin, the friction power, the oil flow and the bearing temperature. Because the oil viscosity falls with temperature and the temperature in turn depends on the friction power, the operating state can only be found iteratively – exactly the procedure of DIN 31652-1 Figure 4 that this calculator runs. It belongs to the design step after the preliminary choice of diameter, width and material: once load, speed and installation space are fixed, it answers which clearance, which oil and which cooling give safe full-film operation.

Where the calculator sits in the design sequence

Journal bearing design proceeds in stages: first diameter and width are fixed from the installation space and the shaft and checked against the material with the specific bearing load p̄ = F/(B·D). Then clearance and oil are chosen – the clearance from fit and tolerance gives ψ₂₀, viscosities come from the data sheet, if necessary via dynamic/kinematic viscosity conversion. Only then does the hydrodynamic calculation proper begin, which this calculator performs in full: it chains the individual steps Sommerfeld number, minimum film thickness and friction power with the heat balance, so the temperature dependence of the viscosity no longer has to be estimated by hand.

Typical questions it answers: is unpressurised lubrication with heat dissipation through the housing enough, or is pressure-fed lubrication with a cooler needed? Which assembly clearance maximises the film thickness? How much oil must the pump deliver? And does the flow in the gap stay laminar so that the standard applies at all?

Where the inputs come from

Load and speeds: F is the resulting bearing reaction from the shaft calculation or load assumption; nJ the operating speed. nB is non-zero only for a rotating shell, nF only for a rotating load direction (unbalance). Geometry: D and B from the design; B without chamfers and circumferential grooves. Clearance: ψ₂₀ from the chosen fit, together with the expansion coefficients for aluminium or bronze shells, because the hot clearance then differs noticeably from the assembly clearance (1.6 ‰ instead of 1 ‰ in the example of the standard). Oil: data-sheet viscosities at 40 °C and 100 °C; the operating temperature should lie between the reference points. Heat dissipation: housing surface from the drawing or the approximations of the standard (eq. 19–21), k = 15 to 20 W/(m²·K) in still air; for pressure feed the inlet temperature of the cooled oil and the feed pressure, usually 0.05 to 0.5 MPa. Guide values: material group per DIN 31652-3 Table 2; custom limits where manufacturer data or roughness values are known.

Formula and variables

So = F · ψeff² / (D · B · ηeff · |ωeff|) → ε = f⁻¹(So, B/D) → hmin = 0.5 · D · ψeff · (1 − ε)

ωeff = ωJ + ωB − 2·ωF (DIN 31652-1, eq. 2)
ψeff = ψ₂₀ + (αlB − αlJ)·(Teff − 20 °C) (eq. 35, 37)
μ/ψeff = π/(So·√(1 − ε²)) + ε/2·sin β (eq. 10)
Pf = F · (μ/ψeff) · ψeff · (D/2) · |ωrel| (eq. 11)
Q₁ = D³·ψeff·|ωJ + ωB|·Q₁*, Q₁* = 0.25·(B/D − 0.223·(B/D)³)·ε (eq. 12; DIN 31652-2 eq. 4)
Qp = D³·ψeff³·pen/ηeff · Qp* (eq. 13; Qp* per DIN 31652-2 eq. 6–15)
Convection: TB = Tamb + Pf/(k·A) (eq. 26) · Pressure-fed: Tex = Ten + Pf/(Q·ρ·c) (eq. 27)
log₁₀ η = C₁/(T + 95 °C) + C₂ (eq. 34) · Re = ρ·U·(s/2)/η ≤ 41.3·√(D/s) (eq. 6)

Symbol / input	Meaning
F, nJ, nB, nF	Bearing load and speeds of shaft, bearing and load direction; they give the effective angular speed ωeff and the relative speed for friction.
D, B, B/D	Nominal diameter, load-carrying width and width ratio; B/D governs the DIN 31652-2 characteristic functions.
ψ₂₀, ψeff, s	Relative assembly clearance at 20 °C, hot clearance at operating temperature and absolute clearance s = ψ·D.
η₁/T₁, η₂/T₂, ηeff	Two viscosity points of the oil and the interpolated effective viscosity at Teff.
So, ε, β	Sommerfeld number, relative eccentricity and attitude angle – the hydrodynamic parameters of the operating point.
hmin	Minimum film thickness in the narrowest gap; must exceed hmin,lim.
μ/ψeff, μ, Pf	Relative and absolute friction coefficient and the friction power, all of which becomes heat.
Q₁, Qp, Q	Oil flow from self-pressure, from feed pressure and in total.
TB, Tex, Teff	Bearing temperature with convection cooling, oil outlet temperature with pressure feed, and the effective temperature governing viscosity.
A, k, Tamb / Ten, pen	Heat-dissipation parameters: housing surface, heat transfer coefficient and ambient temperature, or oil inlet temperature and feed pressure.

Choose the inputs correctly

How to use the calculator

Define load, speed, D, B and the assembly clearance, add the expansion coefficients of bearing and shaft and the two viscosity points of the oil. Choose the heat dissipation: for the first question, whether unpressurised lubrication is enough, use convection cooling with housing surface and ambient temperature; if the temperature comes out too high, switch to pressure-fed lubrication with inlet temperature, feed pressure and feed element. Select the material group or enter custom limits. After calculating, the assessment shows whether film thickness, load, temperature and flow regime meet the guide values. Then vary the assembly clearance in steps to find the optimum for hmin and check the smallest and largest tolerance clearance separately.

Worked example

How to read film thickness, temperature and flow

The minimum film thickness is the central safety quantity: if it exceeds the DIN 31652-3 guide value hmin,lim (Table 1 by diameter and circumferential speed, or (0.5 to 1.0)·(RzJ + RzB) from the roughness), the surfaces do not touch and the bearing runs without wear. The bearing temperature shows whether the heat dissipation suffices – with convection cooling the housing surface decides, with pressure-fed lubrication the oil flow. The flow Q is at the same time the design value for pump and cooler. The relative eccentricity classifies the operating point: near 1 the film becomes thin, below about 0.5 oil whirl threatens. Since clearance, temperature and film thickness are coupled, a parameter variation of the assembly clearance pays off: the standard's Table A.2 shows that hmin has a maximum at an optimum hot clearance and drops steeply towards small clearances – the tolerance range should therefore lie about 1/3 below and 2/3 above the optimum.

Force in N, speeds in min⁻¹, lengths in mm, relative clearance in ‰, viscosity in mPa·s, temperatures in °C, expansion coefficients in 10⁻⁶/K, housing area in m², heat transfer coefficient in W/(m²·K), feed pressure in MPa; outputs in µm, W, cm³/s and l/min. Internally the core works in SI units (m, Pa·s, rad/s).

Convection cooling or pressure-fed lubrication – the heat balance

The friction power in the film turns entirely into heat. DIN 31652-1 clause 4.6 distinguishes two cases, one of which dominates in practice. Bearings without recirculating lubrication (ring oiler, oil bath) pass the heat through housing and shaft to the surroundings: PA = k·A·(TB − Tamb). With pressure-fed lubrication the oil flow absorbs the heat: PQ = ρ·c·Q·(Tex − Ten) with the volumetric heat capacity ρ·c ≈ 1.57 + 0.003·T in J/(K·cm³). Equating with Pf gives the bearing temperature or the oil outlet temperature. Since both paths really act together, neglecting the other one is on the safe side.

For the viscosity the standard recommends the outlet temperature Tex with pressure feed – the least favourable value, giving the largest margin against mixed friction; with little oil heating the mean temperature is also customary. The calculator starts per eq. (31) with Ten + 15/ψ‰, averages the assumed and computed temperature after each step and stops at a 0.01 K deviation. If convection cooling yields an impermissible temperature – as in the example of the standard with 137 °C instead of 80 °C – pressure-fed lubrication with external cooling is required.

Typical applications

Design and verification of plain bearings in gearboxes, turbomachinery, pumps, compressors, rolling mills and large engines; deciding between an unpressurised bearing and pressure-fed circulating lubrication; determining the optimum clearance and its tolerance range; sizing oil pump and cooler from the flow; assessing an oil change for film thickness and temperature.

Assumptions, limits and common mistakes

DIN 31652-1:2017-01 (clause 4.1 eq. 2, 4, 6; 4.3 eq. 7, 9; 4.4 eq. 10, 11; 4.5 eq. 12–15; 4.6 eq. 16–18, 22–27, 31, 34; 4.7 eq. 35–37; Annex A.1 with Tables A.1 and A.2 as reference test), DIN 31652-2:2017-01 with Corrigendum 1:2019-04 (eq. 1–4 characteristic functions, eq. 6–15 feed elements, Tables 1 and 2 as cross-check), DIN 31652-3:2017-01 (Table 1 hmin,lim, Table 2 p̄lim, Table 4 TB,lim). Limits: fully enclosing circular bearing Ω = 360°, 0.125 ≤ B/D ≤ 1, laminar flow, steady load, rigid and aligned sliding surfaces, Newtonian mineral oil; the closed-form eq. (1) deviates from the tabulated Reynolds solutions by up to about 6 %. Half bearings (Ω = 180°), misalignment, elastic deformation, pressure dependence of viscosity and dynamic loads are not covered.

Common mistake: Do not confuse assembly clearance with hot clearance – with aluminium or bronze shells on steel shafts the clearance opens noticeably in operation, with steel on steel it stays the same. Do not take the viscosity at room temperature; enter both data-sheet points and let the iteration determine the operating temperature. Enter the load-carrying width without a ring groove; a 360° ring groove at mid-width splits the pressure field and DIN 31652-1 then requires calculating each half with half the load. For loads above about 2.5 N/mm² consider start-up: slow starts run in mixed friction and may need hydrostatic jacking. Watch the Reynolds number – at very high circumferential speeds the flow becomes turbulent and the standard no longer applies.

Frequently asked questions

Why does the calculation have to iterate?

Because the viscosity depends on the bearing temperature, but the temperature only follows from the friction power, which in turn depends on the viscosity. DIN 31652-1 resolves this loop iteratively per Figure 4: assume a temperature, compute viscosity, So, ε, friction and the heat balance, form a new temperature and repeat until the deviation is below a threshold. The calculator averages the old and new temperature as the standard recommends.

How is the relative eccentricity linked to the Sommerfeld number?

Through the characteristic function So = f(ε, B/D) of DIN 31652-2 eq. (1), derived from solutions of the Reynolds equation. The calculator inverts it numerically: for the Sommerfeld number formed from load, clearance, viscosity and speed it searches the eccentricity at which the function returns exactly that value. The closed-form equation deviates from the tabulated values by up to about 6 %; the standard's example is reproduced exactly with it.

Which clearance is right?

The hot clearance usually lies between 0.3 and 3.5 ‰, 1 ‰ is a common starting value. High load and low speed call for small clearance, high speed and low load for large. The optimum for film thickness is found by variation: in the standard's example hmin rises from 14.5 µm at ψ₂₀ = 0.5 ‰ to 16.1 µm at 2 ‰ and falls again above. The tolerance range is placed about 1/3 below and 2/3 above the optimum.

What does the attitude angle mean for the design?

It gives the position of the narrowest gap relative to the load direction. The oil feed belongs in the widest gap or just ahead of it in the direction of rotation – i.e. opposite the load, offset by the attitude angle. For ωeff < 0 the shaft displaces against the direction of rotation at −β.

Does the calculator apply to half bearings or bearings with a ring groove?

Only to a limited extent. DIN 31652-1 allows the Sommerfeld number for Ω = 180° with the same equation, but friction and flow follow separate functions there (DIN 31652-2 clause 5) that are not included here. A 360° ring groove in the pressure region splits the bearing into two halves that have to be calculated separately with half the load.

What to do if the temperature is too high?

In this order: increase the clearance (more flow, less friction), choose a thinner oil, enlarge the housing surface or air flow and finally switch to pressure-fed lubrication with an oil cooler. In the standard's example the change from convection cooling to pressure feed lowers the bearing temperature from 137 °C to 71 °C.