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Journal Bearings Calculator — Sommerfeld Number, Oil Film Thickness & Lubrication Regime (DIN 31652 / ISO 7902)

Governing standard: DIN 31652 / ISO 7902· DIN 31652 / ISO 7902 (Raimondi-Boyd, L/D = 1) · ASTM D341 Walther viscosity-temperature · ISO 281:2007 rolling-bearing life · SKF grease interval · ISO 4406:1999 contamination

How ISO 7902 works — the method explained

The MechanixCalc journal bearing calculator designs and verifies hydrodynamic plain bearings to DIN 31652 / ISO 7902. Enter the journal diameter, bearing length, radial clearance, applied load, rotational speed and dynamic viscosity, and the tool solves the Sommerfeld number, looks up the Raimondi-Boyd eccentricity ratio and friction variable, and returns the minimum oil film thickness, friction torque, power loss, oil flow rate and adiabatic temperature rise in a single calculation. The companion Stribeck-curve chart shows where your operating point sits on the boundary–mixed–hydrodynamic transition, and the film-thickness-versus-speed sweep lets you quickly find the minimum safe speed for full hydrodynamic lubrication.

It is designed for machine-design and maintenance engineers who need a defensible calculation for gearbox journals, pump sleeve bearings, compressor crankpin bearings, electric-motor sleeve bearings and any plain bearing in rotating equipment. Five integrated sub-tools extend the analysis: an ISO VG viscosity selector with ASTM D341 Walther viscosity-temperature curves, a hydrodynamic film thickness / lambda ratio (surface-roughness) panel, an SKF-method grease relubrication interval calculator, an ISO 4406 oil contamination classifier, and an ISO 281 rolling-bearing L10 life panel — all from one tool.

What this calculator does

  • DIN 31652 / ISO 7902 Sommerfeld number and eccentricity ratio via Raimondi-Boyd tables (L/D = 1, with Petroff asymptote blending)
  • Minimum oil film thickness h_min, friction coefficient, friction torque, power loss and oil flow with full-hydrodynamic / thin-film / boundary regime classification
  • Stribeck-curve chart and film-thickness-vs-speed sweep for operating-point visualisation
  • ASTM D341 Walther viscosity-temperature curves for all nine ISO VG grades (VG 22 to VG 680) with DN-value viscosity recommendation
  • Hydrodynamic film thickness panel with lambda ratio (h_min / composite roughness) and Boundary / Mixed / Hydrodynamic regime classification
  • SKF-method grease relubrication interval with temperature correction and grease quantity estimate
  • ISO 281:2007 rolling-bearing L10 modified rating life and ISO 4406:1999 oil cleanliness analysis — plus a branded PDF engineering report

Method & formulas

Sommerfeld number and Raimondi-Boyd analysis (DIN 31652 / ISO 7902)

The dimensionless Sommerfeld number S characterises the hydrodynamic load capacity of a journal bearing. It combines viscosity, speed, bearing pressure and the clearance ratio into one parameter; a high Sommerfeld number indicates a well-lubricated, lightly loaded bearing that runs near-concentric, while a low Sommerfeld number indicates a heavily loaded or slowly rotating bearing whose journal eccentricity approaches the bore wall.

Given S, the eccentricity ratio ε (= e/c, where e is the distance between shaft and bearing centres, and c is the radial clearance) is read from the Raimondi-Boyd characteristic tables for L/D = 1. MechanixCalc interpolates linearly between the tabulated points and blends continuously to the Petroff concentric-bearing asymptote beyond the last table entry. The minimum oil film thickness follows directly from the clearance and eccentricity ratio.

Sommerfeld number
S = (μ · N_rps / p) · (r / c)²

where S = Sommerfeld number (dimensionless); μ = dynamic viscosity (Pa·s); N_rps = journal speed (rev/s); p = bearing pressure = W / (L · D) (Pa); r = journal radius (m); c = radial clearance (m); W = radial load (N); L = bearing length (m); D = journal diameter (m)

Minimum oil film thickness
h_min = c · (1 − ε)

where h_min = minimum oil film thickness (mm); c = radial clearance (mm); ε = eccentricity ratio (dimensionless, 0 = concentric, 1 = contact); ε is looked up from the Raimondi-Boyd S-table

Friction, power loss and oil flow

The Raimondi-Boyd tables also tabulate the friction variable fR/C (the product of friction coefficient, journal radius and the ratio r/c) and the dimensionless oil flow variable Q/(r·c·N_rps·L), both as functions of S. MechanixCalc interpolates these alongside ε at each Sommerfeld number, so the friction coefficient, friction torque, bearing power loss and volumetric oil flow rate all come from the same table lookup as the film thickness — ensuring internal consistency. The adiabatic temperature rise is estimated from the power loss and the oil flow through an energy balance.

Friction coefficient (from Raimondi-Boyd fR/C variable)
f = (fR/C) · (c / r)

where f = friction coefficient (dimensionless); fR/C = Raimondi-Boyd friction variable (function of S, read from table); c = radial clearance (m); r = journal radius (m)

Petroff friction torque (lightly loaded limit)
T_f = 4π² · μ · N_rps · r³ · L / c

where T_f = friction torque (N·m); μ = dynamic viscosity (Pa·s); N_rps = journal speed (rev/s); r = journal radius (m); L = bearing length (m); c = radial clearance (m). Valid in the full hydrodynamic regime (S > 0.1); the Petroff equation is the concentric-bearing (ε → 0) limiting case.

Viscosity selection (ASTM D341 Walther / ISO VG grades)

Oil viscosity falls sharply with temperature, and selecting the correct ISO VG grade for the operating temperature is critical to achieving the target Sommerfeld number. MechanixCalc models the viscosity-temperature relationship using the ASTM D341 Walther equation: the Walther transform W = log₁₀(log₁₀(ν + 0.7)) is linear in log₁₀(T + 273.15), so it can be anchored to the ISO VG reference viscosities at 40 °C and 100 °C and interpolated to any operating temperature. The DN-value heuristic (bore diameter × speed, in mm·rpm) provides a first-pass viscosity grade recommendation before the Sommerfeld analysis is run.

ASTM D341 Walther viscosity-temperature equation
log₁₀(log₁₀(ν + 0.7)) = A − B · log₁₀(T + 273.15)

where ν = kinematic viscosity (cSt = mm²/s); T = temperature (°C); A, B = constants derived from the ISO VG grade's reference viscosities at 40 °C and 100 °C per ISO 3448

Worked example

Calculate the Sommerfeld number and minimum oil film thickness for a journal bearing with journal diameter D = 80 mm, bearing length L = 80 mm, radial clearance c = 0.060 mm, radial load W = 4000 N, journal speed N = 1500 rpm, and dynamic viscosity μ = 46 mPa·s (ISO VG 46 at ~60 °C).

Given

  • Journal diameter D80 mm
  • Bearing length L80 mm
  • Radial clearance c0.060 mm
  • Radial load W4000 N
  • Journal speed N1500 rpm
  • Dynamic viscosity μ46 mPa·s = 0.046 Pa·s

Result

  • Sommerfeld number S≈ 0.817
  • Eccentricity ratio ε≈ 0.17
  • Minimum oil film thickness h_min≈ 0.050 mm
  • Lubrication regimeFull hydrodynamic (S > 0.3)
  1. Convert units: N_rps = 1500 / 60 = 25 rev/s; r = 80 / 2 = 40 mm = 0.040 m; c = 0.060 mm = 6.0 × 10⁻⁵ m.
  2. Compute bearing pressure: p = W / (L × D) = 4000 / (0.080 × 0.080) = 4000 / 0.0064 = 625 000 Pa.
  3. Compute clearance ratio: r / c = 0.040 / (6.0 × 10⁻⁵) = 666.7; (r/c)² = 444 400.
  4. Compute Sommerfeld number: S = (μ · N_rps / p) · (r/c)² = (0.046 × 25 / 625 000) × 444 400 = (1.15 / 625 000) × 444 400 = 1.84 × 10⁻⁶ × 444 400 ≈ 0.817.
  5. At S = 0.817, interpolate linearly in the Raimondi-Boyd table for L/D = 1 between the rows S = 0.631 (ε = 0.20) and S = 1.33 (ε = 0.10): ε = 0.20 + (0.10 − 0.20) × (0.817 − 0.631) / (1.33 − 0.631) = 0.20 − 0.10 × 0.266 ≈ 0.17 (full hydrodynamic regime — the bearing runs near-concentric).
  6. Minimum oil film thickness: h_min = c · (1 − ε) = 0.060 × (1 − 0.17) = 0.060 × 0.83 ≈ 0.050 mm.
  7. Check against the 0.1 % diameter danger limit: 0.001 × 80 = 0.080 mm. Since h_min = 0.050 mm < 0.080 mm, the engine flags a metal-contact-risk warning (red). Verify the design — consider increasing clearance, raising speed or selecting a higher-viscosity grade to increase h_min above the 0.080 mm threshold.

This example uses round numbers for illustration. The eccentricity ratio is read from the L/D = 1 Raimondi-Boyd characteristic table by linear interpolation — the full tool does this automatically. Verify against your actual geometry, operating temperature (which determines μ via the Walther equation) and surface roughness (lambda ratio).

Frequently asked questions

Which standard does this journal bearing calculator use?

The primary engine follows DIN 31652 / ISO 7902, the standard for hydrodynamic plain bearings under steady-state conditions. The Sommerfeld number and eccentricity ratio are resolved using the Raimondi-Boyd characteristic tables (for an L/D ratio of 1), with a continuous Petroff-asymptote blend beyond the tabulated range. Viscosity-temperature behaviour follows the ASTM D341 Walther equation (ISO 3448 grade classification). The rolling-bearing sub-panel follows ISO 281:2007, grease intervals follow the SKF General Catalogue method, and oil cleanliness is classified per ISO 4406:1999. The governing method and all references are shown in the generated PDF report.

What is the Sommerfeld number and why does it matter?

The Sommerfeld number S is the key dimensionless parameter of a hydrodynamic journal bearing. It groups viscosity, speed, bearing pressure and the clearance-to-radius ratio into one value that determines how well the oil film supports the load. A high S (above ~0.3) means full hydrodynamic lubrication — the oil wedge fully separates the shaft from the bearing, friction is low and wear is negligible. A low S (below ~0.1) means thin-film or boundary lubrication — the film is insufficient, metal contact occurs and wear is severe. Sommerfeld number analysis is the correct way to verify that a proposed combination of oil grade, speed and clearance gives adequate film thickness before committing to a design.

How do I choose the right ISO VG oil grade for my bearing?

The ISO VG grade sets the kinematic viscosity at 40 °C; the actual operating viscosity at your bearing temperature is lower and is predicted by the ASTM D341 Walther equation built into the tool. The DN-value method (bore diameter × rotational speed, in mm·rpm) gives a first-pass recommendation: DN < 25 000 favours higher-viscosity grades (VG 100–220) for film thickness; DN > 200 000 favours lower viscosity (VG 22–46) to limit heat generation. The correct approach is to enter your likely grade into the Sommerfeld analysis and confirm that h_min / composite roughness (the lambda ratio) is above 3 for full hydrodynamic lubrication.

What is the lambda ratio and how does it relate to lubrication regime?

The lambda ratio Λ = h_min / Ra_composite compares the minimum oil film thickness to the composite surface roughness of the journal and bearing surfaces (Ra_composite = √2 × Ra for identical surfaces). When Λ > 10 the film is thicker than the asperities and the bearing runs in the hydrodynamic regime with negligible wear. When 3 < Λ < 10 it is in the mixed regime — some asperity contact occurs and surface finish matters. When Λ < 3 the film breaks down (boundary lubrication) and EP or anti-wear additives are required. The hydrodynamic film panel in this tool reports Λ alongside the Raimondi-Boyd result.

Is the journal bearing calculator free?

You can use it during a free 30-minute preview with no sign-up required. A free 14-day account trial unlocks every calculator on the platform with no credit card. The branded PDF engineering report and saved calculations are part of a paid plan.

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