Hydrodynamic Journal Bearing Design Suite

Content

Estimate finite-length hydrodynamic journal-bearing load, stiffness, leakage, friction, power loss, and ideal lubricant heating.

Engineering screening tool—verify inputs and results independently. Use at your own risk.

Bearing and lubricant inputs

Hydrodynamic load capacity

—N

How to use

  1. Select Oil bearing or Water / elastomer. These presets use the same finite-length hydrodynamic model and only change example values.
  2. Enter D, L, and ΔD in the displayed units. Enter positive n, η, ρ, and cₚ, then set h₀ so 0 < h₀ < ΔR=ΔD/2.
  3. Select Calculate; values also update while editing. Reset restores the selected preset.
  4. Interpret load capacity as the positive radial reaction opposing journal displacement. Review stiffness, side leakage, friction, power loss, and the adiabatic temperature-rise screen together. Correct any red warning before use.
  5. The result becomes increasingly sensitive as ε approaches 1. Do not use this model for mixed/boundary contact, starvation, instability qualification, turbulent flow, severe thermal-viscosity feedback, flexible/misaligned surfaces, or safety-critical bearing approval.

Input and output definitions

D is shaft diameter; L is bearing length; ΔD is diametric clearance and ΔR=ΔD/2; h₀ is minimum film thickness; n is rotational speed; η is dynamic viscosity; ρ is lubricant density; and cₚ is specific heat capacity. Positive eccentricity moves the journal toward the loaded wall. The sign convention takes positive load to oppose journal displacement, positive flow to leave through the bearing ends, and positive power to be dissipated as heat.

F is radial load capacity; S is incremental stiffness; Q is total side leakage; Ploss is viscous power loss; ΔT is an ideal adiabatic bulk-temperature rise; So is the finite-length Sommerfeld/load number; β is the attitude angle; Cq is the flow number; cμ is the friction number; and μ is the friction coefficient. The model requires 0 < ε < 1. Displayed values use up to six significant digits; the calculation itself is not rounded. For compatibility checks, the source-style comparison in the test suite also verifies its stated staged rounding.

Journal bearing geometry and hydrodynamic reactionCross-section showing bearing radius, displaced journal centre, radial clearance, minimum film thickness, rotation, and opposing hydrodynamic load.bearing centrejournal centreeFωΔR = radial clearanceh₀ = ΔR − eε = e / ΔRβ = attitude angle
Finite journal-bearing geometry. The converging film develops a pressure field and a reaction load at attitude angle β.

Equations used

Convert all geometry to metres. R=D/2, ΔR=ΔD/2, ω=2πn/60, L̄=L/D, ε=(ΔR−h₀)/ΔR, and γ=√(1−ε²).

Symbols and intermediate quantities

Geometry and motion: R is journal radius [m]; ΔR is radial clearance [m]; ω is shaft angular speed [rad/s]; L̄ is the dimensionless length ratio L/D; ε is the dimensionless eccentricity ratio; and γ is the dimensionless eccentricity complement √(1−ε²).

Bruker

Force resultant and interpolation: fr∞ and ft∞ are dimensionless radial and tangential long-bearing force components. λs, λm, λt, and λr are dimensionless short-limit, matched, tangential, and radial interpolation parameters. j is an index that selects r (radial) or t (tangential) in the shared attenuation equation. Zr and Zt are dimensionless finite-length attenuation factors, while fr and ft are dimensionless attenuated radial and tangential force components.

Flow interpolation: λe is the dimensionless flow attenuation parameter; Ze and Zm are dimensionless flow attenuation factors; and Cqe and Cqm are dimensionless maximum-film and minimum-film edge-flow coefficients whose normalized difference gives Cq.

Long-bearing force components: fr∞=−12ε²/[(2+ε²)γ²] and ft∞=6πε/[(2+ε²)γ]. Interpolation parameters are λs=√[(2+ε²)/(2γ²)] and λm=√{2(2+ε²)(1+γ)/[4γ²+4γ−γε²]}. Then λt=(λs−λm)exp(−L̄)+λm, λr=√2λt, and Zj=1−tanh(λjL̄)/(λjL̄).

fr=Zrfr∞, ft=Ztft∞, So=√(fr²+ft²), β=atan(−ft/fr), and cμ=2π/[So√(1−ε²)]+εsinβ/2.

Optimol

For flow, λe=1−exp(−2L̄), Ze=1−tanh(λeL̄)/(λeL̄), and Zm=1−tanh(λmL̄)/(λmL̄). Cqe=π(1+ε)−πεZe(ε+1)(ε+2)/(2+ε²), Cqm=π(1−ε)+πεZm(ε−1)(ε−2)/(2+ε²), and Cq=(Cqe−Cqm)/(2π).

Dimensional outputs: F=SoηωLR/(ΔR/R)², μ=cμ(ΔR/R), Q=Cqω(L/D)·2R²ΔR, Ploss=μFωR, and ΔT=Ploss/(Qρcₚ). Incremental stiffness follows the implemented 0.1% forward eccentricity step: S=[F(1.001ε)−F(ε)]/(0.001εΔR).

Worked example

  1. Select Oil bearing and Reset: D=40 mm, L=40 mm, ΔD=40 µm, n=300 rpm, h₀=3 µm, η=0.04 Pa·s, ρ=850 kg/m³, and cₚ=1800 J/(kg·K).
  2. Expected outputs are approximately F=9,164 N, S=3,701 MN/m, Q=0.342 cm³/s, Ploss=9.0 W, and ΔT=17.3 K.
  3. The result indicates a fully hydrodynamic load screen with an ideal adiabatic heat rise; verify viscosity at the resulting temperature and perform a coupled thermal/flow analysis before design approval.

Theory and method

The finite-length interpolation bridges the Ocvirk short-bearing and Sommerfeld infinitely long-bearing limits. It preserves finite-length end leakage while estimating the radial and tangential pressure-resultant components, attitude angle, friction, and flow from geometry and eccentricity.

Assumptions are steady, fully flooded, laminar, incompressible, isoviscous Newtonian flow between rigid, aligned circular surfaces. The pressure field is a screening approximation. Cavitation, starvation, grooves, turbulence, inertia, elastic and thermal deformation, viscosity-temperature feedback, misalignment, wear, mixed lubrication, and rotor-dynamic instability are excluded. Small eccentricity can be dynamically unstable even when the static load calculation is finite; lemon-bore or multilobe designs require separate analysis.

Rtec

The adiabatic relation assumes all friction power is carried away by Q. It is an upper-level screening balance, not a local film-temperature or thermal-equilibrium solution.

References and traceability

The archived method page carries an Andres (1989) attribution, but its author, title, venue, and other bibliographic identity is not exposed there and could not be confirmed in Crossref, OpenAlex, or Semantic Scholar; no DOI is claimed. Independently checkable references are: F. W. Ocvirk, Short-Bearing Approximation for Full Journal Bearings, NACA Technical Note 2808 (1952); A. Sommerfeld, Zur hydrodynamischen Theorie der Schmiermittelreibung, Zeitschrift für Mathematik und Physik 50 (1904), 97–155; K. Sassenfeld and A. Walther, finite journal-bearing design charts (1954); and T. Someya (ed.), Journal-Bearing Databook, Springer, 1989, DOI 10.1007/978-3-642-52509-4. The implemented equations above—not the unresolved shorthand attribution—define this calculator.

Continue in TriboSolver to refine pressure, thermal, deformation, cavitation, and stability behaviour for the actual bearing geometry and duty.