Journal Bearing 3D

Why a shaft floats on oil

In a plain (journal) bearing the shaft does not touch the bearing when it turns fast enough. The spinning surface drags oil into a gap that narrows in the direction of motion; because the oil cannot escape instantly, its pressure rises and carries the load. The gap is only tens of micrometres: the radial clearance c (here 20 to 120 um) against a 40 mm shaft radius, so the pictures enlarge it. Units are SI; the model is an educational one for a single rigid rotor on one bearing.

1. Film thickness and the Reynolds equation

With the shaft centre displaced by (x, y) from the bearing centre, the gap at angle θ is h = c − x cosθ − y sinθ (eccentricity ratio ε = e / c). For an incompressible oil film the Reynolds equation reads

∂/∂x (h³ ∂p/∂x) + ∂/∂z (h³ ∂p/∂z) = 6 η U ∂h/∂x + 12 η ∂h/∂t, with U = ω R.

The right side has a wedge term (shaft surface speed dragging oil into a changing gap) and a squeeze term (the shaft centre moving).

2. Short-bearing solution (used here)

For a bearing short compared with its diameter, flow along the axis dominates and the x-derivative can be dropped. Then p(θ, z) = (6 η/h³) Φ(θ) (z² − L²/4) with Φ = (ω/2)(x sinθ − y cosθ) − ẋ cosθ − ẏ sinθ. Oil cannot pull (cavitation): where the formula gives a negative pressure it is set to zero (Gumbel, or half-Sommerfeld, condition). The forces on the journal are one-dimensional integrals: F = η L³ R ∫ (Φ/h³) (cosθ, sinθ) dθ over the part of the circle where Φ < 0. This is accurate for L/D up to about 0.5 and is a fair trend model up to L/D = 1. The unrolled map shows the pressure on the bearing surface; it vanishes at both ends, which is why a longer bearing carries a load much more effectively (about L³).

3. Statics: the position of the shaft

The shaft settles where the film force balances the load. Newton iteration on (x, y) gives the equilibrium locus as speed grows: the shaft climbs from the bottom (high eccentricity) towards the centre (low eccentricity) and sideways by the attitude angle. The closed-form Ocvirk (short-bearing) results are W = η U L³ ε / (4 c² (1 − ε²)²) · √(π²(1 − ε²) + 16 ε²) and tan φ = π √(1 − ε²) / (4 ε). The Sommerfeld number S = (η N / P)(R / c)² (with P = W / (2 R L)) combines everything into one dimensionless group.

4. Friction, heat and the Stribeck curve

The shear in the oil gives a friction torque T = η ω R³ L ∫ dθ / h (Couette part only; the small pressure-gradient part is neglected) and a power loss T ω. The heat is carried away by the oil that leaks out of both ends, Q = L R ∫ |Φ| dθ, so the film warms by about ΔT = Tω / (ρ cp Q); the mean film temperature is the supply temperature plus half of that. Viscosity falls with temperature as η = η40 exp(−0.034 (T − 40)) (typical slope for mineral oil). When the minimum film is thinner than about 3 times the combined surface roughness, asperities start to touch: an illustrative contact share raises the friction coefficient towards a boundary value of 0.12. Plotted against the Hersey number η N / P this gives the classic Stribeck curve: friction falls as speed rises, reaches a minimum, then rises slowly with viscous shear.

5. Dynamics, oil whirl and stability

The page integrates the journal centre with m ẍ = Fx + m eu ω² cos ωt and m ÿ = Fy − W + m eu ω² sin ωt (fourth-order Runge-Kutta with sub-steps set by the squeeze-film time constant). At low load and high speed the film acts like a spring with a strong cross-coupled stiffness: the shaft is pushed sideways by its own motion and begins to circle in the bearing at a bit less than half the shaft speed (oil whirl). Linearising the film about equilibrium gives stiffness and damping matrices; the Routh-Hurwitz criterion on the characteristic polynomial det(m s² + C s + K) = 0 gives the whirl threshold speed. Above it the orbit grows until it is limited by the nonlinear film (or touches the bearing).

Numerical checks in the page

(1) the load from numerically integrating the pressure equals the closed-form Ocvirk load at eccentricities 0.2 to 0.9; (2) the Newton equilibrium reproduces the closed-form attitude angle; (3) the force balance residual at the equilibrium; (4) a time-domain simulation decays below the Routh-Hurwitz whirl threshold and grows above it.

Simplifications

Short-bearing and Gumbel approximations, not a full finite-difference Reynolds solution; a rigid, perfectly aligned rotor on a single bearing; isoviscous across the film with one mean temperature; Couette friction only; the asperity contact share and the boundary friction value are illustrative, not measured; oil inlet grooves and supply pressure are not modelled; the oil properties are typical mineral-oil numbers. Do not use for design.