Journal Bearings and Hydrodynamic Lubrication

Why journal bearings matter

Nearly every large rotating machine that turns for years without failing rides on plain journal bearings, not rolling-element bearings. A rolling bearing carries load through a handful of tiny Hertzian contact points and eventually fatigues; a hydrodynamic journal bearing carries load through a continuous film of oil that, in principle, never wears at all. That is why they dominate exactly the machines you cannot afford to replace often.

• Steam and gas turbines. Multi-tonne rotors spinning at 3,000–3,600 rpm ride on tilting-pad and cylindrical journal bearings sized for decades of continuous duty.
• Engine crankshafts and connecting rods. Every automotive main and rod bearing is a hydrodynamic journal bearing — Babbitt or trimetal shells fed by the oil pump.
• Large electric motors and generators. Sleeve bearings replace ball bearings once shaft loads and speeds exceed rolling-element fatigue limits.
• Pumps and compressors. Centrifugal and reciprocating machines use journal bearings for radial support and thrust bearings for axial load.
• Ship propeller shafts and hydro turbines. Water- or oil-lubricated bearings support enormous slow-turning shafts.
• Quiet, high-precision spindles. No rolling elements means no ball-pass vibration, so grinding spindles and instruments favour fluid films.

The trade is that they need a lubricant supply, a minimum speed to build lift, and careful control against instability — problems that rolling bearings do not have. Understanding the physics is how engineers keep them alive.

How the oil wedge carries the load — step by step

The whole trick is geometry plus viscosity. Here is the sequence that turns rotation into a load-carrying pressure field:

1. The shaft sits eccentric. The journal (shaft) has radius r; the bore is larger by the radial clearance c (typically c/r ≈ 0.001, i.e. about 1 µm of clearance per mm of radius). Under load the shaft drops off-center, so the gap between shaft and bore is not uniform — it is a crescent that converges on one side and diverges on the other.
2. Rotation drags oil into the gap. Oil wets the shaft surface and, by viscous adhesion (no-slip condition), is carried along at half the surface speed on average. This flow is forced into the converging part of the crescent.
3. The converging wedge pressurizes the film. Oil is essentially incompressible. As the gap narrows, the same flow cannot pass through a smaller cross-section without a pressure rise. Pressure climbs to a peak just before the point of minimum film thickness, then collapses as the gap diverges. This is the hydrodynamic wedge action.
4. Pressure integrates to a lift force. Integrating the film pressure over the projected bearing area gives a resultant force that exactly balances the applied load — the shaft floats. Crucially, the shaft center settles at an attitude angle roughly 90° from the load line, not directly opposite it, because the peak pressure sits ahead of the minimum-gap point.
5. Equilibrium sets the eccentricity. Heavier load or thinner oil pushes the shaft closer to the bore (higher eccentricity ratio ε = e/c, where e is the shaft-center offset), thinning the minimum film until the wedge pressure again balances the load. Full-film operation keeps ε below ~0.9 so a finite minimum film remains.

The governing equation — Reynolds' lubrication equation

Osborne Reynolds simplified the Navier-Stokes equations in 1886 for a thin film where the gap is thousands of times smaller than its length. Inertia and cross-film pressure variation drop out, leaving a single equation for the pressure field. In its steady one-dimensional form (pressure varying around the bearing):

d/dx ( h³/μ · dp/dx ) = 6 U · dh/dx

Every symbol, with SI units:

Symbol	Meaning	Units
p	Local film pressure	Pa (N/m²)
h	Local film thickness (the gap)	m
μ	Dynamic (absolute) viscosity of the oil	Pa·s
U	Surface (sliding) velocity of the journal, = ωr	m/s
x	Coordinate running around the bearing circumference	m
dh/dx	Rate of film-thickness change (the wedge slope)	dimensionless

The right-hand side is the wedge term: pressure is generated only where the film thickness changes. Where the gap converges (dh/dx < 0) the source term drives pressure up; where it diverges, pressure drops (and would go negative, so the film cavitates and ruptures instead — real bearings only carry load over the converging half). The h³ factor is decisive: halving the minimum film thickness increases the pressure needed to pass the same flow by up to eightfold, which is why film thickness collapses so steeply as load rises. The full bearing uses the two-dimensional Reynolds equation, adding a term in the axial direction (z) that accounts for oil leaking out the ends of a finite-length bearing.

The Sommerfeld number and design charts

Rather than solving Reynolds' equation for every case, engineers collapse the problem into one dimensionless group — the Sommerfeld number (also called the bearing characteristic number):

S = (r/c)² · (μN / P)

Symbol	Meaning	Units
r	Journal radius	m
c	Radial clearance (bore radius − journal radius)	m
μ	Dynamic viscosity	Pa·s
N	Rotational speed	rev/s
P	Projected unit load = W / (L·D)	Pa

Here W is the applied load, L the bearing length and D the journal diameter, so P is the pressure the load would exert if smeared over the projected rectangle. A high Sommerfeld number means a lightly loaded, fast, thick-oil bearing that runs with a generous film and small eccentricity (safe, low-wear). A low Sommerfeld number — heavy load, low speed, or thin oil — pushes the shaft toward the bore and thins the film toward metal contact. The classic Raimondi–Boyd charts take S (and the L/D ratio) and return everything a designer needs: the eccentricity ratio ε, the minimum-film-thickness variable h₀/c, the friction variable (r/c)·f, the oil flow, and the temperature rise. Typical automotive main bearings operate around S = 0.02–0.2; lightly loaded turbine bearings run higher.

The Stribeck curve — three lubrication regimes

A journal bearing does not live in the full-film regime all the time. The Stribeck curve plots the friction coefficient against the parameter μN/P and reveals three distinct regimes it passes through as speed rises from zero:

Regime	Condition	Film vs. asperities	Friction μ
Boundary	Very low speed / high load (startup, stall)	Surfaces touch; load carried by asperities + additive films	~0.08–0.15
Mixed	Rising speed; partial film	Film partly separates surfaces; intermittent contact	~0.01–0.08 (falling steeply)
Hydrodynamic	High μN/P; full film	Continuous oil film; no solid contact	~0.001–0.005

Friction drops steeply through the mixed regime and reaches a minimum right at the boundary between mixed and full-film operation — the point where the surfaces just fully separate. Push μN/P higher (more speed or more viscosity) and friction rises again, slowly, because now it is dominated by viscous shear of a thicker film. This is why designers do not simply choose the thickest oil available: excessively viscous oil raises drag, churning, and heat. The safe operating point is comfortably to the right of the friction minimum, on the full-film branch, with margin so that load spikes or a hot, thin oil do not drop the bearing back into mixed lubrication.

Worked example — film pressure and load capacity

Consider a crankshaft main bearing: journal diameter D = 50 mm (radius r = 0.025 m), length L = 25 mm (so L/D = 0.5), radial clearance c = 25 µm (so c/r = 0.001), running at N = 50 rev/s (3,000 rpm) with an oil of viscosity μ = 0.015 Pa·s (a warm SAE 30 at ~90 °C). It carries a radial load W = 5 kN.

First the projected unit load:

P = W / (L·D) = 5000 / (0.025 × 0.050) = 4.0 × 10⁶ Pa = 4.0 MPa

Then the Sommerfeld number:

S = (r/c)² · (μN/P) = (1000)² × (0.015 × 50 / 4.0×10⁶) ≈ (10⁶) × (1.875×10⁻⁷) ≈ 0.19

From the Raimondi–Boyd chart for L/D = 0.5, S ≈ 0.19 gives an eccentricity ratio of about ε ≈ 0.6 and a minimum-film variable h₀/c ≈ 0.4. The minimum film thickness is therefore:

h₀ = (h₀/c) · c ≈ 0.4 × 25 µm ≈ 10 µm

Ten micrometers — about one-seventh the diameter of a human hair — is the entire margin keeping steel off steel. It also explains why bearing surfaces must be finished to a fraction of a micrometer and why a single hard particle larger than the film is catastrophic; the soft Babbitt overlay exists specifically to embed such debris. The chart's friction variable, (r/c)·f ≈ 3–4 for this case, gives f ≈ 0.003–0.004, confirming the tiny full-film friction coefficient.

Common misconceptions and failure modes

• "The oil pump generates the load capacity." No — the load-carrying pressure is generated purely by shaft rotation and viscosity (hydrodynamic). The feed pump only supplies flow and cooling. A hydrostatic bearing is the different case where an external high-pressure pump floats the shaft even at zero speed.
• "Thicker oil is always safer." Only up to a point. Too much viscosity raises friction, churning losses, and film temperature — which then thins the oil anyway. The Stribeck curve shows friction rising again on the full-film branch.
• "Wear happens while running." In full-film operation there is essentially no wear. The damage happens at start-up and shut-down, in boundary/mixed lubrication, before the film is established. Frequent stop-starts, not run hours, kill bearings.
• "More clearance is better for cooling." Excess clearance lowers (r/c)², dropping the Sommerfeld number and film thickness, and invites oil whirl. Clearance is a tuned compromise, not "as much as fits."
• Wiping and seizure. If the film breaks down (overload, oil starvation, overheating), the soft overlay smears — a "wiped" bearing — and in the extreme the surfaces weld and seize.
• Fatigue and cavitation erosion. Cyclic loading (engine firing) fatigues the overlay; film rupture and re-formation can cavitate and erode the surface, especially at oil-hole edges.
• Oil whirl / oil whip. At light load and high speed the shaft center precesses at ~0.42–0.48× shaft speed because the mean film velocity is about half the surface speed. If this sub-synchronous whirl locks onto a shaft natural frequency it becomes destructive oil whip. Cures include elliptical (lemon-bore), pressure-dam, axial-groove, and tilting-pad bearings that break the circumferential symmetry that feeds the instability.