Sliding contact bearings and hydrodynamic lubrication

Why it matters

Every crankshaft main and big-end bearing, camshaft bearing and turbocharger shaft in a car runs on a plain (journal) bearing with a pressurised oil film. These bearings carry very high, fluctuating loads with almost no wear for hundreds of thousands of kilometres, but only while the oil film is thick enough. Understanding when the film forms, what it costs in friction, and what makes it collapse is central to engine design.

Key ideas

Sliding contact (journal) bearing: a shaft (journal) rotates inside a bush or shell with a small radial clearance c, typically c/r ≈ 0.001. Engine bearings are thin steel-backed shells lined with tin- or lead-based babbitt, copper-lead or aluminium-tin alloys; bronze bushes are used for small-ends and lightly loaded pins.

Bearing materials need: compressive and fatigue strength, conformability (to tolerate misalignment), embeddability (to swallow dirt particles), corrosion resistance, good thermal conductivity, and compatibility with the steel journal so it does not seize.

Regimes of lubrication (Stribeck/McKee curve: friction coefficient against the bearing characteristic number Z·N/p):

• Boundary lubrication: at very low speed or high load, the film is only a few molecules thick and asperities touch; friction is high (μ ≈ 0.05–0.15) and wear occurs. This happens at engine start-up.
• Mixed lubrication: partial film, partial contact; friction reaches its minimum near the transition.
• Hydrodynamic (thick-film) lubrication: above a critical Z·N/p, a full film separates the surfaces; friction is low (μ ≈ 0.001–0.01) and rises slowly with Z·N/p because of viscous shear.

How the hydrodynamic film forms: under load the journal runs eccentric in the bush, forming a converging wedge. Rotation drags viscous oil into the wedge, the pressure rises (Reynolds equation) and this pressure supports the load. A film needs relative speed, viscosity, a converging gap and a continuous oil supply. Hydrostatic bearings instead use externally pressurised oil and work even at zero speed.

Design parameters

• Unit bearing pressure p = W/(L·D) on the projected area; take allowable values from the data book (engine main bearings run at several MPa).
• Length-to-diameter ratio L/D, typically 0.4–1 in engines (short bearings save space and tolerate misalignment).
• Clearance ratio c/r: too small causes heat and seizure risk, too large drops the load capacity and increases oil flow and noise.
• Sommerfeld number S = (r/c)²·μ·n_s/p, a dimensionless group that fixes the eccentricity, minimum film thickness h₀, friction variable (r/c)·f and oil flow. Raimondi–Boyd charts give these quantities against S for each L/D; read them from your data book.
• Petroff's equation gives the friction of a lightly loaded, concentric journal. It underestimates friction for heavily loaded (eccentric) bearings but is a good first estimate and is often examined.

Heat balance: friction power becomes heat, carried away by the oil flow and the housing. The oil temperature rise lowers viscosity, which reduces the film thickness, so viscosity is evaluated at the mean film temperature by iteration.

Formulas

p = W / (L·D) Petroff: f = 2π²·(μ·n_s / p)·(r / c) Friction torque (Petroff): T_f = 4π²·r³·L·μ·n_s / c = f·W·r Power loss: P_f = 2π·n_s·T_f Sommerfeld number: S = (r/c)²·μ·n_s / p Bearing characteristic number: Z·N / p

Symbols: W = radial load (N); L = bearing length (m); D = journal diameter (m), r = D/2; c = radial clearance (m); p = unit bearing pressure (Pa); μ = absolute (dynamic) viscosity (Pa·s); n_s = journal speed (rev/s); N = speed (rpm); Z = viscosity (in the McKee curve, often in centipoise); f = coefficient of friction (–); T_f = friction torque (N·m); P_f = power lost (W); S = Sommerfeld number (–). Keep n_s in rev/s in S and Petroff's equation.

Worked examples

Example 1 (standard). An engine main bearing: D = 60 mm, L = 30 mm, radial clearance 0.03 mm, oil viscosity 10 mPa·s at operating temperature, 3000 rpm, load 6 kN. Estimate p, S, the Petroff friction coefficient and the power loss.

1. p = 6000/(0.030 × 0.060) = 3.33 × 10⁶ Pa = 3.33 MPa.
2. n_s = 3000/60 = 50 rev/s; r/c = 30/0.03 = 1000.
3. S = 1000² × 0.010 × 50 / 3.33 × 10⁶ = 0.150.
4. f = 2π² × (0.010 × 50 / 3.33 × 10⁶) × 1000 = 0.00296.
5. T_f = f·W·r = 0.00296 × 6000 × 0.030 = 0.533 N·m.
6. P_f = 2π × 50 × 0.533 = 167 W (Petroff estimate; a heavily loaded bearing at S = 0.15 runs eccentric and the chart value of friction would be higher).

Example 2 (GATE level). In a lightly loaded test, a journal of 50 mm diameter and 50 mm length with 0.025 mm radial clearance runs at 1800 rpm, and the measured friction torque is 0.2 N·m. Find the oil viscosity and the power loss.

1. n_s = 30 rev/s; r = 0.025 m.
2. From Petroff, μ = T_f·c / (4π²·r³·L·n_s) = 0.2 × 0.025 × 10⁻³ / (4π² × 0.025³ × 0.05 × 30).
3. Denominator = 4π² × 1.5625 × 10⁻⁵ × 0.05 × 30 = 9.253 × 10⁻⁴; numerator = 5 × 10⁻⁶.
4. μ = 5.40 × 10⁻³ Pa·s (5.4 mPa·s).
5. P_f = 2π × 30 × 0.2 = 37.7 W.

Common mistakes

• Using rpm instead of rev/s in Petroff's equation or the Sommerfeld number.
• Using the diametral clearance where the radial clearance c is required.
• Taking pressure on the full cylindrical surface instead of the projected area L·D.
• Thinking hydrodynamic bearings suit low speed; a film needs speed, and low speed means boundary lubrication.
• Using Petroff for a heavily loaded bearing as if it were exact.
• Ignoring the fall of viscosity with temperature.

For GATE ME

Expect Petroff's equation (friction coefficient, torque, power loss or viscosity), the Sommerfeld number and its dependence on speed, load and clearance, the regimes on the Z·N/p curve, and properties of bearing materials. Practise unit conversions: mPa·s to Pa·s, rpm to rev/s, mm to m.

Quick check

1. A 40 mm journal, 40 mm long, carries 3.2 kN. What is the unit bearing pressure?
2. If speed doubles and everything else is constant, how does the Sommerfeld number change?
3. In which regime does an engine bearing run at cold start?
4. Name two properties a bearing lining needs besides strength.
5. If the radial clearance is halved, how does Petroff's friction torque change?

Answers: 1. 2 MPa. 2. It doubles. 3. Boundary (or mixed) lubrication. 4. Any two of conformability, embeddability, corrosion resistance, thermal conductivity, compatibility. 5. It doubles.