Sliding contact bearings and hydrodynamic lubrication

Plain journal bearings: materials, Stribeck lubrication regimes, hydrodynamic wedge action, Petroff's equation, Sommerfeld number, Raimondi-Boyd charts, power loss and pV limits.

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Why it matters

Plain (journal) bearings carry the crankshafts of engines, turbine and compressor rotors, large pump shafts and heavy rolling-mill rolls, where they outlast rolling bearings by running on a full oil film with no metal contact. Small self-lubricating bushes are everywhere in mechatronic devices - hinges, linkages, low-speed pivots. Designing them means understanding when an oil film can carry the load and how much power it wastes in friction.

Key ideas

Construction. A journal (the part of the shaft inside the bearing) rotates in a bush with a small radial clearance c (typically of the order of 0.001 times the radius). Bush materials need conformability, embeddability (to absorb dirt), compatibility (resistance to seizure) and fatigue strength: white metal (babbitt), bronzes, aluminium alloys, sintered porous bronze impregnated with oil, and polymers (PTFE, acetal, nylon) for dry or lightly lubricated running.

Lubrication regimes (Stribeck curve). Plotting friction coefficient against the bearing characteristic number μN/p (viscosity × speed ÷ unit load):

  • Boundary lubrication (low μN/p) - surfaces touch through thin adsorbed films; high friction and wear (starting and stopping).
  • Mixed lubrication - partial film; friction falls steeply as μN/p rises.
  • Hydrodynamic (thick-film) lubrication - full film; friction is lowest near the transition and then rises slowly because viscous shear grows with speed. Designs operate well to the right of the minimum-friction point so that small speed drops or load rises do not push the bearing into mixed lubrication.

Hydrodynamic action. The rotating journal drags viscous oil into the converging wedge formed when the journal runs eccentric in the bush. Pressure builds in the wedge and supports the load. Three conditions are needed: relative motion, a converging gap, and a viscous fluid supplied continuously. The journal centre moves to an eccentricity e; eccentricity ratio ε = e/c; minimum film thickness h0 = c·(1 − ε). Higher load or lower speed or viscosity increases ε and thins the film.

Hydrostatic bearings use an external pump to supply pressurised oil, so they carry load even at zero speed (machine-tool slides, telescopes).

Petroff's equation assumes a lightly loaded, concentric journal (no eccentricity): the viscous shear torque gives f = 2π²·(μN/p)·(r/c). It underestimates friction for loaded bearings but shows the key groups.

Sommerfeld number S = (r/c)²·μN/p is the dimensionless design parameter of a full journal bearing. With S and the length-to-diameter ratio L/D, Raimondi-Boyd charts give ε, h0/c, the friction variable (r/c)·f, oil flow and temperature rise. Practical L/D is about 0.5 to 1.5.

Heat balance. Friction power heats the oil; viscosity falls sharply with temperature, so the operating viscosity must be found by iteration with the temperature rise (or from your data book's empirical heat-dissipation equations). Indian textbooks also use McKee's empirical friction equation - take its constants from the data book.

Dry and porous bushes are rated by a limiting pressure-velocity product (pV), which controls frictional heating; check p, V and pV against catalogue limits.

Formulas

p = W / (L·D)

  • p: unit bearing pressure (Pa); W: radial load (N); L: bearing length (m); D: journal diameter (m).

S = (r/c)²·(μ·N / p)

  • S: Sommerfeld number; r: journal radius (m); c: radial clearance (m); μ: dynamic viscosity (Pa·s); N: speed (rev/s).

f = 2π²·(μ·N / p)·(r/c) (Petroff, concentric, lightly loaded)

  • f: coefficient of friction.

T_f = f·W·r · P_loss = T_f·2π·N = f·W·v

  • T_f: friction torque (N·m); P_loss: power loss (W); v: journal surface speed (m/s).

h0 = c·(1 − ε), ε = e / c

  • h0: minimum film thickness (m); e: eccentricity (m); ε: eccentricity ratio.

p·V ≤ (pV)_limit (dry or porous bushes)

  • V: sliding speed (m/s).

Worked examples

Example 1 (standard). A journal bearing, D = L = 50 mm, radial clearance 0.04 mm, carries 5 kN at 1500 rpm with oil of viscosity 0.02 Pa·s. Find p, S, the Petroff friction coefficient and the power loss.

  1. p = W/(L·D) = 5000 / (0.05 × 0.05) = 2.0 MPa.
  2. r/c = 25/0.04 = 625; N = 1500/60 = 25 rev/s.
  3. S = (r/c)²·μN/p = 625² × 0.02 × 25 / 2.0×10⁶ = 0.0977.
  4. f = 2π²·(μN/p)·(r/c) = 19.74 × (0.02 × 25 / 2.0×10⁶) × 625 = 0.00308.
  5. T_f = f·W·r = 0.00308 × 5000 × 0.025 = 0.386 N·m; P_loss = T_f·2πN = 0.386 × 2π × 25 = 60.6 W.

Example 2 (GATE level). A full journal bearing with L/D = 1, D = L = 60 mm, radial clearance 0.03 mm, runs at 1200 rpm with μ = 0.012 Pa·s under 7.2 kN. For this Sommerfeld number the Raimondi-Boyd chart (L/D = 1) gives h0/c ≈ 0.5 and (r/c)·f ≈ 3.22 (read from your chart). Find h0, f, the power loss and compare with Petroff.

  1. p = 7200 / (0.06 × 0.06) = 2.0 MPa; r/c = 30/0.03 = 1000; N = 20 rev/s.
  2. S = 1000² × 0.012 × 20 / 2.0×10⁶ = 0.120.
  3. h0 = 0.5 × 0.03 = 0.015 mm.
  4. f = 3.22 / 1000 = 0.00322.
  5. T_f = 0.00322 × 7200 × 0.030 = 0.696 N·m; P_loss = 0.696 × 2π × 20 = 87.4 W.
  6. Petroff: f = 2π²·S/(r/c) = 2π² × 0.120 / 1000 = 0.00237 - about 26% low, because the loaded journal runs eccentric.

Common mistakes

  • Using N in rpm instead of rev/s in S and Petroff's equation (or vice versa with the chart's convention).
  • Using diametral clearance where the formula needs radial clearance (r/c, not D/c).
  • Projected area is L·D, not π·D·L.
  • Assuming a higher-viscosity oil is always better: friction and heating rise, and viscosity falls with temperature.
  • Applying Petroff to a heavily loaded bearing and trusting the result.
  • Forgetting that hydrodynamic bearings run in boundary lubrication at every start and stop.

For GATE ME

Questions typically ask for the Sommerfeld number, Petroff friction or power loss, the unit bearing pressure, and conceptual points on the Stribeck curve, hydrodynamic versus hydrostatic lubrication and the effect of load, speed and viscosity on film thickness. Practise unit handling (rev/s, radial clearance, Pa) because most errors come from there.

Quick check

  1. Three conditions for hydrodynamic lubrication?
  2. W = 3 kN, L = 40 mm, D = 50 mm. Unit pressure?
  3. If speed doubles with everything else fixed, how does S change?
  4. What lets a hydrostatic bearing carry load at zero speed? Answers: 1. Relative motion, a converging gap, a continuous supply of viscous fluid. 2. 1.5 MPa. 3. It doubles. 4. Oil is supplied under pressure by an external pump.

Interview questions

All Machine Design interview questions

Try answering each one aloud before you open it.

  1. 1.What is a sliding contact bearing, and how does it differ from a rolling contact bearing?Concept

    A sliding contact bearing, also known as a plain bearing, allows relative motion between two surfaces with sliding contact. It typically consists of a shaft rotating within a hole. In contrast, a rolling contact bearing uses rolling elements like balls or rollers between the surfaces to reduce friction. Sliding contact bearings are generally simpler and cheaper but have higher friction compared to rolling contact bearings.

  2. 2.Explain the principle of hydrodynamic lubrication in sliding contact bearings.Concept

    Under load the journal runs slightly eccentric in the bush, forming a converging wedge-shaped gap. Rotation drags viscous oil into the narrowing wedge, which builds pressure that lifts the journal on a full film so there is no metal contact. It needs relative motion, a converging gap and a continuous oil supply; film thickness grows with viscosity and speed and falls with load (the Sommerfeld number captures this), and at start-up and shutdown the bearing passes through boundary lubrication.

  3. 3.Why is hydrodynamic lubrication preferred in high-speed applications?Application

    Hydrodynamic lubrication is preferred in high-speed applications because it provides a full fluid film that separates the surfaces, minimizing friction and wear. At high speeds, the lubricant film is maintained effectively, reducing the risk of metal-to-metal contact and extending the life of the bearing. This results in smoother operation and increased efficiency.

  4. 4.What happens if the lubricant film in a sliding contact bearing breaks down?Application

    If the lubricant film breaks down, the surfaces may come into direct contact, leading to increased friction, wear, and potential damage to the bearing surfaces. This can cause overheating, increased energy consumption, and ultimately, bearing failure. Regular maintenance and proper lubrication are essential to prevent such issues.

  5. 5.How does viscosity of the lubricant affect the performance of a sliding contact bearing?Application

    The viscosity of the lubricant affects the thickness of the lubricant film. Higher viscosity lubricants can maintain a thicker film, which is beneficial for load-bearing capacity and reducing wear. However, too high a viscosity can increase friction and energy loss. The right balance is needed to ensure optimal performance and efficiency.

  6. 6.Explain the role of surface roughness in the performance of sliding contact bearings.Concept

    Surface roughness affects the ability of the lubricant to form a continuous film. Smoother surfaces allow for better film formation and reduced friction. However, some degree of roughness can help retain lubricant on the surface. The ideal surface finish depends on the specific application and operating conditions.

  7. 7.Why are materials like bronze and babbitt commonly used for sliding contact bearings?Application

    Babbitt (white metal) is soft with a low melting point, so it conforms to slight misalignment, embeds dirt particles and resists seizure if the film breaks down, but it has limited fatigue strength and is used as a thin lining on a steel backing. Bronzes are stronger and harder and carry higher loads and temperatures, with good compatibility against steel journals. The choice trades conformability and embeddability against load and fatigue capacity.

  8. 8.What is the effect of load on the hydrodynamic lubrication regime in sliding contact bearings?Application

    The load affects the thickness of the lubricant film in hydrodynamic lubrication. Higher loads can reduce the film thickness, increasing the risk of metal-to-metal contact. To maintain effective lubrication, the bearing design must ensure that the film thickness remains sufficient under the expected load conditions.

  9. 9.A journal bearing with D = L = 50 mm and radial clearance 0.025 mm carries 4 kN at 1500 rpm with oil of viscosity 0.02 Pa·s. Find the Sommerfeld number and the Petroff power loss.Numerical

    Unit pressure p = W/(L·D) = 4000/(0.05 × 0.05) = 1.6 MPa, N = 25 rev/s and r/c = 25/0.025 = 1000. S = (r/c)²·μN/p = 10⁶ × 0.02 × 25/1.6 × 10⁶ = 0.3125. Petroff f = 2π²(μN/p)(r/c) = 0.00617, so the friction torque is f·W·r = 0.617 N·m and the power loss is 0.617 × 2π × 25 ≈ 97 W.

  10. 10.A journal bearing of 100 mm diameter carries a load of 5000 N at 1000 rpm. If the coefficient of friction is 0.01, calculate the power lost in friction.Numerical

    Surface speed v = π·D·N/60 = π × 0.1 × 1000/60 = 5.24 m/s. Friction force = f·W = 0.01 × 5000 = 50 N, so power loss = f·W·v = 50 × 5.24 ≈ 262 W. Equivalently, friction torque f·W·r = 2.5 N·m times ω = 104.7 rad/s gives the same 262 W, which must be removed as heat by the oil.

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