Plain Bearings

A plain bearing supports a journal on the sliding surface of a bush or liner. The preferred operating state is a lubricant film that separates the surfaces and carries the load by hydrodynamic or hydrostatic pressure.

Preliminary checks

For a radial bearing of diameter \(d\) and length \(l\),

$$p=\frac{F}{dl},\qquad v=\frac{\pi dn}{60}.$$

Material screening commonly uses \(p\le[p]\), \(v\le[v]\), and \(pv\le[pv]\). These criteria do not replace a full-film lubrication analysis.

Hydrodynamic film

Journal eccentricity creates a converging wedge. Surface motion draws lubricant into the wedge and generates pressure. The Reynolds equation for an isoviscous steady film is

$$\frac{\partial}{\partial x}\left(h^3\frac{\partial p}{\partial x}\right)+\frac{\partial}{\partial z}\left(h^3\frac{\partial p}{\partial z}\right)=6\eta U\frac{\partial h}{\partial x}.$$

With radial clearance \(c\) and eccentricity ratio \(\varepsilon=e/c\),

$$h_{min}=c(1-\varepsilon).$$

The Sommerfeld number \(S=(\eta\omega/p)(r/c)^2\) characterizes operating regime.

Heat and reliability

Frictional power is \(P_f=fFv\) and must be removed by lubricant flow and the housing. Check minimum film thickness relative to roughness, oil viscosity at operating temperature, start–stop wear, misalignment, contamination, cavitation, seizure, and fatigue of the lining. Materials include bronzes, babbitts, sintered metals, polymers, and multilayer composites.

Bearing geometry and clearance

The length-to-diameter ratio affects load capacity, side leakage, alignment sensitivity, and heat removal. A longer bearing has a larger projected area but is more vulnerable to edge loading. Diametral clearance must permit an oil film, thermal expansion, manufacturing error, and lubricant flow without causing excessive vibration.

Lubrication regimes and the Stribeck curve

At low speed or high load, boundary contact dominates. As the parameter \(\eta v/p\) increases, the bearing passes through mixed lubrication into full-film operation. Friction first decreases as surfaces separate and then rises gradually because viscous shear becomes dominant.

Oil-film adequacy

The film parameter \(\lambda\) compares minimum film thickness with combined surface roughness. Full separation requires an adequate margin not only at steady rated speed but also during start-up, shutdown, load reversal, and transient overload. Shaft bending and housing distortion can reduce local film thickness at the edges.

Oil supply and groove design

Supply grooves should introduce oil in a low-pressure region without interrupting the principal load-carrying film. An incorrectly placed circumferential groove can divide the bearing and sharply reduce capacity. Feed pressure, flow, drain arrangement, aeration, filtration, and seal leakage are part of the lubrication system.

Hydrostatic bearings

Externally pressurized bearings carry load even at zero speed. Flow restrictors provide pocket stability and distribute pressure. They offer high stiffness, low starting friction, and precise motion but require a reliable pump, filtration, pressure control, and emergency response to supply loss.

Thermal equilibrium

Viscosity falls as temperature rises, reducing film thickness but also viscous loss. The operating point is obtained iteratively: assume temperature, determine viscosity and film behavior, calculate friction and oil flow, then solve the heat balance. Cooling or a larger oil supply may be required for high-speed or heavily loaded bearings.

Calculation sequence

1. Define load vector, speed range, duty, starts, temperature, and required life.
2. Select geometry, material, clearance, surface finish, and lubricant.
3. Check preliminary \(p\), \(v\), and \(pv\) limits.
4. Perform hydrodynamic or hydrostatic analysis for pressure, flow, and \(h_{min}\).
5. Check heat balance, stability, wear, edge loading, and start–stop operation.
6. Design oil feed, grooves, seals, filtration, monitoring, and maintenance.