Lubrication and Journal Bearings
Hydrodynamic lubrication from viscosity to design: Newton's viscous law, Petroff's friction equation, the Sommerfeld bearing characteristic number, Reynolds' theory of the converging film, and the Raimondi-Boyd charts that relate film thickness, friction, flow, and temperature rise for journal-bearing design.
12–1Types of Lubrication
Learning objectives
- Recognize hydrodynamic, hydrostatic, elastohydrodynamic, boundary, and solid-film regimes from operating cues.
- Relate viscosity, shear, clearance, speed, load, and temperature to journal-bearing behavior.
- Calculate Petroff friction, Sommerfeld number, film geometry, and boundary-bearing \(PV\).
- Use chart or supplied performance data without presenting empirical values as universal constants.
- Iterate viscosity and temperature while checking convergence, stability, film margin, flow, and power loss.
- Assess materials, tolerances, supply systems, start-stop duty, wear, and manufacturability.
Five Regimes
Lubrication is classified by how the surfaces are kept apart. Hydrodynamic (thick-film) lubrication fully separates the surfaces with a pressure film generated by the motion itself — no external pump needed. Hydrostatic lubrication supplies pressurized fluid from an external source. Elastohydrodynamic lubrication occurs where surfaces contact in rolling/sliding (gears, rolling bearings) and the film pressure elastically deforms the surfaces. Boundary lubrication exists when the film is too thin to fully separate the surfaces (films of molecular thickness). Solid-film lubrication uses graphite or molybdenum disulfide where liquids fail (high temperature).
Physical meaning
A lubrication regime identifies what physically carries the normal load. A full fluid film transmits load through pressure without nominal solid contact; a boundary film shares load with asperities, so wear and surface chemistry become central. The same bearing can cross regimes during startup, shutdown, overload, contamination, or loss of supply.
Problem-solving strategy
- Draw the journal, load direction, rotation, supply, and clearance; solve the bearing reaction from shaft equilibrium.
- Identify the likely regime and distinguish fluid-film performance from boundary wear checks.
- Use viscosity at the estimated film temperature, not automatically at the supply temperature.
- Compute \(P\), \(r/c\), and \(S\); obtain chart quantities only from stated geometry and traceable data.
- Iterate temperature and viscosity, then check film margin, friction power, flow, pressure, materials, and tolerances.
12–2Viscosity
Newton's Viscous Law
Absolute (dynamic) viscosity \(\mu\) is the proportionality between shear stress and velocity gradient in a fluid — a measure of internal frictional resistance. Fluids that obey this linear relation are Newtonian. The ips unit is the reyn, \(1\ \text{reyn}=1\ \text{lbf}\!\cdot\!\text{s}/\text{in}^2\); the SI relation is \(1\ \text{reyn}=6890\ \text{Pa}\!\cdot\!\text{s}\). The cgs poise (and centipoise) are still common in lubricant data.
Kinematic Viscosity and Saybolt
Kinematic viscosity \(\nu = \mu/\rho\) is what the Saybolt Universal viscometer measures (seconds for 60 mL to drain). The conversion from Saybolt seconds uses the Hagen-Poiseuille relation. Lubricant viscosity falls sharply with temperature, which is central to journal-bearing analysis: as the oil heats in the loaded zone, its viscosity drops, requiring an iterative solution.
Viscosity and data provenance
Dynamic viscosity has dimensions of stress multiplied by time; kinematic viscosity divides that resistance by density. Always state which one is tabulated and at what temperature. A viscosity grade names a range, not an exact operating value, so released analyses need a supplier viscosity-temperature curve and aging or contamination allowances.
12–3Petroff's Equation
Friction of a Lightly Loaded Journal
Petroff (1883) explained bearing friction by assuming the journal is concentric (lightly loaded). With a journal of radius \(r\) rotating at \(N\) rev/s in a bearing of radial clearance \(c\), the shear of the oil film gives the friction coefficient directly. Though it ignores the eccentric load-carrying film, Petroff's equation matters because it identifies the two governing dimensionless parameters \(\mu N/P\) and \(r/c\).
The Sommerfeld Number
The bearing characteristic number, or Sommerfeld number \(S\), bundles the designer-controlled parameters into one dimensionless group. It is the master variable of journal-bearing design: every Raimondi-Boyd chart plots a performance variable against \(S\). Here \(P = W/(2rl)\) is the unit load on the projected bearing area.
Applicability of the dimensionless model
Petroff's concentric model estimates viscous drag for a lightly loaded, full journal with a Newtonian, isoviscous film and negligible end effects. The Sommerfeld number organizes full-film performance, but it is not itself a safety factor. Low \(S\), cavitation, turbulence, starvation, misalignment, elastic distortion, or non-Newtonian response can invalidate simplified results.
12–4Stable Lubrication
The Stick-Slip Boundary
Plotting friction \(f\) against the parameter \(\mu N/P\) shows two regimes. To the right, in thick-film (stable) lubrication, a rise in temperature lowers viscosity and hence \(\mu N/P\), which lowers friction, which cools the bearing — a self-correcting, stable loop. To the left of a minimum (around \(\mu N/P \approx 1.7\times10^{-6}\)), in boundary (unstable) lubrication, lower viscosity raises friction and heat, a runaway toward seizure. Designers keep the operating point well to the right of this minimum.
12–5Thick-Film Lubrication
Eccentricity and the Converging Wedge
Under load the journal shifts off-center; the offset between journal and bearing centers is the eccentricity \(e\), and the eccentricity ratio is \(\epsilon = e/c\). This creates a converging-diverging film. The minimum film thickness \(h_0 = c - e = c(1 - \epsilon)\) occurs on the line of centers; the position of \(h_0\) and the point of maximum pressure shift with load. Maintaining an adequate \(h_0\) against surface roughness is the central design goal.
12–6Hydrodynamic Theory
Tower's Experiment, Reynolds' Equation
Beauchamp Tower's 1880s friction experiments accidentally revealed film pressures high enough to blow out a lubricator plug — direct evidence of a self-generated pressure film. Osborne Reynolds explained it: the lubricant adheres to both surfaces and is dragged by the moving journal into a converging wedge, generating pressure. The governing Reynolds equation (1-D form shown) balances the pressure-driven and shear-driven flows; Sommerfeld's closed-form solution expresses the performance variables as functions of the eccentricity ratio.
12–7Design Considerations
Given vs. Sought Variables
Two groups of variables organize the design. Controlled (given) variables: the viscosity \(\mu\), the load per unit area \(P\), the speed \(N\), and the bearing dimensions \(r, c, \beta, l\). Dependent (sought) performance variables: the coefficient of friction \(f\), the temperature rise \(\Delta T\), the oil flow rate \(Q\), and the minimum film thickness \(h_0\). The general Sommerfeld number uses the journal speed \(N_j\); a minimum film thickness floor of roughly \(h_0 \ge (0.0002 + 0.00004d)\) in guards against asperity contact.
12–8The Relations of the Variables (Raimondi-Boyd)
The Design Charts
Raimondi and Boyd solved Reynolds' equation numerically and published ~45 charts plotting each performance variable against the Sommerfeld number \(S\) for several \(l/d\) ratios. Key chart outputs: the minimum-film-thickness variable \(h_0/c\) (and its position \(\phi\)), the coefficient-of-friction variable \((r/c)f\), the flow variable \(Q/(rcNl)\) with side-leakage fraction \(Q_s/Q\), and the temperature-rise group. A designer computes \(S\), reads each variable off the chart at the appropriate \(l/d\), and solves back for the physical quantity.
The Iterative Temperature Solution
Because viscosity depends on temperature, and temperature rise depends on viscosity, the solution iterates: estimate \(\Delta T\), read \(\mu\) at the average film temperature, compute \(S\) and the chart variables, recompute \(\Delta T\), and repeat until the assumed and computed \(\Delta T\) converge. This trumpet-curve viscosity behavior makes the temperature loop the heart of journal-bearing analysis.
12–9Steady-State Conditions in Self-Contained Bearings
Heat Balance with No External Cooling
A self-contained bearing (pillow block, ring-oiled) has no external lube circulation; it dissipates the friction heat to the surroundings by convection and radiation from its housing. Steady state is found by equating heat generated by friction (\(H_{gen} = 2\pi T N = f W r \omega\)) to heat dissipated by the housing (\(H_{loss} = \hbar_{CR} A (T_b - T_\infty)\)). The equilibrium film temperature is solved iteratively, since both viscosity and the heat balance depend on it.
12–10Clearance
The Clearance Trade-Off
Radial clearance \(c\) is one of the most consequential and difficult choices. Small clearance raises load capacity and film stiffness but runs hotter and is sensitive to manufacturing tolerance and thermal growth. Large clearance runs cooler with more flow but lowers capacity and can become noisy. Because clearance grows in service from wear, designers analyze a range of clearance (tolerance band) and ensure acceptable performance — adequate \(h_0\) and acceptable \(\Delta T\) — across the whole range, not just the nominal value.
Specify diametral versus radial clearance unambiguously. Manufacturing capability, roundness, shaft finish, alignment, press-fit closure, and differential thermal expansion determine the operating clearance; analyzing only a nominal drawing value confuses manufactured geometry with the hot running condition.
12–11Pressure-Fed Bearings
Forced Circulation to Limit Temperature
To cut the film temperature rise, lubricant is pumped in under supply pressure \(p_s\) — a pressure-fed bearing. A central circumferential groove with a supply hole opposite the load zone effectively creates two half-bearings of smaller \(l/d\). Assuming laminar flow and negligible groove pressure drop, the pumped side flow carries away the friction heat, so the temperature rise is set by the energy balance between friction power and the heat capacity of the through-flow.
12–12Loads and Materials
Unit Loads and Bearing Alloys
Typical allowable unit loads \(P = W/(2rl)\) range from a few hundred psi (light machinery) to a few thousand psi (engine bearings). A good bearing material embeds dirt, conforms to misalignment, resists scoring, and has fatigue strength: babbitt (tin/lead based) is the classic soft, conformable lining; bronze and aluminum alloys carry higher loads; bimetal and trimetal bearings combine a strong backing with a soft overlay. The journal is normally harder than the bearing so wear concentrates in the replaceable bearing.
12–13Bearing Types
Full, Partial, and Fitted
A full journal bearing wraps the journal 360°; a partial bearing (e.g. 120°) covers only the loaded arc, reducing friction and flow where the unloaded arc is unnecessary. A fitted bearing has equal journal and bearing radii (zero nominal clearance). Bushings may be solid or split for assembly. Selection balances load, speed, alignment, and serviceability.
12–14Thrust Bearings
Hydrodynamic Thrust Films
A hydrodynamic thrust bearing carries axial load on a film generated by tilted or stepped pads that form converging wedges as the collar slides over them — the same Reynolds mechanism in linear form. Tilting-pad (Kingsbury/Michell) thrust bearings let each pad pivot to its optimum film angle automatically, carrying very high axial loads (e.g. hydro-turbine shafts). Fixed-incline and step pads are simpler alternatives.
12–15Boundary-Lubricated Bearings
When a Full Film Cannot Form
At low speed, high load, or frequent start-stop, no hydrodynamic film develops and the bearing runs in boundary lubrication — load carried on thin adsorbed films and asperity contact. Design shifts from film theory to a wear/PV limit: the product of unit pressure \(P\) and sliding velocity \(V\) must stay below a material limit (\(PV \le (PV)_{max}\)) to keep frictional heating and wear acceptable. Self-lubricating materials — porous bronze impregnated with oil, PTFE composites, graphite — are used for these "boundary" or "bushing" bearings.
Recognition strategy
Uniform parallel spacing and specified Newtonian viscosity indicate simple Couette flow, so Newton's viscous law applies before any journal-bearing pressure-wedge model is needed.
Given information
- \(h=0.500\ \text{mm}=5.00\times10^{-4}\ \text{m}\), \(U=3.00\ \text{m/s}\), \(A=0.0400\ \text{m}^2\), and \(\mu=0.0850\ \text{Pa}\!\cdot\!\text{s}\).
Model and interpretation
Let positive \(x\) follow upper-surface motion and \(y\) cross the film. No slip, constant viscosity, steady laminar flow, and a linear velocity profile are assumed; pressure flow is absent.
Verification
Units: \((\text{Pa}\!\cdot\!\text{s})(\text{s}^{-1})=\text{Pa}\); pressure times area gives newtons, and newtons times meters per second gives watts.
Signs and direction: The computed values are magnitudes; shear on the moving plate acts in negative \(x\), while the drive supplies positive force and power.
Plausibility: Halving the film thickness would double gradient, shear, force, and power, which agrees with the inverse-clearance behavior of viscous drag.
Engineering interpretation
The calculation isolates viscous drag and demonstrates why small clearance raises heat generation. A real bearing also has pressure-driven flow, changing temperature, end leakage, surface roughness, and possible turbulence that must be evaluated separately.
Recognition strategy
The request explicitly calls for a concentric light-load screening estimate. Petroff's equation gives friction, while the same inputs form the Sommerfeld number used for later chart analysis.
Given information
- \(r=0.0400\ \text{m}\), \(l=0.0600\ \text{m}\), \(c=5.00\times10^{-5}\ \text{m}\), \(P=1.80\times10^6\ \text{Pa}\), \(N=25.0\ \text{s}^{-1}\), and \(\mu=0.0300\ \text{Pa}\!\cdot\!\text{s}\).
Model and interpretation
The journal rotates positively and the bearing reaction magnitude defines \(P\). The film is Newtonian, concentric, isoviscous, fully flooded, and laminar; eccentric pressure support is not predicted.
Verification
Units: Both \(S\) and \(f\) are dimensionless because \(\mu N/P\) is dimensionless; load, radius, and angular speed then give torque and watts.
Signs and direction: Friction torque opposes positive rotation, so the drive must supply \(+2.27\ \text{N}\!\cdot\!\text{m}\) to maintain speed.
Plausibility: The friction coefficient is much less than dry sliding values, and \(2.27(2\pi\times25)=357\ \text{W}\) independently confirms power.
Engineering interpretation
This is a drag estimate, not proof of adequate film thickness or pressure. A professional design would use the correct finite-length chart or validated computation, iterate operating viscosity, and check heat rejection, cavitation, flow, roughness, and clearance tolerance.
Recognition strategy
Frequent stops prevent dependable hydrodynamic separation, so this is a boundary-wear screening decision. Compare projected pressure times sliding speed with traceable supplier data.
Given information
- \(d=0.0500\ \text{m}\), \(l=0.0600\ \text{m}\), \(W=3200\ \text{N}\), \(N=180\ \text{rev/min}=3.00\ \text{rev/s}\), supplied \((PV)_{\max}=0.650\ \text{MPa}\!\cdot\!\text{m/s}\), and \(f_b=0.080\).
Model and interpretation
The reaction is uniformly projected over \(dl\), surface speed is positive, and \(PV\) uses magnitudes. Edge loading, oscillation, shock, temperature derating, and wear life are outside this screen.
Verification
Units: Projected load over area gives pascals, circumferential speed gives meters per second, and multiplying friction force by speed gives watts.
Signs and direction: Pressure and \(PV\) are positive magnitudes; friction acts opposite sliding, while \(120.6\ \text{W}\) is positive heat generation.
Plausibility: The computed \(PV\) is below the supplied limit but not by an order of magnitude, so calling the result a conditional pass is appropriate.
Engineering interpretation
The explicit decision passes only the supplied steady \(PV\) screen. Release still requires supplier temperature and oscillation derating, peak-load history, shaft finish and hardness, installation tolerance, heat rejection, wear-life evidence, lubrication schedule, and testing.
Chapter recap
- Choose the lubrication regime before choosing an equation; full-film strength and boundary wear are different failure models.
- Use operating dynamic viscosity and projected unit load to form \(S\), then trace chart or computed data to its source.
- Iterate viscosity and temperature and assess clearance as a manufactured, thermally changing range.
- Before moving on, be able to compute shear, friction, \(S\), and \(PV\), then check units, direction, plausibility, film, heat, flow, wear, and provenance.