Lubrication and journal bearings

Lubrication reduces friction and wear and carries away heat by putting a film of lubricant between sliding surfaces.

Types of lubrication

1. Hydrodynamic (thick film): the surfaces are completely separated by an oil film whose pressure is generated by the motion itself (wedge action). Friction is very low (coefficient 0.001 to 0.01) and there is no wear. Example: journal bearing running at normal speed.
2. Hydrostatic: oil is supplied under pressure from an external pump, so the film exists even at zero speed. Used for heavy slow machines such as telescopes and large turbines.
3. Elastohydrodynamic (EHL): in rolling contacts such as gears and rolling bearings, the high contact pressure deforms the surfaces elastically and raises the oil viscosity so a very thin film is formed.
4. Boundary: the film is only a molecular layer, and the surfaces touch at asperities. Friction coefficient 0.05 to 0.2. Occurs at low speed, high load or starting and stopping. Additives (anti-wear, extreme pressure) protect the surface.
5. Mixed (partial): a combination of boundary and thick film lubrication.

Friction against $ZN/p$

   f
   |\
   | \                 /
   |  \      mixed    /  thick film
   |   \_           _/
   |boundary \_____/
   +------------------------> ZN/p
              ^
          minimum friction
          (critical value)

Here $Z$ is viscosity, $N$ speed and $p$ bearing pressure. To the right of the minimum, the friction rises slowly with $ZN/p$ and the bearing is in stable lubrication: if the temperature rises, viscosity falls, $ZN/p$ falls, friction decreases and the bearing recovers. To the left, a fall in viscosity increases friction and heat further, leading to seizure (unstable). Journal bearings are designed so that $ZN/p$ is about three times the critical value.

Viscosity

• Dynamic (absolute) viscosity $\mu$ (or $Z$): the ratio of shear stress to velocity gradient, $\tau = \mu\,\dfrac{du}{dy}$. SI unit Pa.s (N.s/m$^2$); 1 poise = 0.1 Pa.s; 1 cP = 1 mPa.s.
• Kinematic viscosity $\nu = \mu/\rho$. SI unit m$^2$/s; 1 stoke = $10^{-4}$ m$^2$/s.

Viscosity falls with temperature; the viscosity index measures how little it changes.

Hydrodynamic theory (Osborne Reynolds, 1886, after Tower's experiments) explains how a pressure is generated in a converging oil film, so that a shaft can float on oil.

Film formation in a journal bearing

   at rest        starting        running
   _____           _____          _____
  (  O  )         ( O   )        (  O  )
 shaft on        climbs up      floats on a
 bottom          the bearing    wedge of oil
                 wall           (offset by e)

1. At rest, the journal rests on the bottom of the bearing, with oil squeezed out.
2. When the shaft starts to rotate it climbs the bearing wall in boundary lubrication.
3. As speed increases, the rotating journal drags oil by viscosity into the narrowing space (converging wedge). The oil is pushed along the wedge and pressure builds up.
4. At a certain speed the pressure supports the load; the journal lifts and takes an eccentric position, with its centre displaced by $e$ from the bearing centre and slightly to one side of the load line.

Assumptions of Reynolds

• The lubricant is Newtonian and incompressible, with constant viscosity.
• Flow is laminar; inertia and body forces are neglected.
• Pressure does not change across the film thickness.
• Film thickness is very small compared with the radius, so curvature is neglected.
• Surfaces are rigid and smooth, and there is no slip at the boundaries.
• (For the simplest case) no side leakage, i.e. infinitely long bearing.

Terms

• Radial clearance $c = R - r$ (bearing radius minus journal radius).
• Eccentricity ratio $\varepsilon = e/c$.
• Minimum film thickness $h_0 = c - e = c(1 - \varepsilon)$.
• Sommerfeld number (bearing characteristic number)

$$S = \left(\frac{r}{c}\right)^2 \frac{\mu N}{p}$$

where $\mu$ is viscosity (Pa.s), $N$ speed (rev/s) and $p = W/(ld)$ the projected-area pressure. $S$ is dimensionless. For a given $l/d$, the design charts (Raimondi and Boyd) give $\varepsilon$ (or $h_0/c$), friction variable, flow and temperature rise as functions of $S$. A high $S$ (viscous oil, high speed, light load) means a thick film and small eccentricity; a low $S$ means large eccentricity and a thin film.

Petroff's law

Petroff assumed a lightly loaded journal running concentric with the bearing, with the clearance $c$ filled with oil, so that the shear of the oil film gives the friction. The law states that the friction torque is proportional to viscosity and speed and inversely proportional to the clearance.

Derivation

Journal radius $r$, length $l$, speed $N$ rev/s, load $W$.

Surface velocity $U = 2\pi r N$. Shear stress in the film $\tau = \mu U/c = \dfrac{2\pi\mu r N}{c}$.

Friction force $F = \tau \times (2\pi r l)$ and torque $T = F\,r$:

$$T = \frac{4\pi^2 \mu r^3 l N}{c}$$

Also $T = f W r$ and $W = p\,(2r)\,l$ (projected area pressure). Therefore

$$f = \frac{T}{W r} = \frac{4\pi^2\mu r^3 l N}{c \cdot p\,(2rl)\, r} = 2\pi^2\,\frac{\mu N}{p}\,\frac{r}{c}$$

This is the Petroff equation.

Numerical

Data: $r = 37.5$ mm, $l = 75$ mm, $c = 0.075$ mm, $r/c = 500$, $N = 1000/60 = 16.67$ rev/s, $\mu = 0.025$ Pa.s, $W = 3000$ N.

$$p = \frac{W}{ld} = \frac{3000}{0.075 \times 0.075} = 0.533\times10^6\ \text{Pa}$$ $$\frac{\mu N}{p} = \frac{0.025 \times 16.67}{0.533\times10^6} = 7.81\times10^{-7}$$ $$f = 2\pi^2 \times 7.81\times10^{-7} \times 500 = 0.00771$$ $$\begin{aligned} T &= f W r = 0.00771 \times 3000 \times 0.0375 = 0.867\ \text{N.m} \\ \text{Power loss} &= T\omega = 0.867 \times 2\pi \times 16.67 = 90.8\ \text{W} \end{aligned}$$ $$S = \left(\frac{r}{c}\right)^2\frac{\mu N}{p} = 500^2 \times 7.81\times10^{-7} = 0.195$$

Answer: f = 0.0077; T = 0.867 N.m; power loss = 90.8 W; Sommerfeld number = 0.195.

Data: $d = 60$ mm, $l = 60$ mm, $r = 30$ mm, $c = 0.06$ mm, so $r/c = 500$; $N = 1500/60 = 25$ rev/s; $W = 4500$ N; $\mu = 0.02$ Pa.s.

Bearing pressure

$$p = \frac{W}{ld} = \frac{4500}{0.06 \times 0.06} = 1.25\ \text{MPa}$$

Sommerfeld number

$$S = \left(\frac{r}{c}\right)^2\frac{\mu N}{p} = 500^2 \times \frac{0.02 \times 25}{1.25\times10^6} = 0.10$$

Minimum film thickness

$$h_0 = 0.35\,c = 0.35 \times 0.06 = 0.021\ \text{mm}$$

Coefficient of friction

$$f = \frac{2.9}{r/c} = \frac{2.9}{500} = 0.0058$$

Friction torque and power loss

$$\begin{aligned} T &= f W r = 0.0058 \times 4500 \times 0.03 = 0.783\ \text{N.m} \\ P_{loss} &= 2\pi N T = 2\pi \times 25 \times 0.783 = 123\ \text{W} \end{aligned}$$

Oil flow

$$\begin{aligned} Q &= 3.6\,(r\,c\,N\,l) = 3.6 \times 0.03 \times 0.06\times10^{-3} \times 25 \times 0.06 \\ &= 9.72\times10^{-6}\ \text{m}^3/\text{s} = 9.72\ \text{cm}^3/\text{s} \end{aligned}$$

Trumpler's criterion

$$h_{0,min} = 0.005 + 0.00004 \times 60 = 0.0074\ \text{mm}$$

The actual $h_0 = 0.021$ mm is nearly three times this, so the film is thick enough to cover the surface roughness and small particles: the design is safe. (The oil temperature rise, from the heat generated, should still be checked.)

Answer: p = 1.25 MPa; S = 0.10; h0 = 0.021 mm (greater than 0.0074 mm); f = 0.0058; power loss = 123 W; oil flow = 9.7 cm$^3$/s.