Lubrication - Example Calculation |

The standard assumptions are made in this calculation, see lubrication notes.

The design procedure used in this example makes use of dimensionless groups plotted against the Sommerfeld number on a series of charts (A A Raimondi and J Boyd, "A Solution for the Finite Journal Bearing and its Application to Analysis and Design", parts I, II and III, Trans. ASLE, vol 1, no. 1, 159-209) to determine other values for the bearing. These charts are reproduced in many texts on Mechanical Engineering Design. To follow this example you should have a copy of one of the texts that include these charts - marked * in the References.

A plain bearing with hydrodynamic lubrication is to be designed to carry a load of 2500 Newtons at a shaft speed of 30 revs per second = N.

**Initial assumptions** are that a SAE
grade 40 oil will be used and the mean oil temperature in the bearing is 68.3^{o}C
(which corresponds to 155^{o}F). At this temperature the oil has an absolute
viscosity of 4 micro reyn (i.e. 27.5 mPa.s or 0.0275 Pa.s). The density is assumed to be
850 kg/m^{3} and the specific heat 1800 J/kg ^{o}C

As a starting point it will be assumed that the: length / diameter ratio = 1 and
for a first calculation length = diameter = 2 radius = 40 mm will be used. l = d = 2 r = 40mm

It is also assumed that the bearing / shaft is between a 'free running fit' - H9/d9 and a 'close
running fit' - H8/f7 with a radial clearance of 0.04 mm.

**2. Calculation**

First check that the 'unit load' (load / projected area of bearing) is acceptable:

The unit load = load / (length x diameter) = 2500 / (0.04 x 0.04) = 1.56 MPa. This
is perhaps on the high side for machinery (but low for IC engines) but this value is accepted
for this example.

Bearing designs are commonly based on charts using the 'Bearing Characteristic Number' or Sommerfeld
Number, this is defined (in a non - dimensional way) as:

S = ( r / c )

where r is radius, c is the radial clearance, mu is the absolute viscosity, N is shaft speed in revs/s and P is the unit load.

As there is no information given about rotation of the bush or load vector, it is assumed that these are not rotating and that S = S

Substituting values: S = (0.02 / 0.00004)

**The first chart** in the series plots the minimum film thickness variable h_{o}/c
where h_{o} is the minimum film thickness against the Sommerfeld number for a variety
of l/d ratios. From this chart h_{o}/c = 0.41 so the minimum film thickness

It is important that the minimum film thickness be significantly greater than the maximum surface roughness. Keep in mind the fact that surface roughness is usually given as an average roughness amplitude (Ra) and the extreme peak to valley depths may well be 3 - 6 times greater than the Ra.

The minimum film thickness = radial clearance - eccentricity

h

The eccentricity of this design is e = c - h

Two lines indicating optima, maximum load and minimum power loss, are also marked on the first chart and the zone between these lines is normally considered to be a recommended operating region.

**The second chart** needed plots the coefficient of friction variable f r/c
where f is the coefficient of friction, against the Sommerfeld number. From this chart the
value of *f r/c* is 3.3 so

The torque to overcome the friction = coefficient of friction x load x radius, substituting values

The power lost is: torque x angular velocity, substituting values

**Rise in Oil Temperature.** It is assumed that the oil that is retained in the bearing
rises in temperature by deltaT and the oil that leaks out of the side of the bearing
rises in temperature by deltaT/2. First the volume flow rates need to be converted to kg/s.

Total oil mass flow rate = 4.128 E-6 x 850 = 0.003509 kg/s

Side leakage mass flow rate = 0.66 x 0.003509 = 0.002316 kg/s

Mass flow rate remaining in bearing = 0.34 x 0.003509 = 0.001193 kg/s

Carrying out an energy balance assuming that all the energy put into the oil raises only the oil temperature:

62.2 = 1800 (0.002316 deltaT/2 + 0.001193 deltaT)

The rise in oil temperature: deltaT = 14.7

As well as the above charts there are also charts to provide additional information such as the maximum film pressure, the angular position where the oil film thickness is a minimum and where it terminates.

**3. Interactive Java Applet**

This problem can also be solved using the
Java Applet to assist with journal bearing design given here

Comparison of Raimondi and Boyd chart results with Java applet (above) click here

David J Grieve. Revised: 23rd February 2010. Original: 24th June 2003.