Journal Bearing Design - with hydrodynamic lubrication
for parallel shaft and journal axes |

The Java applet below calculates a number of values relating to the design of this type of bearing.
**1. Assumptions**

i) The pressure distribution is approximately 'half Sommerfeld', ie p = 0 at theta = 0 and 180^{o} and p = 0 between 180^{o} and 360^{o}.
While this gives a slightly different pressure distribution to some alternatives, the overall load carrying capacity for this method
is very similar to the alternative methods.

ii) The oil inlet hole is located in the middle of the bearing at the point of maximum film thickness where the oil film pressure is zero.
Values must be entered for the oil supply pressure and the diameter of the oil inlet hole. The local
film thickness is the dominant factor determining the oil flow, see 'Basic Lubrication Theory', by Alastair Cameron, 3rd Ed., 1981, Ellis Horwood.

**2. Using the applet**

The solution of the pressure distribution is iterative and uses 'relaxation'. However if
over relaxation is used (ie an SoR factor greater than 1) the solution often fails to converge. This can normally
be overcome by using a 'relaxation' factor (SoR factor) less than 1, such as 0.9 which should be entered in the appropriate box. It must be
noted that using a factor of less than 1 means that a large number of iterations will probably be needed, normally more than 200 and possibly more than 1000.
If the solution does not converge with an SoR factor of 0.9 after 1000 iterations, further increase the number of
iterations, if the solution is diverging, reduce the SoR factor and increase the number of iterations.

The text area on the right hand side first displays the film thickness round the bearing (m) for the specified eccentricity ratio and then lists the
total radial force (N) after each iteration. If a sufficient number of iterations has been chosen, then the total
radial force should be almost constant.

An even number of circumferential intervals and length intervals **must** be chosen.
About 42 circumferential and 14 length intervals usually gives a reasonably good solution. However to obtain more accurate values,
particularly a reasonably accurate value of the maximum
oil film pressure in the centre of the journal, 180 intervals round the circumference should be specified with 16 or more intervals specified along the journal length

For some of the inputs, suggested values are shown in brackets that are typical for mineral oils.

The eccentricity ratio 'epsilon' = 1 - h_{o}/c

The text area at the bottom of the applet displays the pressure distribution (MPa) in the bearing.

There is a large empty area at the bottom of the page (this simplifed the program).

**3. Suggested Procedure for Design**)

A number of parameters may be under the control of the designer, but
there are another group which are dependent on the first group and can be used to
define operational limits for the bearing. This applet assumes that the oil viscosity
is constant throughout the film, which is not very accurate as the oil temperature rises
as it passes through the bearing and as the viscosity is strongly dependent upon
the temperature, it means that bearing design normally involves some iteration.

Given or controlled by the designer:

- lubricant viscosity
- load per unit projected area
- speed, N
- dimensions: r (bearing radius), c (radial clearance), l (journal length) and beta (the angle subtended by the load bearing portion of the bearing).
(For the methodology here it is assumed that all the load is carried by approximately 180
^{o}of the bearing).

The following are dependent upon the first group:

- coefficient of friction
- temperature rise
- oil flow rate, Q
- minimum film thickness, h
_{o}

Many charts use US units for viscosity, ie reyns (usually plotted as micro-reyns). To convert reyns to Pa.s, multiply by 6890.

In the absence of specific information, it may be assumed that mineral lubricating oil has a density of about 850 kg/m^{3} and a specific heat of about 1675 - 1800 J/kg^{o}C.

For hydrodynamic bearings, a length / diameter ratio of about 1 (say 0.8 to 1.3) is believed to be a good compromise. l/d ratios of <1 may be used where a compact design is important, such as in a multi-cylinder automotive engine. Reducing the l/d ratio increases the flow out of the bearing ends, which aids cooling.

The minimum film thickness acceptable depends upon surface finish and should allow expected particles to pass through without causing damage. For some applications, eg in automotive engines, filtering is provided to remove particles whose size would be likely to exceed the minimum film thickness. The following h_{o} values have been suggested:

Maximum oil temperatures should not be allowed to be excessive as oxidation and degradation become rapid. For general purpose machinery, an oil operating temperature of 60^{o}C should give a good long life. Above 100^{o}C the rate of oxidation increases rapidly. Temperatures of 120^{o}C should be avoided in industrial equipment. In automotive engines lubricant temperatures can reach 180^{o}C, but automotive oils are specially formulated (and may even be fully 'synthetic') to withstand such conditions.

The list below gives typical values of bearing 'nominal' pressures (load/length x diameter).

Electric motors, steam turbines, gear reducers, centrifugal pumps about 1 MPa

Automotive engines - main bearings 4 - 5 MPa

Diesel engines - main bearing 6 - 12 MPa

**4. Design Procedure**

1 Select an l/d ratio, 1 is probably a good starting point.

2 Using the specified load and an appropriate 'nominal' pressure, select the bearing length and diameter.

3 Specify an appropriate radial clearance, c, probably based on either a close (H8/f7) or free (H9/d9) running fit.

4 Decide on an initial lubricant viscosity. Because viscosity varies considerably with temperature, it is normally necessary to carry out the calculations, below, at two values of viscosity, one slightly below and the other slightly above the anticipated final value.

5 Enter the various inputs in the text boxes adjacent to the green labels, using the units specified.

6 Select and enter the first eccentricity ratio value in the appropriate box

6 Click the yellow 'Read Data' button.

7 Check the output values. If the 'Resultant Force' does not closely match the specified load, change the eccentricity ratio (if the resultant force is less than the Load specified, increase the eccentricity ratio) and run the calculation again.

8 Repeat the above calculations as needed, varying the inputs as appropriate. .

David J Grieve, 10th December 2009.