# Mechanical Engineering Design Notes

 Design Contents

 COMPUTATION OF A JOURNAL BEARING WITH HYDRODYNAMIC LUBRICATION

Introduction
During the latter half of the 20th century the design of journal bearings that required hydrodynamic lubrication was facilitated by charts developed from finite element analysis (FEA) by Raimondi and Boyd [ref. 'A Solution for the Finite Journal Bearing and its Application to Analysis and Design: III', Trans. ASLE 1, 1958, 194-209, ]. As computers became widespread and more powerful, more software has been written to simplify the design procedure. These web pages contain two similar programmes written for this purpose. One approach makes use of a Java applet and the other is written in Scilab (which is a Free and Open Source Programme with many similarities to Matlab). These two approaches are discussed below.

For both programmes a finite differences approach is used with iterative solutions as the author believes that this is easier than writing FEA with direct solution.

The Java language (as well as being object orientated) requires very formally structured programming, eg all variables must be declared before use. Applets (that can run in a browser) and more complex Java programmes that do not run a browser, but require the Java run time environment to run, have to be compiled before they can be run. This conveniently means they run reasonably quickly. Java is available free and the author wanted the programmes to run over the www it was best best to use Java.

Scilab is a less structured language than Java (variables do not have to be declared before use) and instructions are interpreted on running. This means programmes in Scilab usually run much slower than Java. This may well be an issue when iterative solutions are being computed.

Both the Java and Scilab programmes use the Jacobi method for the iterative solution click here.

For many problems an iterative solution can be speeded up by overcorrecting the adjustment at each step of the iteration by multiplying the correction to be made by a factor greater than 1, eg 1.5. However this hydrodynamic bearing problem will not converge if 'over relaxation' a factor greater than 1 is used (nor if 1 is used). However a solution can be reached if 'under relaxation' is used, 0.9 seems to work. However this means that 1000+ or tens of thousands of iterations will be needed and as convergence is very slow, care is needed to ensure the solution is sufficiently accurate.

An advantage of Scilab programmes being interpreted is that they can be quickly modified and also that after a programme has run the user can display the (final) values of variables that were not originally printed which saves time with finding errors and validation.

It should be noted that the Java programme and one of the Scilab programmes make no allowance for the change in viscosity of the lubricant as the oil heats up while it is in the bearing. The second Scilab programme uses the heat generated while the oil is being sheared within the bearing to calculate the increase in temperature of the oil and then the viscosity in each 'slice' of the oil film. This has to be part of the iterative solution. A difficulty has been to develop an equation that is a reasonable fit to published data of how absolute viscosity varies with temperature - consequently the only data included in this first version of the program for viscosity as a function of temperature is for SAE grade 30 (engine) oil. I plan on providing more data for other grades soon.

A Java Applet to assist with journal bearing design is given here This assumes constant oil viscosity.

Assuming that you have Scilab installed on your computer, the way to access the two programmes below, is to click the appropriate link to display the listing in your browser. You can then highlight the text on the screen, press Ctrl and C to copy it, open 'Notepad' and press Ctrl and V to paste the text of the programme into 'Notepad'. Then save this with the same name but with .sce after the fullstop not .txt, in the location you use for Scilab programme files. Start Scilab and the programme can then be executed. (The browser I used to check this was 'Firefox').

Listing of Scilab journal bearing programme - constant viscosity.

Listing of Scilab journal bearing programme - with viscosity varying as a function of temperature.
Initially the programme displays the requires constants to calculate the viscosities of SAE (engine) oils grades 10, 20, 30, 40, 50 and 60. The user should enter the appropriate values and the temperature at which the oil enters the bearing. Changes can be made to the default data and the programme restarted by clicking the 'OK' button at the bottom of the data entry form.

A .pdf version of the second programme, which is easier to read, is given here
The assumption is made that after the (hotter) oil has left the journal, it will be cooled to the stated entry temperature before it re-enters the bearing.

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

I plan on updating the notes to give information about how the increase in temperature in reducing the oil viscosity effects the load carrying capability and the minimum oil film thickness.

David J Grieve, Revised: 9th March 2015