Wedge action and the journal bearing

I think that it is important to have in mind the magnitude of the diametral clearance in a well-made bearing. It will be about 1/500 of the diameter. Any diagrams will give entirely the wrong impression of the relative sizes.

 

When a plain bearing is running steadily and supporting a constant load it will run eccentrically in a position where the force produced by the wedge action is the equilibrant of the load. There is no great advantage from a design point of view to being able to quantify the eccentricity or the pressure distribution unless the stiffness of the bearing is crucial to the performance of some other part of the design. However it all helps to build a mental model of the behaviour of the bearing if just a few graphs are drawn using the pressure equation.

 

I have brought figure 16-15 from earlier in this chapter as figure 16-19. The pressure distribution is shown as a family of radial vector quantities. The shaft will be subject to equal and opposite pressures. The shaft will run eccentrically and, in figure 16-20 I have magnified the position of axis of the shaft relative to the axis of the bush. This then shows the eccentricity  and the angle of the offset  that puts the net force on the shaft into line with the vertical load.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 16-19 is the outcome of measurement and it can be used to test the physics.

 

In figure 16-19 the red shaded area is clearly symmetrical about a diameter that passes through the points of maximum and minimum gap. The convergent side lifts the load the divergent side plays no active part. The convergent side is effectively identical with a plane surface even if it is wrapped round the shaft.

 

This means that we can use the equation  but we shall have to make some adaptations because it makes sense to change to polar plots and the variable is then an angle  and , the distance round the bush from the point of maximum gap is given . If the speed of rotation is  rpm the peripheral speed of the shaft will be . For this bearing there must be some geometry that relates the angle , the length , the radial clearance  and the eccentricity .

 

In figure 16-20 I have shown the half of the lubricant film that is under pressure due to wedge action. If the radial clearance is denoted  and the eccentricity is  the maximum thick ness of the film is  and the minimum thickness is  and the length of the film round the shaft between two points is . It follows that the notional .

We also have  from which . Then we can substitute in the equation for pressure and create a programme to explore it.

 

I have made this substitution using Mathcad and it is self-explanatory. I have inserted practical values for all the several terms and any of these can be changed easily.

I have drawn a pair of graphs 16-8 for 500 rpm and 16-9 for 1,000 rpm. The black circle represents the shaft in each case and pressures are dawn radially from this circle. This then matches figure 16-15 except for  the orientation of the graphs. The point that I want to make from these graphs is that, if the same load is supported at both speeds, we can find two areas that are more or less equal on the two graphs. If, at 500 rpm, the load could be supported by the bearing with an eccentricity of , at 1,000 rpm, it could be supported with an eccentricity of . This means that, as the speed increases the axis of the shaft moves towards the axis of the bush.

In graph 16-10 I have changed the orientation to a position that, the largest trace might correspond to a bearing supporting a vertical load

 

Inspection shows that, had it been any of the other traces, a different orientation would have been required.

 

It follows that, if we go back to figure 16-20 both the eccentricity and the offset angle  change with load and speed.

 

One might reasonably wonder which of these traces would best match the pressure distribution in a real bearing. The answer would come from practical testing because it is clear that much depends on the properties of the lubricant. The viscosity is temperature dependent and power is always lost in the bearings by viscous shearing so when at the running temperature any temperature-dependent properties of the lubricant may be much lower that the values at room temperature. There is a great deal of data on the performance of bearings of all types and soundly-based empirical expressions relating all the relevant physical quantities. 

 

It seems to me that if you understand the physics of plain bearings you stand a better chance of selecting the one best suited to your needs.