Chapter 16 Lubricated bearings and fluid dampers
Introduction
Bearings of all sorts are
often crucial components of engineering devices. Sometimes they determine the
life and reliability of those devices. The basic application is that of
supporting some device that is mounted on and rotates with a circular shaft.
This covers innumerable devices of all sizes running at all speeds. Perhaps the
best known to most people and not just engineers is the electric motor where
the armature rotates in bearings that are built into the frame. Figure 16-1 is
of a small electric motor that has been dismantled. On the left is the armature
on its highly-polished shaft in the part of the frame carrying the brushes and
on the right, is the main body of the frame with the two permanent magnets. The
bronze oil-less bearing is visible in the main body. The device being carried
by the shaft is the armature with its windings and the commutator. Here we are
concerned with the bearings and it is the high quality of the rubbing surfaces
that should be noted. Electric motors may run in plain bearings or in ball
races. I have shown some small ball and roller races in figure 16-2. In every
case the balls or rollers are captured in a cage. It is often claimed that the
cages are fitted to prevent the balls or rollers rubbing each other. There is a
much more important reason for using a cage.
With the exception of bearings that are constructed by sintering a powder of perhaps leaded bronze to give a bush of the correct shape but with voids, up to 30% of the volume, that can be filled with liquid lubricant or graphite to give a long lasting “dry” bearing. All bearings use a lubricant that may be liquid or grease and it is that fact that leads to this chapter where the main thrust will be about bearings that use liquid lubricants.
Lubrication
I suppose that the first bearings that were invented were probably runners for land sledges and ice sledges. Applied lubrication would have been impossible but some woods have natural lubricants in them and are more durable than others. The arrival of wheels must have shown the need for lubrication to be applied to the hubs and the earliest lubricants were animal grease and organic oils produced by rendering such greases. There are very few of these organically derived lubricants and none, with perhaps the exception of oil from whales, was as good as the lubricants derived from mineral oils when petroleum became readily available and refined. The evolution of internal combustion engines depended on the success of these oils. More recently lubricants with desirable combinations of viscosity, film strength, resistance to high temperatures and low absorption of water have been synthesised by chemical methods and, using these lubricants, modern gas turbine engines for aeroplanes can be run for total times of 100,000 hours (=11 years) without being dismantled provided that the synthetic lubricant used in the ball bearings is monitored and changed at prescribed intervals.
So we have a range of liquid lubricants that can be used to go between the sliding or rolling surfaces of our bearings and give us reliable performance over a very long time.
As
I worked through that chapter the physics of plain bearings showed that
usefully high pressures would only be developed in thin films of lubricant and
it seemed to me that I should go back and make a few simple practical checks. I
have an accurately machined block of tungsten carbide with two flat annular
faces. It is shown in figure 16-3. When it is resting on that face it exerts an
average pressure of or about 1.2 psi. This is a very modest
pressure by engineering standards. I found a thick piece of duraluminium plate
in the as-rolled condition and polished it. I placed it on a table that was not
quite level and put some synthetic oil on the plate to form a film. I then
rested the block on this film. It gently slid down the slope but it was also in
electrical contact with the aluminium plate. That contact resistance was low
enough to carry current to light a car bulb from a battery.
So the film was thick enough to float the block but thin enough for imperfections in the surfaces to be in electrical contact. I wondered just how flat the plate was and rubbed a diamond hone over it and this showed a random pattern of closely spaced high spots that must be almost undetectable by ordinary measurement.
I left the block on the plate for half an hour reasoning that the pressure in the oil film would push the oil out until the block and the plate came into contact unless any strength that the film might have kept them apart. After the half hour the block was not easily moved but, when it did move, it continued to move easily. My impression was that there was solid friction initially but, almost instantly, the oil separated the two and permitted viscous sliding.
When I separated the bock and the plate both were uniformly covered with a very thin film of oil. I rubbed my finger over the film and could scarcely see the oil on my finger pad. The film was very thin.
I tried rolling a steel, bearing ball of 7/16 inch diameter weighing 5.5 gram across a fairly thick film of oil and the ball cut a path through the film but carried oil away to dry patches to continue the trail. I made several trails in all directions and after a while the oil had parted at these trails to form islands of oil. It was not wetting the duraluminium. I changed to glass with the same result. I changed to a mineral oil intended for lubricating my lathe. This did wet the duraluminium and floated the tungsten carbide block. When a ball was rolled over a film of this oil it produced the same trails but these gradually closed. The oil wetted the duraluminium. I changed to a glass mirror with the same result.
After doing these tests I felt that for almost no effort I had learnt a great deal that is useful to an engineer. Anyone can repeat them.
Clearly bearings that are lubricated can only be successful if the lubricant is always in place when the bearing is operating. It follows that there must be a steady supply of lubricant and some system to deliver it to the sliding or rolling surfaces. This is a part of the design but it must not be supposed that we can lubricate every mechanism that we can devise. We cannot always find ways to introduce the lubricant especially in reciprocating bearings like those of engine cross heads and in oscillating bearings like those in vehicle suspensions. Indeed vehicle suspensions have only become long lasting with the introduction of bonded rubber bushes that distort and do not slide.
By implication there are two groups of bearings, one is the plain bearing where surfaces slide one relative to the other and rolling bearings typified by ball bearings where balls roll between two lubricated tracks. Each group of designs subdivides again many times to adapt to all the possible engineering applications and we must see how they work. I will start with plain bearings and then deal with rolling bearings.
It turns out that in order to understand these plain bearings we must start with flow between parallel flat plates.
Flow between parallel plates
Figure
16-4 is a drawing of a fitting that could be made and fitted to the side of a
tank so that liquid could flow through the slit formed between two flat plates
and two spacers. The resulting duct has a width of and a length of .
The rate of flow will depend on the pressure drop between inlet and outlet. We
know enough about flow of this sort to know that if is large compared to the effects of the sides on the flow is
likely to be very small and, if is long when compared with the effect of the entry to the duct will be
small.
Further, even for water with its low viscosity, the flow will be laminar if is of the order of 1 mm.
So,
if we treat the flow through this slit as steady and one dimensional the
outcome is likely to be quite accurate.
Figure 16-5 represents an element of the flow moving from left to right though a slit made up of the two parallel plates of width and length set at a distance apart. As always the velocity of the fluid in contact with the plates is zero and, as there is a flow and it is laminar, the velocity must increase in some orderly way to a maximum in the middle. Within this flow we can consider an element of thickness , of width and at a distance from the lower face.
As the flow is steady this element must be in equilibrium under the forces caused by the pressure difference and the shearing forces due to viscosity. Over the distance the pressure changes from to . Of course the pressure will fall but this gives a direction to the mathematics.
Then the force on the element in the direction of motion is given by :-
.
The shearing force on one face of the element is given by Newton’s expression :-
where is the coefficient of viscosity of the fluid, is the effective area and is the velocity gradient.
Later,
I want to consider the possibility that the flow is between two surfaces when
one of the surfaces is moving as might be the case in a dashpot. As a result I
need to deal with the velocity gradient as a general case. In figure 16-3 I
have shown the lower face and a curve that is entirely arbitrary representing
the velocity distribution through the flow between the plates. On the diagram I
have shown the element of thickness at distance from the plate and two tangents to the
velocity curve at and at .
Then the velocity gradient at is and at is .
This gives two velocity gradients on equal areas and then the net resisting force on the element is given by :-
.
Equating forces:-
, from which :-
.
As we are looking for the velocity distribution we must integrate twice. Rearranging:-
and then .
Integrating again gives .-
.
We know that at and at so . Then :- from which and
In this equation will be constant if is the same throughout the flow so can be replaced by where is the pressure drop between inlet and outlet.
A
good example of this simple system is the Hele-Shaw apparatus. Figure 16-7
shows my home built version. The working section is made up of two glass plates
stuck to 1 mm thick ply at the sides to form a duct with a cross section of a
slit. The combination of the two plates is attached to the water reservoir and
water is fed into the reservoir at the same rate as it flows out. It is used to
produce flow patterns, in this case round the profile of a sail. The pattern is
produced in dye injected through 20 or so holes in the feed box.
I have used typical dimensions for such a piece of apparatus to find the velocity profile.
I
let the depth in the feed box be 300 mm, the distance between the plates be 1
mm, the length of the plates be 500 mm and the viscosity of the water be 0.001
kg/ms.
In graph 16- I have plotted the two terms separately and the sum of the terms. The velocity distribution is in black and is parabolic with a maximum value of about 0.75 m/s.
Obviously it would be valuable to have an expression for the mean velocity as in the Poiseuille expression for flow in pipes. This involves another integral.
Then the flow
I have already pointed out that if is constant. This means that we can either write as required.
This can be compared with the Poiseuille expression for laminar flow in a pipe :-
Flow between parallel plates where one is moving
This is interesting because we have, in
effect, a combination of Newton’s conceptual model with which he defined his
coefficient of viscosity and steady viscous flow. In figures 6.5a, b, and c, I
bring the diagram from Chapter 6 that shows this model. It is not practical as
I said in Chapter 6 but it shows us that the flow between parallel plates where
one is moving can either be regarded as a case of Newton’s model with a flow
superimposed on it or as a flow with Newton’s model superimposed on it.
In this treatment of the system we already have an expression for the flow between fixed plates.
The derivation above for stationary parallel plates can be adapted for the case of parallel plates when one moves at a constant speed.
We can go back to this equation:- and note that now when , and, when , where is the speed of the moving plate and is supposed to be in the same direction as . Then as before but now :-
Then and
.
Substituting gives from which:-
and finally :-
.
This
is clearly the first term is the former equation for a parabolic distribution
and the second term is a new term that adds or subtracts a proportion of the
velocity in the ratio as in Newton’s model. It gives a skewed
velocity profile. In graph 16-2 I have shown the velocity distribution for the
same figures as I used for the previous graph. The water behaves as if the new
velocity is the sum of the velocity for fixed plates with a linear proportion
of the speed of the upper plate added.
It is possible for the upper plate to move in the direction of the flow or against the flow and a graph can be drawn to show the result It is graph 16-3 where I have let take values from by increments of 0.25 m/s. The skewing is progressive and the maximum velocity moves towards the moving plate.
It is clear that, for the same pressure gradient, the flow will be reduced as the moving plate changes from moving with the flow to moving against it.
It is possible to find the flow.
This is an interesting expression because can be negative. This means that despite the pressure difference acting on the flow the net flow can be zero or indeed be in the opposite direction to the pressure difference.
We can find the condition for the flow to be zero. We can write :-
and then
If this is evaluated for water flowing between plates, as in my example above, zero flow will occur when and the velocity distribution for can be plotted as in graph 16-4.
The outcome is intriguing because we now have, in a distance of 1 mm, flow in one direction near to the stationary plate and a flow in the opposite direction near to the moving plate. The cross-hatched areas must be equal.
However there is no reason why the value of could not be much greater so that the flow in the direction of the pressure difference becomes very small.
My figures are for water and a very wide gap by the standards of lubrication between the plates but the viscosity of air is lower than that of water by a factor of abut 50 and the viscosity of oil can be several thousand times that of water. All sorts of combinations of velocity and gap will be needed in practice for various applications. The worked examples at the end of this chapter will give some idea of magnitudes.
The
dashpot
When
I first thought of including the dashpot in this chapter I thought that it
belonged too far back in time to be justified now but on reflection, it is the
fundamental version of modern dampers and shows us something about the
characteristics of such devices. I first met the dashpot in use on a steam
engine as part of the Porter governor to control the speed of the engine. The Porter governor is shown in figure 16-9.
Its function is to open and close a steam valve and so control the speed of the
engine. Such valves are, and were, sensitive to the movement of the valve stem
and non-linear. The valve was linked to a lever mounted on a pivot and that
lever was moved by the system of weights and levers shown in the upper part of
the figure. Two fly-weights were mounted on the two upper links and rotated on
a spindle driven by the engine. These fly-weights swung in and out in response
to changing engine speed. Their motion was stabilised by the two lower links
that were attached to a collar that could move up and down and lift the dead
weight. Two spigots on the main link ran in a groove in the collar and so the
fly-weights lifted the lever.
This whole moving system could bob up and down and cause the engine speed to fluctuate and the dashpot was fitted to damp out this motion.
Figure 16-10 shows the essential features of the simple dashpot. There is a cylinder that is fixed somehow, a piston with a significant radial clearance and oil to fill the cylinder. In practice there would be some system of guidance to keep the piston coaxial with the cylinder.
If
a force is exerted on the piston this force will
produce a pressure in the oil under the piston and cause a flow through the
annular gap. There must be a relationship between the force, the viscosity of
the oil, the relevant dimensions and the speed of the piston.
In this device the oil flows between two parallel surfaces but they are two cylindrical surfaces of length and, if is small when compared with the effective width and the oil flows through an area .
Now, if the force is applied suddenly, the piston will start to move and, because of the high viscosity of the oil, conditions will quickly become steady with the piston moving at a velocity . Then the work done by the force will be continuously lost to the internal energy of the oil. We now have a case of oil flowing between two parallel plates where one is moving. The expression that we have is:-
and the pressure in the oil under the piston = then :-
.
This easily reduces to :- .
If is small when compared with we can write .
Then, once the dimensions are chosen, the speed is proportional to .
Of
course when a dashpot is in use the force is always changing and it is worth
calculating the relationship between force and velocity for a typical case. I
have taken the case where the diameter of the piston is 50 mm and the length is 30 mm with the oil
having a viscosity of 0.08 kg/ms. Then, in graph 16-5 I have plotted speed
against force for three values of ,
1 mm, 0.5 mm and 0.25 mm.
Cleary, for a given oil the speed response of the dashpot is critically dependent on the value of .
It is evident that, by a trial process, coupled to some calculation, a dashpot can be matched to an application.
However there is one important fact to note. Dashpots have to work in both directions and on the down travel in figure 16-10 the pressure will rise to resist any force. On the upward travel the pressure below the piston can only fall to zero absolute and, if the upward force is too great, voids will appear in the oil under the piston. Those voids will make the oil foam. One might consider capping the cylinder and fitting a seal between the cap and the piston rod. Then, if the space above the oil is filled with compressed nitrogen the dashpot will work in both directions equally well without foaming. This is the basic concept of the so-called shock absorber[1] used in vehicle suspensions.
In the shock absorber the piston fits tightly in the cylinder so that is small and, instead of the oil flowing through this very narrow space round the piston a hole in made in the piston and energy is dissipated into the oil from the jet of oil produced by the hole. In this design it is easier to make the hole reproducible than the radial clearance between the piston and the cylinder.
The most common application of the shock absorber must be in the spring suspensions of vehicles. There the shock absorber must be cheap to make, last a long time and be of a shape that suits the application. On top of this the oil must circulate so that it can be cooled to carry away the heat generated by the continual loss of energy into the oil. This leads to the complexity shown in figure 16-11.
The shock absorber is telescopic and a system of holes and non-return valves in both the piston and the end cap that joins the inner cylinder and the outer annular volume is designed to circulate the oil. A typical shock absorber is shown in figure 16-.
Flow between plates set at a small angle when one moves
We need some relatively simple physics to help us to understand bearings. The model is for two plane surfaces, one fixed and the other moving at some constant speed , that make a small angle . There must be some physical restraint for the fixed surface because it cannot extend to the point where the surfaces meet. We also need a constraint at the open end. The result looks like a block with a tapering gap making an angle to a second moving surface in figure 16-12. The diagram is grossly out of scale because the angle is too large. A practical angle will be in the region of one second of arc or a gap of less than one tenth of a millimetre.
I have let the fixed surface have a length and be inclined at . This surface can be at any height above the moving surface and, if we extend the fixed surface to cut the moving surface at point the intersection will be at some distance from the leading edge of the fixed surface. We now have the two surfaces located relative to each other by three dimensions, .
What we need to find out is how the pressure varies with distance along the fixed surface and we see immediately that, if the surface is of finite width, there will be leakage of lubricant from the sides. We cannot handle that so we must simplify and consider only two-dimensional flow that might well prevail along the centre line. If we accept that, we have to decide what to do about the velocity distribution between the two surfaces. We have already done this for parallel plates and we found that the velocity distribution can be found from :-
Now we have converging surfaces and we have to deal with a pressure that will rise from zero at the leading edge of the block and fall to zero again at the trailing edge. We are not looking for the velocity distribution but the variation of pressure with distance .
The best solution comes from defining the distance between the plates at distance from the leading edge of the fixed plate and supposing that the velocity distribution is given by the equation for parallel plates. Then if we imagine the pressure changes to take place in increments it turns out that it is best to use not the equation for velocity but the equation for volume flow:-
to find the incremental change in pressure.
We can write to give flow per unit width and then :-
which can be changed to :- where is the pressure drop in distance at distance from the leading edge. Then :-
In this case is a function of and given the geometry in figure 15- we can say that :-
. Then :-
.
This can be rearranged for integration to give :-
and .
So
Now we have to decide what to do with this equation. We know that the pressures at are both equal to the absolute pressure of the surroundings. So we have two conditions that will let us find a value for . However we must also recognize that is fixed if are fixed. This means that we could use the two pressure conditions to find and then eliminate . This is hideously long but it yields :-
We need a better mental picture of the behaviour of the lubricant between the surfaces and all we have is this equation. What we are talking about is lubricant being dragged by viscous forces between two plates that might be less than 0.1 mm apart at the entry and closer at exit. This is a thin film with a thickness about equal to that of 80 gram copying paper and certainly thicker than the films that I found in my simple testing.
It seems to me that the important thing to find out is how the pressure varies between inlet to the plates and exit. We can find that from a plot of pressure against . The expression for pressure shows that the pressures throughout will be proportional to the value of the coefficient of viscosity and that the pressures will increase directly with the speed of the moving plate. There will be no such simple relationship for the remaining variables where is the gap at inlet. These three are linked by the geometry and we must think about how the geometry can be exploited in preparation for an exploration using Mathcad.
We
must have some way to tackle this because we shall want to know how the
pressure distribution varies with the angle for a given value of at inlet and how it varies with for a given value of angle .
I have said that the geometry of this arrangement of a surface at a fixed angle
and a second moving surface can be defined from a knowledge first of the angle ,
then of the gap at the leading edge and then of the length of the fixed surface.
If distance is defined from figure 16-12 it follows that or the expression for pressure will become
invalid if .
So we need combinations of and that give .
It is usual an engineering to explore relationships between the quantities in equations by plotting graphs of the relationship between any two with all the other quantities constant. Here all we can do is find the pressure distribution for a fixed value of with varying and vice versa.
Figure 16-13 gives the programme for drawing the graphs in graph 16-6 and 16-7. I simply chose likely values for viscosity, velocity and length and fiddled out suitable values for the angle and the gap at inlet. They are given in figure 16-13 where I also give the expression reworked in terms of and instead of .
The information for graphs 16-6 and 16-7 is given in the legends on the graphs. I have used the same vertical scale in both graphs just to show that changing the inlet gap for a given angle gives much greater pressures than changing the angle for a given inlet gap. It is also evident that in both graphs the peak values move towards the exit as the pressures rise.
The gaps and angles are so small that in any practical application it is most unlikely they would be designed quantitatively. For the important applications of this physics we must exploit the fact that one surface is free to move relative to the other to find a geometry that gives equilibrium operation.
The journal bearing
Journal bearings are usually used in pairs to support a shaft that might be an axle for, say, the armature of an electric motor or perhaps a large fan that it drives. The shaft has two coaxial journals that are accurately turned with a good finish. These journals rotate in a bearing bushes housed in blocks. The bushes are made of some alloy that has suitable properties for letting the shaft rotate freely when suitably lubricated and for resisting wear during the instant of starting. It may be that the bushes are lined with expensive alloys having particularly good mechanical properties.
The bearing, as a device, must have diametral clearance between the shaft and the bearing bush to accommodate the lubricant and this diametral clearance will have a minimum value at which the bearing is made and some maximum value at which the bearing is regarded as being worn out and must be refurbished. A typical ratio of diameter to clearance is 500. So a bearing of 50 mm diameter will run on a bush with a diametral clearance of about 0.1 mm.
Bearings,
generally, lose lubricant by leakage and, in the case of the journal bearing,
this results from axial flow of lubricant. This loss must be made good and a
journal bearing can be fed with lubricant from a cup in the cap that must be
refilled at intervals or by a ring resting on a part of the shaft exposed by a
cutaway in the top brass and lifting lubricant from a well in the bearing
block. It can also be fed by continuous pumping, that is, by forced
lubrication.
We need to know what happens during starting and when running. Obviously any diagram that one might draw must grossly exaggerate the clearance. Figure 16-14 shows three states for a journal bearing in which I have exaggerated the clearance.
In the first state the shaft is at rest and in metal-to-metal contact. The space between the shaft and the bush is, or should be, filled with lubricant as I have shown in red. There will be a gap in the lubricant at the lowest point but once solid contact is made, there will still be tiny spaces in the region of contact that are filled with lubricant.
When the shaft starts to move the film quickly becomes continuous and then the shaft tends to “climb” up the right hand side of the bearing because of the viscous drag. I very much doubt whether it moves through the angle I show but, again, I need to distort to carry the message. This offset to the right reaches some maximum and then, as the speed of rotation increases, lubricant is dragged faster and faster by viscosity into the tapering space on the right hand side and the pressure of the lubricant in that space increases. This pushes the shaft to the left and, as it moves the high-pressure region creeps under the shaft and lifts it until ultimately, when the speed becomes steady, it runs in the position shown in the third diagram supporting the load on the lubricant with the solid surfaces not in contact.
Here
we have the fundamental mode of operation of all of our successful plain
bearings. It is this action of lubricant being dragged by viscosity into the
reducing space between two solid surfaces set at a small angle and creating a
high pressure in the film of lubricant to separate the moving and stationary
surfaces. It is called wedge action.
In figure 16-15 I have drawn a typical measured pressure distribution in the lubricant when the shaft is rotating. The radial arrows represent the magnitude of the pressure acting on the inside of the bush. Equal and opposite pressures act on the shaft to push it to the left and, more especially, upwards.
These high pressures lead to axial flow in both directions and the greater the clearance the greater the loss of lubricant sideways. There is another incentive here to keep the clearance to a minimum.
There is yet another incentive and that is to do with the “stiffness” of the bearing but we need more information and I will deal with it later.
The thrust pad
There was a need for a thrust bearing in flour-milling machinery. The moving millstone was driven by a vertical shaft that was supported by some sort of footstep bearing. I have shown an arrangement for a footstep bearing in figure 16-16. The sliding surfaces are just the flat end of the shaft and the face in the base. There can be no wedge action with such an arrangement and lubrication was from a reservoir in the base with a radial groove in the shaft to carry oil to the centre of the shaft where it might find its way between the two surfaces. Wear must have been inevitable.
When
the screw propeller began to be used on ships in about 1850 there was a need
for a thrust bearing to transfer the force produced on the propeller, that had
been transmitted through the shaft, on to the structure of the ship and, of
course, bearings to support this long shaft. For 50 years the thrust bearing
was the plain annular bearing shown in figure 16-17 derived from the footstep
bearing possibly with several bearing collars on the same shaft. At the end of
the 19th century the idea of wedge action was evolved by Navier and
others and around 1900 the problem of the thrust bearing was resolved by an
Australian called Michell. (He pronounced it Mitchel.) He invented the
self-aligning thrust pad.
In
essence his invention was to divide the annular thrust pad into eight separate
sectors and allow each sector to pivot about a radial axis. The arrangement is
shown in figure 16-18 which is taken from a photograph. The substantial
back-plate has radial grooves and the thrust pads have elongated nibs, shown on
the inverted pad top left, that fit in these grooves to act as pivots. The
shaft with a plain collar goes through the hole in the middle so that the
collar rests on the thrust pads. The pivots on which the pads move are offset
to one side so that when lubricant is fed to the rotating shaft it goes under
the pads and they rock until they make a small angle and produce a wedge of oil
and high pressures. If the pivots are in the right position the pads are
self-aligning. Such a bearing is unidirectional. Michell bearings are very much
smaller than plain bearings for the same duty and much more reliable. They
appear to be easy to supply with lubricant. Given the very thin films involved
these bearings require great accuracy in manufacture.
I do not know for sure but presumably these are used in pairs one for “ahead” one for “astern”.
Wedge action
It is clear that bearings like the footstep bearing where there is no wedge action are always difficult to lubricate. In engineering there are plenty of sliding bearings like the cross-heads supporting the little ends of the connecting rods in steam engines and double-acting marine diesel engines. All sorts of schemes were used to get the lubricant between the surfaces but mainly they depended on having grooves cut in the faces and drip-feeding oil into the grooves. It is just a practical matter that might be made easier by careful choice of the properties of the lubricant.
But it is wedge action that makes the plain bearing reliable and successful. The physics of the angled surfaces with one sliding is there to use to find out more about such bearings. In the journal bearing the two surfaces are wrapped round a shaft but, as the clearance is so small the physics is directly applicable. There can be no question that the Michell thrust pad is also a direct application of the physics. Let me look at the journal bearing first
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.
Wedge action and the thrust pad
Figure
16-22 is figure 16-18 brought forward to use again. If we consider one pad when
the whole assembly is in use we can see that lubricant will flow into the space
between the pad and the thrust collar and immediately the pressure will start
to rise and also immediately the lubricant will start to flow radially in both
directions to leak out. We might suppose that there will be one circular path
through the space between the pad and the collar where the flow is actually
along this circular path. Then we can treat this as one-dimensional flow and we
can apply equation:-
to this path to find the pressure distribution and go on to find a position of the thrust so that the position of the pivot can be determined.
In figure 16-19 I have drawn part of a Michell thrust pad bearing. The pads have been simplified to sectors of an annulus. I have also shown dotted a possible path to which we might apply our physics.
In the lower diagram I have drawn the section through a pad where this notional path lies. The pad is at some angle and its leading edge has a gap with the moving face of . The length of the path is .
Ignoring the small difference between the length of the arc and the
length this is identical to the diagram for the journal bearing but this time we shall require a Cartesian plot and not a polar plot..
Again we can explore this system if we let be constant and vary and vice versa for fixed values of the other relevant quantities.
Figures 16-24 and 25 give the details of the Mathcad programmes for graphs 16-11 and 12. The pad might fit in a bearing assembly for a collar of about 750 mm overall diameter and turning at 120 rpm. The figures for the inlet gap appear to be realistic and there is not much scope for change. The pressures are quite significant and it is easy to see why this design is successful especially on ships where the shaft may turn non-stop for several days at a time.
This
pad has to be mounted on a pivot and part of the design will be the location of
the pivot. We can go one more step and extend the programme to find the force
on the pad per unit width and to find the point of action of the force from,
say, the inlet edge. In figure 16-23 the first integral will find the force and
the second the distance from the leading edge.
There is no point to plotting graphs so I give typical values in table 16-1
|
Angle = 0.000075 radians |
Inlet gap z =0.04 mm |
||||
|
z |
Force |
Force |
|||
|
0.03 mm |
90 kN/m |
0.114 m |
0.0001 rad |
50 kN/m |
0.114 m |
|
0.04 |
29 |
0.109 |
0.00008 |
3.2 |
0.11 |
|
0.06 |
6.7 |
0.106 |
0.00006 |
2 |
0.107 |
|
0.08 |
2.5 |
0.104 |
|
1.1 |
0.104. |
These forces are trivial by engineering standards and large forces will be associated with smaller values of and . It is worth looking at these. However it is the value of that is of interest and that varies very little for any combination of and . This means that the position of the pivot is not far away from the radial centreline of the pad and would, no doubt be found by testing.
This analysis has given a better understanding of the way in which the Michell thrust bearing works. One thing that is clear is that the bearing must be made very accurately if the eight pads are to share the load.
Rolling bearings
Introduction
When I started to write this section I had various preconceptions that really hinged round my view that it was easy to see the basic principle of the rolling bearing but it was not easy to see why they seemed to be so effortlessly successful. I think that I came to see that, despite the simple principles, the rolling bearing depends for its success on a great deal of well-documented empirical data. I also came to have a better idea of the mechanism of lubrication that made it all look so simple. I also decided that ball and roller bearings were so different that I could deal with only one and I chose the ball bearing.
It
is hard to see any obvious origin for rolling bearings that corresponds to
runners on sledges or wheels on carts. It looks to be an invention. The first
patent for a ball bearing was taken out in 1791. However the ball bearing
requires ingenuity and adequate means of manufacture.
In figure 16-26 I have drawn the basic features of a ball bearing. It comprises two tracks called the inner and outer races, a set of eight balls (in this case.) and a cage, that is not shown on the diagram, to separate the balls. They form a closely-fitted assembly. The inner race fits snugly on a shaft and the outer race fits snugly in a housing. Even if the balls are perfect spheres and the races are perfectly cylindrical there must be clearance. If a shaft is at rest and is supported on two such bearings figure 16-26 might represent one of the bearings. In the position shown, the inner race will, in fact, be supported by two of the three lower balls B, C and D in the bearing and be out of contact with the rest. Let us suppose that the inner race is in contact with C and D. As soon as the shaft starts to rotate in the anticlockwise direction balls C and D rotate clockwise and start to move their centres anticlockwise. The shaft and the inner race “topple” about ball C and come into contact with ball B. At the same time D goes out of contact. As the shaft continues to turn balls A, B and C will move to the positions of B, C and D as shown and the switch will take place again. This is repeated over and over again as the inner race rotates.
We need to look more closely at surfaces that are separated by balls or rollers.
Suppose
that you had two flat steel plates with surfaces that had been ground and
lapped to the standard of surface plates. Suppose that one plate was set up
horizontally and four steel balls set up in a line as shown and the second
plate rested on them also as shown. (If you feel perturbed that the top plate
may drop sideways, add, mentally, a fifth ball some way behind these four balls
to stabilise the plate.) I have shown the arrangement in fig 16-27. Two of the
four balls would not be in contact with the top plate because these balls will
have been made by some manufacturing process and there will be some variation
in diameter from one ball to another. This difference may be very small but,
because the error is of the same magnitude as the deformation of the two balls
that are in contact under the weight of the top plate, it becomes significant.
We need to think more about this simple system.
The balls that move are in contact with both plates and will initially have point contact and this inevitably means that there is a high pressure on the balls at these points of contact. Both plates and the rollers will deform. If the force being supported by the balls is not excessive, the deformation is elastic and, if it is large enough, the top plate may drop enough to allow it to come into contact with one or more of the other two balls. These will not be subject to the same forces as the first two rollers. So the behaviour of these four rollers is critically dependent on the dimensions of the balls and, if the top plate were to start moving and rolling the balls the points of contact would move and the elastic deformation move along the plate with the balls and round the balls.
In my supposition above I have described the material as steel. Now I have to be more specific. Steels are alloys of iron and other elements and there are hundreds of useful alloys.
All
of these alloys can be deformed elastically and, surprisingly, Young’s modulus,
that is a measure of the relationship between applied stress and elastic
strain, does not vary much at around .
What does vary is the stress that can be applied to a given steel before it
deforms permanently and the nature of this deformation. The stress when the
steel deforms plastically and permanently is often called the yield stress.
This is around for mild steel. However for a hardened steel
alloy there is no plastic deformation, only a sudden brittle fracture at about .
No one would think of making a rolling bearing from mild steel because the
balls would soon develop flats from overload. One of the alloy steels would be
used in a hardened condition and then the safe working load would be nearly
three times that for a bearing in mild steel. However there is a snag. Where
mild steel would acquire flats the hardened steel would chip and pit in brittle
failure. The tungsten carbide roller in figure 16-3 is very hard and it has
been damaged by excessive stress. Figure 16- 28 shows the damage. There is a
feint line on the highly polished surface that is a line of cracks and the
black mark is a chip out of the roller, as is the smaller black mark. This
roller worked with a second identical roller to squeeze stainless steel wire of
about 0.5 mm diameter into a flat ribbon about 2 mm wide that was made into
pot-scourers. A very large chip that extended to the edge of the roller fell
out of the second roller. My guess is that the stress was close to the limit
for this material and one or other or both rollers were turning slightly
eccentrically. It is a typical brittle failure and the piece that came from the
other roller could be fitted exactly back where it came from. The failure shown
in figure 16-28 could have been made by a ball rolling on the surface when the
ball would be damaged as well.
The manufacture of bearing balls
There is a further practical problem for ball bearings and that comes from the methods used to manufacture the balls. Most metal artefacts are made by machines that cut or grind and usually the work piece is held in a chuck or a vice or some special fitting whilst the cutting is performed. The trouble with making balls is that there is no way to hold the ball to work on it and balls are required by the thousand. Steel balls are made in an unlikely way that may go back for centuries to the manufacture of spherical marble balls for architectural use. The method that is actually used does not involve holding the ball but employs a random process that produces balls that are very nearly spherical but in fact are the outcome of chance.
I once came across the process in action in a river. At a weir there was a standing wave at the foot of the weir wall. By chance a block of wood from the trunk of a tree about 500 mm in diameter and about 500 mm long had become “caught” in the back roll. It could never get through the standing wave as it went through a cyclic motion of being carried under at the weir wall bobbing up due to its buoyancy and then back-rolled on to the wall by the wave and thrown back to the weir wall. It was getting battered on the wall and the apron. When I saw it, it was two thirds of the way to becoming spherical.
Bearing
balls are made from steel wire that is cropped accurately to a required length
and them formed into a ball by sudden squeezing between two hardened cups. This
produces a blank ball that is slightly over size and has a small flash all
round it. Hundreds or thousands, according to size, of such ball blanks are
fettled to get rid of the flash. Then the balls are fed continuously and
repeatedly through a ball-making machine. This machine looks remarkably like a
machine for milling flour in that it has two cast-iron, cylindrical plates of
about the same proportions as millstones. One is flat and the other has
circumferential grooves as shown in figure 16-29. They are set up so that the
flat plate would rotate anticlockwise if fed as in figure 16-29. The balls are
fed into the open segment with a water slurry and travel round the grooves
colliding with each other and the solid surfaces and gradually being worn to a
spherical shape. The balls make many passes through the grooves and come out
looking remarkably spherical.
There is a great deal more to the process including hardening etc. but our interest is in the fact that this is a random process.
In practice the balls resulting from the manufacture of a batch will vary slightly in shape. One variation will in deviation of the surface from a true sphere because of tiny surface imperfections. Then if one could imagine this surface having some mean shape it might not be quite spherical and it will certainly vary slightly in mean diameter. This is hardly surprising.
It is normal to grade the balls from a batch and use the best for top quality bearings and the worst for cheap bearings for which there are many applications. Other grades fit between.
The races by comparison with the balls can be made very accurately because they can be held and their shapes generated on machines.
The net result is that ball races of ordinary standard have clearances that vary somewhat from ball to ball and this means that the design clearance must be chosen to accommodate the largest ball that is likely to be inserted and that the smallest ball will have greater clearance than is desirable. Only the very best ball races will have uniform clearance.
The object of designing a ball bearing is to support a shaft so that its axis of rotation remains stationary relative to the bearing block. We have seen that a ball bearing with perfect balls but with clearance will not achieve this if it is dry. However if the bearing is lubricated with oil or grease this problem can be reduced to the point of being eliminated.
Wedge action in ball bearings
It is essential to lubricate ball and roller bearings; they will not run dry. No one seems to take much notice of the lubricant and appear to regard it as incidental as one might the lubricant applied to the hinges of a door. Yet the lubricant is trapped by the rolling action to produce a wedge action like that in the plain bearing and in the thrust pad.
In
figure 16-26 I said nothing about the profile of the races. They could just be
two plain, cylindrical rings. I have drawn such an arrangement in figure 16-30.
If these balls run in, say, a liquid lubricant, lubricant in the spaces that I
have coloured in red would be pushed out of the way be the rolling action of
the ball just as the bows of a ship push the water aside to form the bow wave.
Inevitably the pushing process will be resisted by the viscosity and by the
inertia of the lubricant. The necessary acceleration will be imposed on the
lubricant by a distributed force acting on the lubricant and there will be
equal and opposite distributed forces acting on the ball and on the race. The
magnitude of these forces will depend on the shape of the space in which the
ejection of the lubricant takes place. It is always an action that is of the
same character as wedge action in plain bearings and thrust pads. The
distributed force creates a pressure that tends to separate the ball and the
tracks.
In
the plain bearing and the thrust pad we found that high pressures were developed
in converging surfaces that were close together and only converging slowly.
This is clearly not possible for the arrangement in figure 16-30. However we
have some control over the shape of the space in which this action takes place.
In practical ball bearings grooves are cut in both races. They are most likely
to have the profile of an arc of a circle. I have drawn the arrangement in
figure 16-31. The arc has a larger radius than the ball and the closer that
this radius becomes to that of the ball the less space that there will be to
accommodate the lubricant being ejected from the space between the balls and
the tracks in the races. This can lead to very high pressures in the lubricant
and separate the balls and the tracks to eliminate metal-to-metal contact and
perhaps accommodate for variation in the clearance resulting from the random
variation in the size of the balls. But it must be remembered that the
clearances involved are commensurate with the deformation of the balls and the
track when under load and also commensurate with the clearances needed to
produce high pressures.
Fortuitously
the provision of the tracks in the races makes possible the construction of
ball bearings as single units if the dimensions are chosen so that, if the
inner race is placed inside and touching the outer race, the balls can be
fitted into the space between them and then the inner snapped across to
centralise it with the balls around it. Then the balls can be spaced when the
cage can be fitted. See the photograph in figure 16-32. The cage shown is one
of two identical pressings. They are fitted one each side and spot welded
together. The two races, the balls and the cages form a robust unit that is
just called a ball race.
Lubricating ball bearings
Ball and roller bearings are lubricated with oil or grease, either of which might be mineral or synthetic, and it is extremely difficult to decide just how the lubricant behaves. Many bearings are “sealed for life” like those on motor-car wheels. I have a drilling machine with sealed-for-life bearings that I have used regularly for 50 years. The bearings show no signs of wear or deterioration. Just how the grease manages to find its way around the tracks and the balls is difficult to either imagine or observe and the sheer durability of the grease is remarkable.
Other ball races can only function if they are force lubricated and then the way in which the lubricant gets into the spaces is not in question.
It becomes a design to be finalised by making and testing but whilst dimensions can be chosen and realised in hardware it is not so easy to choose a lubricant. It is not immediately obvious what properties are required for the lubricant. We know from the previous work that we need a high viscosity to achieve high pressures in the gap but we also know that we want low viscosity to let the lubricant flow quickly into the spaces between the balls. Density is not very important. But the ability to wet the balls and the tracks is important.
Whatever means of introducing the lubricant is used we ultimately have a set of balls that roll along a track covered with lubricant. We must look for some help to think about the flow pattern. I thought that I might get some help from simple experiments. The following is the outcome of those experiments.
Figure
16-33 shows a table tennis ball being rolled through a thick film of steam oil.
Steam oil is very viscous and it obviously wets the celluloid of the ball.
Even
this light ball cuts through the film of oil and leaves a wake, like banks on
either side of a country road, that immediately starts to close under the
action of gravity and possibly the intermolecular forces and certainly surface
tension but is resisted by viscosity.
The oil clings to the ball and stays on to take part in the motion ahead of and under the ball. The process of displacing the oil in front of the ball can be seen in figure 16-34 as a brown collar of oil and the profile of the oil clinging to the ball can be seen to change as it is carried over the ball. Figure 16-35 shows that the oil that is pushed to the sides is “stretched” between the ball and the surface and tries to form the sort of web that is shown in fig 7-5 of Chapter 7. The web of oil breaks to form a trail of oil over the ball and a trail along the plate. I know that, in a real ball bearing the oil would be in a track that is hollow, but it seems to me that the oil will still form this web and break to let some oil be carried round the ball into the path of the next track. Centrifugal action will tend to bring this oil back to the crown of the ball. I have put this together in figure 16-36 to give a mechanism for lubrication ball bearings. There I have suggested that the oil that is carried round by a ball joins with oil that is already on the track in a process that must involve surface tension and then the oil is squeezed and pushed sideways to form the parallel wake that appears in figure16-35. Some of the oil is carried on the ball to the next point of contact. I have shown the two balls in figure 16-36 with a considerable gap between them. I think that the raised profile of the parallel wakes changes quickly in the way that is shown in figure 16-37 and there must be some minimum gap that lets these wakes spread ready for the next ball to arrive. It seems like a viable mechanism to me and I think that the way that the lubricant is carried round with the ball may be the key to an understanding these bearings and especially sealed-for-life bearings. It might certainly be the case for bearings lubricated with grease.
I thought about grease and wondered whether it would be worth using the same ball with a slurry of icing sugar in water. Figure 16-37 shows that the same web as in 16-35 forms in the icing sugar only now it is a semi solid and we can see the process more clearly Figure 16-38 shows the front of the ball and the icing sugar that has come over the ball amalgamating with the wave forming under the ball
If this ball were to be one of a pair with the second following closely behind the first there would be nowhere for trails of oil or icing sugar to change shape to lubricate the second ball. A balance must be struck between the number of balls to share the load bearing and the need for spacing between the balls to give successful lubrication.
The behaviour of lubricant is clearly complex but it does seem to be the case that, more or less by chance it works very well. It raises the question of the ideal combination of properties for a lubricant.
Effect of linear speed of rolling
I think that it is easy to just think of speed of rotation as the dominant quantity in the selection of a ball bearing. Yet it is the speed of rolling that matters. It matters for two reasons, the first is that this is the speed at which the cyclic deformation of the race and the balls takes place and secondly the speed of rolling affects the lubrication. On both counts lower speeds are to be preferred.
Ball
races are used at speeds at least up to 30,000 rpm but they run mainly at much
lower speeds. For instance the wheels of a small car at 60 mph will rotate at
about 1,000 rpm. The ubiquitous single-phase motor runs at 3,000 rpm. The ball
races associated with these two applications will have typical diameters for
the path of the balls. For wheel bearings this will be about 40 mm and electric
motors about 25 mm or smaller. In these applications the linear speed of
rolling is quite low.
However, one must not overlook the fact that the linear speed of rolling in a ball race is less than half that of the linear speed of the moving track where the track is the line of potential contact between the ball and the race. This is evident from figure 16-27 where the centres of the balls will move at half the speed of the upper block.
It is easy to make a plot of the rolling speeds in terms of the rotational speed and the diameter of the path of the balls allowing for the balls rolling at about a half the rotational speed. I have done this in graph 16-13. It is the graph of rolling speed where is the diameter of the inner track and is the speed of rotation of the inner race. Note the rotational speeds are in geometric progression. There is a dotted line for a rolling speed of 3 metres/sec. This is about twice unhurried walking speed and a speed that I can visualise and think of as low. Many of the races in common use have much lower rolling speeds.
Rolling bearings in general
I have established the general mode of operation of ball bearings. It could be extended to roller bearings but this book is about fluids not mechanical design. There are dozens of configurations for all sorts of applications. The roller bearing is an obvious complement to ball bearings because the potential area of contact of a roller is so much greater that that of a ball. However, roller bearings are much more sensitive to accuracy of alignment than ball bearings. This leads to self-aligning roller bearings with barrel-shaped rollers.
Stiffness of bearings
Anyone who has worked on lathes, both old and new, will know that effective turning is not possible if the work-piece in its chuck and mounted on the head stock (the main spindle) is able to jump up and down because the bearings are worn or just badly adjusted. One solution shown in figure 16-38 is used on my lathe. The main bearing is tapered and the lubricated spindle can be moved in or out until it has almost no movement in response cutting loads. What it amounts to is that the clearance is a minimum consistent with the bearing not seizing even at the highest speeds.
Most bearings are not tapered and the shaft can move in the bush. The shaft will move and its movement depends on the magnitude and direction of the load applied to it. If that load is steady, as it might be for an electric motor driving a shaft through a coupling, the shaft will not move erratically but run with an eccentricity that is constant in magnitude and direction. If the bearing is a main bearing for an engine the shaft will be subject to a force that varies cyclically and the shaft will always be changing its eccentricity and the direction of this eccentricity. How much it moves depends on the stiffness of the bearing.
The idea of stiffness has its roots in springs where it means the ratio of the force to the extension. Here we have a shaft moving within the limits of its clearance and resisting the forces applied to it by viscous effects and inertia forces in the lubricant. It has stiffness and I have heard lathes with low stiffness being described as “lively”. It seemed very apt.
We need to understand stiffness in connection with plain bearings and also with ball bearings.
What we do know from the work on wedge action is that for a bearing that is well-designed and well-made the clearance will be very small and small changes in the position of the shaft relative to the bush will lead to very large resisting forces. A good plain bearing is stiff. We also know that the faster that a plain bearing runs the closer will be the axis of the shaft to the axis of the outer race. Such bearings are used up to 20,000 rpm in high performance engines for racing.
There
are some applications where ball bearings having a high stiffness must be used.
Figure 16-39 is of a turbo fan aircraft engine. It is complicated. The front
part comprises the fan and the compressor section and the back part is the
turbine section. In between is the combustion chamber. The fan and the first
stages of the compressor are driven by the last stages of the turbine through a
hollow shaft that runs right through the engine and runs in ball bearings
mounted in the frame of the engine. The high-pressure section of the compressor
is driven by the first stages of the turbine and these two also run on a hollow
shaft but this shaft is mounted on ball races that run on the central shaft.
Here we have ball bearings running one inside the other with a requirement to use up the minimum radial space. The pressures in gas turbine engines now exceed those on diesel engines and this is only possible because the compressor blades have the smallest possible clearance in the casing. Now we have bearings that must be stiff to prevent the blades rubbing on the case. The only way is to reduce the clearances to a practical minimum. We have seen that bearing balls are produced by a process that leaves a random element to their size and roundness and the only way to get bearings of the standard required for aircraft engines is to select from a very large batch of balls and match balls for diameter and accuracy of shape and surface finish. Such bearings are very expensive but very stiff.
Gears
These days almost all gears use the involute profile or a derivative of the involute profile. A modern gear box for a road vehicle can be expected to work without repair for at least 150,000 miles (200,000 km). The working faces of the teeth on gear wheels are curved and slide one over the other during contact. The gears work in a bath of oil and inevitably there will be wedge action to keep the teeth mechanically separated. We now know that these long working lives result from accurate profiles for the teeth and very good finishes.
Conclusion
I think that it is clear that plain and rolling bearings depend for their success on being made accurately and being continuously lubricated with a lubricant with appropriate properties for the application. It seems that stiffness comes from using small clearances and I understand that high performance racing engines cannot be turned over at all if they are cold.
I did not describe roller bearings because this is a book on the use of fluids in engineering and not about mechanical design but it seems to me that a great range of roller bearings have been designed although invented might be a better word. They must all, in their final form, be the outcome of testing.
Chapter 16 Worked examples.
Q1 Calculate the flow of oil of viscosity 0.08 kg/ms through a slit of width 50 mm and thickness 1 mm under a pressure difference of 2 bar. The length of the slit is 150mm. Treat this as a one-dimensional flow.
Q2 The diagram shows a
fitting in which oil of viscosity 0.08 kg/ms flows at a steady rate between two
annular surfaces set 0.5 mm apart. If the pressure drop between inlet and
outlet is 1.5 bar calculate the steady rate of flow. Suppose the flow to be
laminar throughout.
Q3 For the fitting in Q2 show that ,if the oil discharges at atmospheric pressure, the pressure at radius is given by gauge
Q4 Calculate the force exerted by the oil on the lower plate in Q2.
Q5 The solution to Q4 is very tedious. Solve it using a mathematics package.
Q6 The
figure shows a cylindrical pot filled with oil. The pot contains a cylindrical
steel block of diameter 100 mm and length 100 mm. The radial clearance between
the pot and the block is 0.5 mm. Keeping in mind that there is a buoyant force
on the block suppose that the block descends through the oil co-axially with
the cylinder and calculate a value for the speed of descent.
Take the density of the oil and of steel to be 7,860 kg/m3 and 850 kg/m3. Take the viscosity of the oil to be 0.08 kg/ms.
Q7 For the system shown in Q6 plot graphs showing :-
(i) how the speed of descent varies with the clearance for values of clearance from 0.5 to 2 mm,
(ii) how the speed varies with an additional applied force acting downwards for a clearance of 0.075 mm. Use a range of force of 0 to 2000 N,
(iii) how the speed varies with diameter for a clearance of 0.5 mm and an added force of 1000 N and,
(iv) how the speed varies with temperature over the range 10°C to 30°C if the diameter and the length are both 50 mm and a total force on the piston of 1,000 N. Take the relationship between temperature and kinematic viscosity over this range to be where is in °C and .
Q8 A thrust pad is 150 mm long and inclined so that the clearance at the leading edge is 0.01 mm and at the trailing edge is 0.003 mm. The collar that works with the pad moves at 1.5 m/s. If the viscosity of the lubricant that is fed continuously to the pad is 0.09 kg/ms :-
(i) use a maths package to plot the graph of the pressure distribution in the lubricant and structure the programme so that the gaps at the leading and trailing edges can be changed at will,
(ii) find the force exerted on the pad per metre width of the pad,
(iii) find the position of this force.
Q9 Once the programme in Q8 is available it is easy to find out the effect of changing the dimensions. The interesting changes are in the clearances. Try changing leading edge clearance whilst keeping the same trailing edge clearance. Then try vice versa.
Solutions
Q1 This computation serves only to get some idea of the numbers involved for these dimensions and for oil of this typical viscosity. The system has no practical use.
We have .
Substituting
This is 0.069 litres/s.
Q2 It will be clear that the solution depends on the use of the
expression In this case the value of varies with radius with all constant. An integration is needed. The
diagram shows an annular ring of radius and width .
Oil flows through the annular space associated with this ring and sustains a
pressure drop of .
Then we can write :- .
So the total pressure drop is from which we can find a value for the flow.
. Then
Q3 We know the volume flow from Q2. We can use the expression to find the pressure drop over the radial distance . Then, if is the pressure at the inlet radius 0.025 m the pressure at radius is given:-
from which :-
In gauge pressure and then :-
and
.
Q4 The expression above can be further simplified to give :-
and
.
This pressure acts on an annular area and exerts a force
.
From this we can find the force on the upper face by integration.
.
Now and the first integral becomes :-
. The second integral becomes:-
The force on the annular area becomes
There is a force on the inner circular area given by .
The net force is 779+294=1073N
Q5 We have and if we multiply by we get the force on the elemental area. So the elemental force is given:-
This can be integrated using Mathcad.
Add the force on the central area to give answer.[2]
Q6 The net downward force = volume of cylinder times the difference in the weight densities.
Using
This block will fall very slowly indeed – it would take 5 minutes to fall though 100 mm.
(iv)
Q7 This is an
exercise in using where is the downward force that will be the sum of
the net weight in oil and the added force.
Q8 This is an application of for converging flat plates.
We need a value for . Clearly, from the exaggerated diagram, .
By
proportion
The programme and graph are given below. You have to keep a track of the units but I have substituted in an obvious way.
The total force per unit width is just an integral and the position of the resultant force comes from integrating the moment of every elemental force about the leading edge and dividing by the force.
The force per unit width has little meaning but, if the pad were to be 75 mm wide the force calculated from this one-dimensional treatment would be 3.54 kN.
(We know that there will be leakage and the pressure will vary across the pad. If I wanted to find a better figure for the force I would let the transverse pressure distribution be parabolic and rework it. It is better than guessing.)