The lubrication of roller bearings subjected to couples

The lubrication of roller bearings subjected to couples R. GOHAR Roller bearings are generally designed to carry mainly radial loads with additional bearings employed to carry any axial loads. Occasionally, limited space prevents the use of these additional bearings, so that the roller bearing itself must carry the axial load. This must be taken by flanges on both the inner and outer races. The effect on the rollers themselves is to modify the load distribution on their generator contact surfaces because the rollers are now subjected to a couple in addition to the radial load. A similar effect is produced if t w o flanged (or unflanged) roller bearings support a loaded shaft. As its ends are nominally direction fixed, the rollers must be subjected to terminal couples, thus causing a modification to the load distribution at the contact surfaces. The effect of such forces on the footprint shape and on the bearing lubrication is investigated.

NOTAT ION xyz W

Wl w2 T l CI

P Pmax

R1 R2 R E P Q C e

hs

x~

1,1o ho a

U

co-ordinates (Fig 2) total load per unit length a t y (load intensity) radial load intensity maximum load intensity due to couple applied couple on a roller effective length of roller (observed footprint length) footprint half width at y (Fig 2c) pressure at x, y (Fig 2b) maximum pressure at y outer race inside radius roller radius reduced radius (1/R = l/R2 - l/R1) reduced Young's modulus of glass and steel eccentric load on load pan rib reaction eccentricity of P distance between rib reactions gap height aty, x wave length of sodium light in air viscosity of oil at atmospheric pressure entry film thickness in elastohydrodynamic lubrication pressure viscosity coefficient of oil rolling speed (U = Urace + Uroller) at contact

Department of Mechanical Engineering, Imperial College, London, SW7 2BX, England

ASSU MPTI ONS Flange lips are small compared with roller diameter so that equal and opposite axial rib loads are applied at the respective rib geometric centres, see Fig 1. It is these loads which cause the couple.

-q I _

Q

I

i

(~1

I I I I kI

__J

Fig 1 Flange forces or couples cause a modified load distribution on the roller

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2 The couple is small enough to allow us to assume a linear modified load distribution at the roller-race contact surfaces. 3 The rollers have been suitably blended at their ends so as to cause the radial load distribution, along the roller length, to be constant right to the roller ends 1'2. Stress concentrations are therefore neglected in the simple theory to be described, although Ref 2 shows they can be significant in all but the most carefully profiled rollers. 4 The modification to the load distribution is small enough to employ 2-dimensional elastic (plain strain) and lubrication theory in the contact region.

ANALYSIS

Assume that one o f the rollers of a cylindrical roller bearing is subjected to a uniform radial load, Wl per unit length, together with a couple T caused either by an axial load on the inner race or by a moment. Fig 1 shows the effect on an axial load on the outer race. This is supplied to a roller through rib reactions which cause a couple T = In accordance with assumption 2, let T, therefore, modify the distribution of w on the top of the roller in the manner of Fig 2a. The maximum and minimum load intensities are now (w 1 + w2) and (Wl - w2) acting diametrically across the roller. There is no nett change in radial load. Then:

Qe.

2y w(y) = w I + ~-w 2

(1)

Z

Where: +a

(2)

--(l

) =f

a

In accordance with assumption 4, let the pressure distribution be Hertzian in the x-direction, ie:

/J

/

P =Pmax(1- x2/a2)1/2

(3)

p(x)

where is the pressure and a(y) is the footprint half width, see Figs 2b and c.

X

From Equations 1, 2 and 3: 7r

w I + 2w2y/l = 2aPmax

(4)

Also, following assumption 4 and Ref 4:

a=l8-~EIwI+2@w21f 1/2 x-.-./////

(s)

l/, ~ /, ~/ / /

/

for line contact. From equations 4 and 5 :

y

Pmax=l LIWl+2--fw2) I 1/2

(6)

X

and: l

a= Q

t,.( ~

wl +

w2

)i

(71

where:

C

Fig 2(a) Load intensity~ (b) assumed transverse pressure distribution and gap shape. (c) footprint shape

22

TRIBOLOGY international February 1975

l --~
1 2

Equations 6 and 7 give the modified pressure distribution and footprint shape caused by a couple T acting on a single roller.

As T must be resisted by linearly distributed reactions on the top and bottom of the roller: +U2 T =2 I

wydy

i,I

-4/2

Also: w2

l

( w - Wl) 2y a SO:

w2P

T=--

(8)

3

In the experiment to be described, T is supplied by an eccentric load P, distance c(y) from the z axis of the roller, ie: T = Pc

(9)

If 2y/l w2 > Wl, the footprint will become foreshortened and Pmax will become negative in Equations 6 and 7. It is assumed that the assumption of a linear load intensity breaks down before these conditions are reached.

iii)

b

Deformed region just outside the footprint

As plain strain is assumed, it is possible to estimate approximately the modified gap shape, in the x-direction, just beyond the footprint. Referring to Fig 2b, an approximate value for hs is s: 1)3/2

h s - aPmax 3.81 (x/a E

(1 O)

or, from Equations 4 and 10:

x/a = 1 +

hs

) ] I 2/3

(11)

C

Equation 11 gives the gap shape at any value ofy. It also enables theory to be compared with experiments which utilize optical interferometry to measure the region of contact. Thus, using the well-known theory for fringes of equal thickness s, i f h s is measured at the first dark fringe outside the contact area then: hs -

x~ 2#o

(12)

Fig 3a shows a typical fringe picture of the end of a blended commercial roller. The light used is sodium (Xa = 5.89 X 10 -8 m) and the medium is oil (#o = 1.5). The first dark fringe is clearly visible. As the load is radial, there being no couple present, the footprint and surrounding fringes are rectangular over most of their length. The dark central region, however, has a finite hs along its edge and does not indicate accurately the extent of the footprint. This is the

d Fig 3 Fringe pictures: glass outer race and blended steel roller," medium: oil 07o = 28.7poise at 26°C) Wl = 3. 72 X 105 N m - 1. (a) radial load only, (b) radial load and couple (T = 1.5 Nm), (c) rotating bearing, radial load only (h o = 0.88 X 10-7m), (d) rotating bearing, radial load and couple, (T= 1.5Nm, h o = O . 8 8 X I 0 - 7 m )

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reason why the first dark fringe is the one measured in these experiments. Fig 3b shows the same roller end when a couple is present in addition. The modification to the load intensity has caused this end to widen considerably, although the nett load is the same. EXPERIMENTAL DETAILS The main object o f the design of the apparatus was to simulate a roller journal bearing in which provision was made for measuring roller to outer race oil film thickness. The outer race was, accordingly, made of crown glass polished to 5.08 × 10 - 8 m cla and having a 200 A chrome layer on its inner surface, whilst its outer surface was bloomed. This was in accordance with the usual practice when employing optical interference techniques to measure the oil film. The rollers used were all 0.012 m l o n g and 0.0127 m in diameter. They were made o f steel from bearing manufacturers' stock but polished to better than 5.08 X 10 8 m cla. There was a small radius at each end or there was blending at each end over ¼ of the length.

Fig 5

View of the apparatus from above

steel cables round [he sleeve supported a pan which could be loaded either concentrically for a radial load, or with a small addition load at high eccentricity if couples were to be supplied to the rollers.

Referring to Fig 4, for a sectional view, and Fig 5 for a view from above, the inner race was a hardened steel shaft 0.0508 m diameter with a groove 0.003178 m deep (the rib) for axial location of the rollers. The groove base was ground and lapped to a surface finish of 1.27 × 10 - 7 m. Axial clearance for the rollers was 1.27 X 10 5 to 2.54 X 10 . 5 m. The inner race shaft was supported on two externally pressurized double row orifice air bearings and was axially located at each end by single orifice externally pressurized air thrust bearings. The glass outer race was itself the inner member of a single row orifice externally pressurized air bearing, the outer member of which was a steel sleeve used for attaching the loading mechanism. Two

The roller assembly was pitch located by a ptfe cage attached to the stationary sleeve of the loading mechanism air bearing. With the above arrangement the roller bearing could be used for static tests or operate in a counter rotating mode. The glass outer race was axially located by two ptfe pads just clearing its faces. These offered negligible resistance to outer race rotation during the dynamic tests. The outer race linear velocity normally was never less than 98% of that of the driven inner race. At the moderate speeds employed, outer and inner race angular speeds were measured by stop watch, although, for very high speeds,

Correction

Air supply to the slave journal bearing

Viewing window

Thrust air bearing

/

r ~ /

/

/

/

Driven pulley

I

Hor d en ed s reel shaft/ inner race

• Air supply to the thrust bearing

]2 ____Tacho - c o u n t e r

J

...,,ing sorew

.

="

....

.

.

IIV

.

~b ~

T

I

..

Orw,ng belt

I

y/l (/,/ tli/~//,l"//(/l/",,/ //l//I/I///I//A Loading c o b l e s ~

~kRigidsteel table

.o.,o,t.% G~=T=~Couple

Fig 4

24

wheel

beoring

Sectional view of the apparatus

TRIBOLOGY international February 1975

ql

I i Scole 9, ........ ! 2 in r J i i J [ 0 I0 20 30 40 50 mm

there was a photo electric pickup for the outer race, and a magnetic pickup for the inner race shaft, connected to a decade counter. Optical access to the contact between the outer race and the top roller was through a viewing window located in the top of the sleeve surrounding the outer race, see Fig 2. The window was a glass plug, so shaped as to correct the outer race to roller cylindrical lens effect. Interference fringes were viewed through the window by means of a low power, long working distance micrometer microscope having light coming from above the objective. For the purpose of the experiments, there were four equipitched rollers. Maximum radial clearance between the inner race and bottom roller was 2.54 × 10 -5 m under zero load. This was sufficient to cause all load to be taken by the top roller. The weighing pan, attached to the outer race, had an extension arm which enabled the couple -+T to be applied by means of a small weight P, (P ~ Wll) at variable eccentricity +-c. RESUL TS A N D DISCUSSION

The effect of a couple T on the footprint shape has been shown in Fig 3 for a blended roller. Clearly the end of the footprint is seen to be enlarged due to the additional load on the roller end. However, because of the large ratio of footprint length to breadth, the actual footprint shape over its whole length cannot easily be shown in a fringe photograph. In order, therefore, to check the accuracy of the theory, the geometry of the first dark fringe contour for a roller with small end radii was measured directly with the micrometer microscope, and then drawn to a distorted scale. Fig 6 shows the half contour of the first dark fringe for three different values of T, the highest value being near the assumed limit of the theory, that is when the footprint is about to become foreshortened. Because the test roller might not have been quite symmetrical in its end radiusing, the couple was first applied towards one end. The roller was reversed and the couple was applied towards the other end. The mean footprint width at each value o f y was calculated. As the couple increases and the maximum load intensity becomes almost twice the radial contribution, so edge stresses become significant, especially at the loaded end of the roller. This is shown in Fig 6c, for the experimental points represented by A, which have exceeded the theory (full line) before falling to zero width. The effect is less pronounced in Figs 6a and b. The maximum footprint width is seen to occur near the start of the roller end radiusing, which occupies only about 1/5 o f the roller length between faces. The bulge in the footprint can also be seen on Fig 3b. The dotted line in Fig 4 is the theoretical footprint outline shown for comparative purposes. The assumption of a linear load distribution, suggested in Ref i and giving a maximum pressure intensity of up to twice the radial value, appears to be accurate. The theory therefore enables the maximum contact pressure to be calculated from Equation 6, having first estimated the couple T on the roller due, for example, to an axial load on the inner race. The existence of a couple will also exacerbate edge stress concentrations which Ref 2 claims can exceed 7 times the purely radial value of maximum Hertz pressure. Obviously, the method of reducing the effect of the additional couple and edge stress concentrations, is to blend the rollers over a considerable part of their length, and then

0 X

a

o'i 12

o

8

x X

b

o

0 X

/

4"~///

g, ""

J

f

0

o I

I

1

I

1

I

2

4

6

8

I0

12

ym x 10 -3

Fig 6 Contact shape: glass outer race and steel roller; medium: oil, ( % = 28poise at 26°C) w I = 3. 72 X 105 N m - I) Experimental results at first dark fringe: (a) O, T = 0.75 Nm, (b) A, T = + I . 2 Nm;Fq, T = - 1 . 2 Nm, (c) O, T-+2. 07 Nm; V, T = - 2 . 07 Nm --: theory at first dark fringe, .... : theoretical f o o t p r i n t shape

supply an additional radius at the end. Assuming the blend is taken up completely by the direct load (giving approximately a uniform axial pressure distribution) then the addition of a couple will cause the roller to tip on to the end radius. The stress concentration caused by a sharp end digging into the race is therefore alleviated. This practice is often followed in the bearing industry for first quality roller bearings. When the bearing is run in a counter rotating mode under lubricated conditions Fig 3c is obtained for a purely radial load, and Fig 3d for the same load with a couple added. The oil film thickness near the roller end is barely seen to change, despite the couple nearly doubling the load intensity there. The central region has remained dark at a f'dm thickness of 0.88 × 10 - 7 m, except at the very end where the characteristic end closure horseshoe shape has appeared 7. This shape is synonymous with highly loaded bearings and

T R I B O L O G Y i n t e r n a t i o n a l February 1975

25

has been achieved here with no nett increase in the applied load. The observation that oil film thickness (excepting that due to end closure) is fairly insensitive to load intensity, is in accordance with the Grubin theory of ehl. This gives the film thickness in the dark region of Figs 3c and d asS: 1.28(a~oLOO.727R 0.455

ho =

(13)

a0.182

h o is seen to be practically insensitive to a moderate change

why roller bearings subjected to axial loads still appear to run satisfactorily. ACKNOWL EDGEMENTS

The author wishes to thank the Scientific Research Council for helping to fund the project, and also Mr H. Bahadoran who designed and made the test rig, took the photographs, and supplied the sectional view. REFERENCES

in the value of a. (From assumption 4, a can be taken as approximately half the local width of the dark region in Figs 3c and d.)

l Iko, O. and Orte, S. 'Axial load carrying capacity of cylindrical roller bearings', Ball Bearing J, No 15 1 (1967) 13-20 and 21 26 2 Rodzevich, N. V. 'Design and selection of optimum shape of bearing rollers', Russian EngingJ, 50, No 7 (1970) 37-42 3 Thorp, N. 'Oil films and friction in elliptical contacts', PhD Thesis, University ot Londo n ( 1972) 4 Bahadoran, H. and Gohar, R. 'Oil film thickness in lightly loaded roller bearings', to be published

CONCL USI ONS

The assumption of a linear load intensity variation has been shown to be accurate for rollers subjected to couples. The theory enables the maximum direct stress, due to the combination of the couple and the radial load, to be calculated. To this stress should be added a safety factor to allow for end stress concentrations. These in their turn can be reduced by proper blending of the rollers.

5 Gohar, R. 'Oil films under elastohydrodynamic conditions', PhD Thesis, University of London (1965) 6 Lundberg, G. 'Elastic contact between two semi infinite bodies', Forsch Gebiete Ingenieur (1939) 201-211 7 Bahadoran, H. and Gohar, R. 'End closure in elastohydrodynamic line contact', to be published

The redistribution of the load intensity due to a couple hardly affects the oil film thickness, and probably explains

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