FRICTION AND LUBRICATION 265 
233. Friction of Pivots and Collars.—A thrust along the axis of a 
‘shaft is taken up by a pivot or collar bearing. A pivot must be on the 
_ end of a shaft, but a collar may be at any part of the length of the shaft. 
_ The rubbing surface of a pivot or collar may be any surface of revolution, 
the axis being the axis Of the shaft. In the case of a pivot, the rubbing 
surface is generally either flat or conical. In a collar, the rubbing surface 
_ is generally flat. 
In the present state of knowledge on the subject of friction, it is 
at to determine a correct expression for the friction of a pivot or 
. There is first of all the question of the distribution of the pressure 
on the rubbing surface to consider. When the bearing is new and there 
is perfect contact over the whole of the bearing surface, it is probable that 
the pressure is uniformly distributed, but since parts of the surface are at 
different distances from the axis, they must be moving with different 
velocities, and there is therefore, very probably, unequal wear, which will 
at once cause a redistribution of the pressure, and unequal distribution 
of pressure accompanied by different velocities will almost certainly 
result in variation in the coefficient of friction at different distances 
from the axis. ‘ 
In what follows expressions will be found for the friction of pivots 
and collars on the assumption that the coefficient of friction is constant, 
and that either the pressure is uniformly distributed, or that the wear is 
uniform over the rubbing surfaces, and is directly proportional to the 
pressure and to the velocity. To say that the wear is uniform and 
directly proportional to the pressure and to the velocity is equivalent to 
stating that the product of the pressure and velocity is constant, or that 
the product of the pressure and radius is constant, because the velocity is 
proportional to the distance from the axis. 
P= total axial load carried by pivot or collar. 
p=intensity of normal pressure on rubbing surfaces when uniform, 
or at radius 7 when variable. 
r=radius of an indefinitely narrow ring of the surface, and dr 
its width. 
M=moment of friction on pivot or collar. 
Case I. Flat Pivot (Fig. 404)—(a) Uniform 
pressure p= ——7- Load on ring of radius 7 and width 
2 ; 
dr=2prrdr. Moment of friction on ring = 2pprrdr, 
M= aoe rdr= Spats - seer’ = aePr; ; 
0 
(6) Uniform wear. Let pr=c. Load on 
ring = 2rprdr = 2redr. 
Total load = P = 27re ‘ dr = 2rer,, therefore ¢ = Ser, 
Fia. 404. 
QprPrdr_ pPrdr 
en 
Moment of friction on ring = 2reprdr = 
