July 2, 1915.

THE COLLIERY GUARDIAN.

11

An approximate result would be obtained by making
the line from end to end of the drum surface a circular
arc of about 46 or 48 degs. Such a result is shown in
fig. 4, which represents a shape of drum that forces both
ropes to coil closely and evenly from one end of the
wind to the other. This is one form of the ungrooved
spiral screw tread described in a previous paper.*

(3) If, however, it is thus possible to get even coiling
without grooves and without lateral friction, during the
first half of the wind, why not go one step further,
place the pulleys in tandem, centre each rope at the end
of the wind, and thus avoid lateral friction at the drum
during the whole of the journey? This result is also
plotted, to its theoretical values, in fig. 5, in which the
convexity between (b) and (c) is hardly perceptible;
but, in practice, it might be nearly attained by arranging
the curvature from one end of the drum to the other
in a circular arc of 60 degs. (as shown by fig. 6), giving
at each extreme end of the drum a maximum inclination
of 30 degs. from a line drawn parallel to the axis, or
26 to 27 degs. from the normal surface of the barrel
drum of theory.

This sketch, however, in common with fig. 4, is not
the exact shape of the theoretical drum, nor yet a very
convenient shape for the practical reel. Fig. 4 illus-
trates a fairly good form, if the ropes centre at middle
wind. But the inclination at each end of the drum is
rather more than theory requires for close even coiling,
whilst any curvature of the middle portion is only
required to balance in some measure the loads against
the engine.

In fig. 6, however, as will be seen by comparing it
with fig. 5, the inclination at the end of the drum is very
much more than sufficient, and might possibly impose
some slight pressure on the coiled rope as one coil after
another came into position. But this slight pressure,
let it be particularly noted, would be practically

Fig. 4.—Practical Form of Spherical Drum
to Ensure Even Coiling without the Use
of Grooves, Ropes Centring at Mid-wind.

negligible in comparison with the abrasive force insepar-
able from any system of coiling the rope upon itself in
two or more layers. On the other hand, the curvature
at the middle portion of the drum might not be quite
enough for its purpose, and a little splaying of the rope
might possibly result; nevertheless, whenever the rope
reached that faulty part of .the drum, the obliquity of
the rope would vary in the same direction, although not
in exactly the same ratio, as the inclination of the drum
varied, so that the actual spread of the rope, if any,
would certainly be small.

It will now, probably, be clear that, if the rope centre
at the end of the wind, the theoretical shape, regard
being had only to the diminution of friction, will be
nearly attained, and the practical purpose completely
served by tapering the drum surface in a straight line,
and at a uniform inclination to the axis of (A + a) degs.;
where A = an angle slightly exceeding the angle of
repose, and a = the maximum angle which, during its
lateral travel, the rope makes with its centre line or with
the plane of its pulley. The value of a may be ascer-
tained by simple measurement. The value of A, the
angle which slightly exceeds the angle of repose, will
depend upon the state of the surfaces in contact—that
is, upon the kind of rope used, the material forming the
surface of the drum, and the state of the lubrication
which obtains between them. Each engineer may
estimate the value of A most suitable for his own
particular case.

It seems probable, however, that the coefficient of
friction obtaining between the lubricated surfaces will
seldom, if ever, exceed 0-2, which is the approximate
tangent of lljdegs.; but, if the angle of dip on the
drum and the angle of obliquity in the rope always pull
in opposite directions, as they should do, the safety of

*“ The Lateral Friction of Winding Ropes,” by H. W.
G. Halbaum, Trans. Inst. M.E., 1915, vol. xlix., p. 125;
Colliery Guardian, February 19, 1915.

the arrangement would hardly be affected even if the
angle of inclination should appreciably exceed the actual
value of A.

With respect to the maximum angle a of obliquity
(still considering the case of ropes centring at the end of
the wind), that angle should hardly exceed 3, or at the
most, 4degs., and, therefore, the total angle of inclina-
tion, relatively to the axis of the drum, will not, in
the majority of instances, exceed 11| + 3| = 15 degs.
This gives a slope of 1 vertical in about 3J horizontal, or
about 3|in. per ft. Such an inclination, judged by
mechanical principles, would be sufficient to prevent
spread of the rope at any point in the wind, and, judged
again by mechanical principles, it would be an absolutely
safe inclination, provided always that the ropes are so
centred that the angle of obliquity and the angle of
inclination invariably pull in opposite directions.

Of course, where the rope centres at mid-wind, the
middle part of the drum should, theoretically, be
parallel; otherwise, for half of the wind the two angles

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Fig. 6.—Spherical Drum.

Pulleys in tandem, even coiling, no grooves, and no
lateral friction.

defined above will both be exerting their forces in the
same direction. Nevertheless, for ropes centring at
mid-wind, fig. 4 probably shows the best practical form
of drum; because, in the first place, the angle a is small,
and, in the second place, if only the ends of the drum
are tapered, one would still be obliged to round off the
angle where the tapered and parallel surfaces meet.

Returning, however, to the case where the ropes
centre at the end of the wind, fig. 7 shows the practical
shape of the drum required, assuming that A = 11| and
a = 3Jdegs. No doubt the actual values of A and a
might be determined for each particular case, and the
drum tapered at the precise inclination of (A + a) degs.
But the writer will submit hereafter such practical con-
siderations as go to show that an angle of slope very
much greater than (A + a) degs. may be adopted with
the view of increasing the efficiency, without at all
diminishing the safety, of the arrangement. At present,
however, it is safer, and also most instructive, to proceed
step by step.

In the case just described, the pulleys must be in
tandem, or nearly so; each rope in turn, as its cage
comes to bank, will then come into the same vertical
plane as that which contains the pulleys and the middle
diameter of the drum. During the entire wind, the
rope coils closely and evenly by the influence of gravity
alone, whilst the pressure exerted in the line of the
radius vector entirely prevents lateral friction at every
stage of the wind.

Pulleys thus set represent no new nor novel idea; and
as for the inclined drum surface dictated by theory and
modified by practical considerations, there is no novelty
about that either. The same principle is already illus-
trated in practice by the well-known C pulley, which,
however, meets a much more difficult situation. In the
case of the winding drum, as in that of the C pulley, the
only practical difficulty is that of estimating beforehand
the actual coefficient of friction, and even that difficulty
is modified in the present case by the fact that a very
approximate estimate will fulfil the object. For pur-
poses of illustration only, a provisional value of about
0-2 has been assumed as likely to cover all cases of the
angle of repose where ropes are moderately well lubri-
cated. But it must be always remembered that, in the
arrangement against friction already proposed, and also
in that dealing with balance to be proposed hereafter,
the angles always exert their influences in opposite
directions; and always wax and wane together as they
swing between their respective maxima and minima.
Therein lies the safety of a much greater inclination on
the drum than that dictated by the exact angle of
repose. And therein also lies the mechanical secret of
the success of the tapered drums which still survive,
whilst so many others of similar shape have been
discarded as absolutely dangerous.

Such an arrangement of centring ropes at the end of
the wind, however fine for the drum, doubles the angle
of friction at the pulley, and the question is whether the
total abolition of lateral friction at the drum would
sufficiently compensate the additional friction at the

Fig. 5.—Theoretical Shape of Tapered Drum
without Grooves.

Pulleys in tandem, no slip, spread or lateral friction.

2jFig. 7.—Same as Fig. 5, but a More Practical
Shape.

All straight lines as seen in section; surface of drum
inclined to axis at the uniform dip of 15 degs.

pulley. It will be interesting to learn members’ views
on that point. But it may now be asked whether it is
not possible to abolish lateral friction at both ends of
the line—that is, at the drum and at the pulley alike.

Rolling Friction.

The writer is reasonably well prepared to defend all
that he has written above, but will not be surprised in
the least if members disagree with what he is now about
to suggest. Accepting that risk, he suggests the substi-
tution of rolling friction for lateral friction at the pulley.
If this method were found practicable, it would trans-
form lateral friction at the pulley into rolling friction
at a binding sheave placed in the headgear. The
question is whether such an arrangement is really
feasible, and the writer, so far as he is concerned, is
quite unable to see why it should not be so.

It has been seriously objected that the pressure on a
binding sheave so placed would inevitably overturn the
headgear! A complete answer to that objection is fur-
nished by the fact that the pressure is already active
on a rubbing surface at the pulley itself, and yet does
not overturn the headgear. Why, then, should the
same amount of pressure overturn it if applied at a
rolling contact?

Besides, what, after all, is the measure of this lateral
pressure ? The entire oblique line of pressure (contained
in the radius vector of the angled rope) is the resultant
of two components at right angles. Then, if A degs.
= the angle of maximum obliquity, and W = the total
weight in tons of the cage, tubs, coal, chains, and rope
suspended from the pulley, the maximum lateral pres-
sure on the pulley will be rather less than P tons; and
(7 A

since400 > 8inA

Thus, if W = 20 tons, and A = 3degs., P = 21 cwt.

As for the size of such a guide pulley, although
diameter is certainly a consideration, it is perfectly