Placing formula =1= = formula =2=,
(_Wₘ_+_Wₖ_)_R_ - (_Wₖ_+_Wᵣ_)_r_ = (_Wₘ_+_Wₖ_+_Wᵣ_)_r_ - _Wₖ__R_,
and
(_Wₘ_+ 2_Wₖ_+ 2_Wᵣ_)
_R_ = _r_ ------------------- (=3=)
(_Wₘ_ + _2Wₖ_)
Since the diameter of a drum is generally given instead of the radius,
it follows that if _D_ = larger diameter, _d_ = smaller diameter, and
then, since _D_ = 2_R_ and _d_ = 2_r_, formula =3= may be written
(_Wₘ_ + 2_Wₖ_ + 2_Wᵣ_)
_D_ = _d_ ---------------------- (=4=)
(_Wₘ_ + 2_Wₖ_)
Formula =4= gives only approximate results, which are, however,
sufficiently accurate for the mine superintendent’s use, and for this
reason friction has been omitted, as it would make the formula much
more complex. It may be expressed as a rule as follows:
=Rule.=--_To find the large diameter of a conical drum, multiply the
small diameter by the sum of the weight of the material to be hoisted,
twice the weight of the cage and car, and twice the weight of the rope;
divide this product by the sum of the weight of the material, and twice
the weight of the cage and car._
[Illustration: FIG. 10]
Applying this rule to the problem given in Art. =14= and omitting
friction,
7(4,000 + 12,000 + 6,000)
_D_ = ------------------------- = 9.6 feet
(4,000 + 12,000)
The drum would then be 7 feet in diameter at the small end and 9 feet
7¼ inches at the larger end.
=18=. Fig. 10 shows a special form of combined conical and cylindrical
drum designed for hoisting a total balanced load of 25 tons through a
vertical height of 550 feet.
Fig. 11 shows a combined conical and cylindrical drum; an unusual
feature is the rope reel shown at each end of the drum, which permits
of properly storing a few hundred feet of extra rope, allowing the rope
to be lengthened, when needed, without splicing.
[Illustration: FIG. 11]
=19. Comparison of Cylindrical and Conical Drums.= The disadvantages
of the cylindrical drum lie entirely in the fact that the load on the
engines is variable, but it is possible to overcome this disadvantage
by adding a tail-rope to the cages to balance the weight of the rope.
This system gives its best results where hoisting is done from one
level only, but in deep hoisting it is impracticable because of the
extra weight added and because of possible excessive swaying of the
rope.
The conical drum has two strong points in its favor: first, the load
on the engine may be nearly equalized during the entire hoisting
period; and, second, the starting of the engines with the load requires
less power.
The disadvantages of the conical drum are as follows: To maintain a
certain average speed of hoisting, the speed toward the end of the
hoist is of necessity higher than the average and comes at a time
when a slowing up should be taking place, so that more care must be
exercised when making the landing. To prevent the rope from being
drawn out of the grooves, the latter must be made deep and with a
large pitch, thereby increasing the width of the face or length of
the drum. In making a landing, when the rope is on the conical face,
the rope must be kept taut, as any slackness will permit the rope to
leave the groove, with the result that all the rope will pile up in the
bottom grooves of the drum allowing the cage to drop into the mine,
unless it is resting on the chairs. If there are several levels to be
hoisted from, the equalizing of the load on the engines can only be
realized for one level; for all other levels this advantage will be
lost. For large depths, conical drums become very long and require
correspondingly long leads from head-frame to drum. To hold the same
amount of rope, conical drums are heavier than cylindrical ones, and as
a result, the power required in starting the load is somewhat increased
owing to the greater inertia of the rotating parts.
Some of these disadvantages have been overcome by making a combination
of cone and cylindrical drums. The drums are so designed that the
landing takes place only when the rope is on the cylindrical portion
of the drum. For deep hoisting, the greater diameter of the drum and
its length must be inconveniently large if the load is equalized.
The length and diameter can be reduced by making one-half of the
drum cylindrical and by having the rope from each end wind on the
same cylindrical portion of the drum. In all cases, however, these
modifications are made at the expense of the equalization of the load
on the engines, and it is not possible to obtain the latter without
including some serious disadvantage.
There are certain objections to both cylindrical and conical drums:
their great size and weight, for large hoists, make them very
expensive; their width necessitates placing the engines far apart,
which adds to the cost of the engines, foundations, and buildings; the
great weight of the drums is also objectionable, because it forms a
large part of the mass to be put in motion and brought to rest at each
hoist.