=12.= In hoisting in balance from deep shafts with cylindrical drums,
if no tail-rope is used, or in hoisting from a single shaft with an
unbalanced cage, the hoisting engine is not loaded equally at different
points of the hoist owing to the gradually changing weight of the
unbalanced rope. The following illustrations will further explain this.
=13. Hoisting With a Cylindrical Drum.=--Suppose that, from a
single-compartment vertical shaft 1,000 feet deep, it is required to
hoist each trip a load, including friction, of 11,000 pounds made up as
follows:
POUNDS
Weight of material 4,000
Weight of car 3,000
Weight of cage 3,000
Friction, 10 per cent. 1,000
------
Total 11,000
If a 1⅜-inch cast-steel rope weighing 3 pounds per foot is used,
winding about a drum 7 feet in diameter, the weight of rope is then 3 ×
1,000 = 3,000 pounds and the load on the rope, when the cage is at the
bottom, is 11,000 + 3,000 = 14,000 pounds, while at the top the load on
the rope is only 11,000 pounds. The moment of the load at the bottom
is then the load 14,000 multiplied by the radius 3½, or 14,000 × 3½ =
49,000 foot-pounds; and at the top, 11,000 × 3½ = 38,500 foot-pounds.
This shows that the load against the engine is much greater at the
beginning than at the end of the hoist.
=14.= Take now a double-compartment vertical shaft of the same depth
as in Art. _13_ and assume the same amount of material hoisted at a
trip, in the same mine car and on the same cage; but that an empty car
and cage are lowered in one compartment while the loaded car and cage
are hoisted in the other. The two cars and the two cages will balance
each other, and the loads will be as follows: At the beginning of the
hoist, when the loaded car and cage are at the bottom, the gross load
is 14,000 pounds, made up as follows:
POUNDS
Weight of material 4,000
Weight of mine car 3,000
Weight of cage 3,000
Friction, 10 per cent. of above 1,000
Weight of rope 3,000
------
Total 14,000
Multiplying this by the radius of the drum, the gross turning moment is
14,000 pounds × 3½ feet = 49,000 foot-pounds, as before, but there is a
counterbalancing load of 6,000 pounds, made up as follows:
POUNDS
Weight of mine car 3,000
Weight of cage 3,000
-----
Total 6,000
Less friction, 10 per cent. 600
-----
5,400
This means a counterbalancing load moment of 5,400 pounds × 3½ feet =
18,900 foot-pounds. The net load moment to be overcome by the engine
at the beginning of the hoist is, therefore, 49,000-18,900 = 30,100
foot-pounds.
At the end of the hoist there is a gross load on the loaded side of
11,000 pounds, made up as follows:
POUNDS
Weight of material 4,000
Weight of mine car 3,000
Weight of cage 3,000
Friction, 10 per cent. 1,000
------
Total 11,000
This is equal to a gross load moment of 11,000 pounds × 3½ feet =
38,500 foot-pounds, but there is a counterbalancing load of 8,100
pounds, made up as follows:
POUNDS
Weight of mine car 3,000
Weight of cage 3,000
Weight of rope 3,000
-----
Total 9,000
Less friction, 10 per cent. of 6,000 600
-----
8,400
This is equal to a counterbalancing load moment of 8,400 pounds × 3½
feet = 29,400 foot-pounds, and leaves a net load moment against the
engine of 38,500-29,400 = 9,100 foot-pounds. In other words, the load
moment that the engine has to overcome varies from 30,100 foot-pounds
at the beginning of the hoist to 9,100 foot-pounds at the end of the
hoist.
=15. Hoisting With Conical Drums.--Conical drums= are designed to
make the work of the engine as nearly uniform as possible throughout
the hoist. To accomplish this, when the cage is at the bottom of the
shaft, and the load is therefore heaviest, the rope winds on that part
of the drum having the smallest diameter. As hoisting continues, the
rope winds on a gradually increasing diameter of drum, and when the
cage is at the top of the hoist, and the load therefore least, the
rope is winding on that part of the drum having the greatest diameter;
in this way, the moment of the load at every point of the hoist is
approximately the same. The great difference in the loads at different
parts of the hoist is due mainly to the variation in the weight of the
rope hanging from the drum; hence, the less the weight of the rope in
proportion to the total load on the engine, the more nearly uniform is
the load on the engine.
[Illustration: FIG. 9]
=16.= Fig. 9 (_a_) shows the condition at the beginning of the hoist
when conical drums are used. Cage _a_ is at the bottom and carries a
loaded car; cage _b_ is at the top and carries an empty car. The net
moment that the engine must overcome is the sum of the weight of the
material to be hoisted, weight of the cage and car at _a_, and the
weight of the rope attached to _a_, multiplied by the small radius _r_
of the drum, minus the weight of the car and cage at _b_, multiplied by
the large radius _R_ of the drum.
Fig. 9 (_b_) shows the condition of things at the end of the hoist,
when the cage _a_ is at the top and cage _b_ at the bottom. The
loaded car and cage _a_, whose rope in Fig. 9 (_a_) was winding on
the smallest diameter of the drum, is now at the top and the rope is
winding on the largest diameter of the drum. The cage _b_ with the
empty car is now at the bottom and the rope is unwinding from the
smallest diameter of the drum. The net moment that the engine must
overcome in this position is equal to the sum of the weight of the
material hoisted, the weight of the cage _a_ and the car, multiplied by
the larger radius _R_ of the drum, minus the sum of the weights of the
cage _b_, the car, and the rope, multiplied by the small radius _r_ of
the drum.
=17.= If the moment of the load against the engine at the beginning of
the hoist is to equal that at the end of the hoist, it is possible to
determine what relative diameters of drum will produce such an effect,
as follows:
Let _Wₘ_ = weight of material hoisted;
_Wₖ_ = weight of cage and car;
_Wᵣ_ = weight of rope;
_R_ = large radius of drum;
_r_ = small radius of drum.
The load moment may be calculated by including friction as ⅒ of the
total weight hoisted, except the weight of the rope, as shown in Art.
=14=; or the friction may be disregarded without serious error. Then,
under the conditions shown in Fig. 9 (_a_), and disregarding friction,