CONICAL DRUMS
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.
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,
Load moment = (Wₘ + Wₖ + Wᵣ)r - WₖR (1)
and under the conditions shown in [Fig. 9(b)],
Load moment = (Wₘ + Wₖ)R - (Wₖ + Wᵣ)r (2)
Placing formula 1 = formula 2,
(Wₘ + Wₖ)R - (Wₖ + Wᵣ)r
= (Wₘ + Wₖ + Wᵣ)r - WₖR,
and
| (Wₘ + 2Wₖ + 2Wᵣ) | ||
| 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 = 2R and d = 2r, formula 3 may be written
| (Wₘ + 2Wₖ + 2Wᵣ) | ||
| D = | d ———————— | (4) |
| (Wₘ + 2Wₖ) |
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.
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.
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.