The following rules will enable any one who understands arithmetical operations to make the calculations necessary for designing a set of cone pulleys in such a manner that the belt can be shifted from one pair to another, and be equally tight in every position. There are six cases to be considered.

Cone Pulleys, Fig. 1

Cone Pulleys, Fig.2

Case 1

Crossed belt passing over two continuous cones. (Fig. 1.) In this case, it is only necessary to use two similar conical drums, with their large and small ends turned opposite ways.

Case2

Crossed belt passing over two stepped cones that are equal and opposite. (Fig. 2.) Draw vertical lines, A B, CD, etc., to the axes of the pulleys, at distance apart equal to the face of a pulley. Lay off, on each side of the axis, distances, ab, a c, equal to the radius of the largest pulley, and d e, d f, equal to the radi us of the smallest pulley. Draw a straight line, LM, through hand e, and .N O through c and f. The points in which these lines cut the verticals determine the radii of the intermediate pulleys. Case 8. Grossed belt passing over any two stepped cones Assome values for the radii of one driving-pulley and the corresponding driven pulley. Then, for any assumed radius of a second driving-pulley, the radius of the driven pulley must have such a value that the sum of these two radii is equal to the sum of the first two. The same must be true for every pair of pulleys in the two stepped cones.

Example.-Suppose the radius of the first driving-pulley is 15 inches, and of the first driven pulley 5 inches. Now, if there are five steps in the driving-cone, having radii of 15, 12, 9, 6, and 3 inches respectively, the corresponding steps of the driven cone will have radii of 5, 8, 11, 14, and 17 inches, since the sum of the radii of each pair of pulleys must be equal to the sum of the radii of the first pair, or 20 inches. It will be evident from the foregoing that, in the case of crossed belts, the construction of cone-pulleys is very simple, since it is only necessary to observe the directions given above, no matter what the distance between the centres of driving and driven pulley may be.

Cone Pulleys, Fig. 3

Case 4

Open belt passing over two continuous pulleys. (Fig. 3.)

For this case equal and similar conoids must be used. Assume the largest radius, AF, and the smallest, B D, and calculate, by the rule on page 111, the length of belt required for pulleys with the given radii, the distance, K L, between their centres being given. Then the middle radius, C H, is found by the following- rule:

Subtract twice the distance between centres from the length of the belt, and divide the difference by the number 6.2832.

Having found the middle radius, draw circular arcs through the points F H D and G I E, thus determining the section of the conoid.

Example.-Suppose that the largest radius is 24 inches, the smallest 6 inches, and the distance between centres of conoids 3 feet. What should be the middle radius?

First find the length of belt: 2 diminished by 0.5 equals 1.5. This divided by 3 equals 0.5, and the corresponding number in table of factors", page 111, is 1.047-(1). 1.047 multiplied by 1.5 equals 1.571-(2). 2 added to 0.5 equals 2.5, which multiplied by 3.1416 equals 7.854-(3). 3 multiplied by 3 equals 9, which less 2.25 equals 6.75. 1.5 multiplied by 1.5 equals 2.25. The square root of 6.75 is 2.6. which multiplied by 2 equals 5.2-(4). The sum of 5.2 and 1.571 and 7.854 equals 14.625, which is the length of belt. Then find the middle radius by the preceding rule: 3 multiplied by 2 equals 6. 14.625 less 6 equals 8.625, which divided by 6.2832 equals 1.373 feet, or about 16-1/2 inches middle radius required.

Cone Pulleys, Fig. 4

Case 5

Open belt passing over two stepped cones that are equal and opposite. -(Fig. 4.)-The construction will be evident from the figure, it only being necessary to form two continuous conoids, as explained above, and divide them into the required number of steps.

Case 6.- Open Belt Passing Over Any Two Stepped Cones

The rules for this case, originally demonstrated by J. B. Henek, are presented below in a simplified form. First assume the radii of one driving-pulley and the corresponding driven pulley, measure the distance between their centres, and find the length of belt required. Then assume values for the radii of the successive pulleys on the driving-cone, and calculate the values of the corresponding radii on the driven cone by the following rules: I. Having assumed the value of one radius, it is first necessary to ascertain whether the one to be calculated is larger or smaller. (1) Multiply the assumed radius by the number 3.1416, and increase the product by the distance between the centres of the pulleys. (2) If the quantity obtained by (1) is greater than half the length of the belt, the assumed radius is greater than the one to be determined. (3) If the quantity obtained by (1) is less than half the length of the belt, the assumed radius is less than the one to be determined. II. When the assumed radius is the greater of the two, to find the other one. The distance between centres and the length of belt are supposed to be given. (1) Multiply the as sumed radius by the number 6.2832, subtract this product from the length of the belt, and divide the remainder by the distance between centres. (2) Add the quantity obtained by (1) to the number 0.4674, and extract the square root of the sum. (3) Subtract the quantity obtained by (2) from the number 1.5708, and multiply the difference by the distance between centres. (4) Subtract the quantity obtained by (3) from the assumed radius; the remainder will be the required radius. III. When the assumed radius is the smaller of the two, to find the other one. (1) Same as (1) of preceding rule. (2) Same as (2) of preceding rule. (3) Subtract the number 1.5708 from the quantity obtained by (2), and multiply the difference by the distance between centres. (4) Add the quantity obtained by (3) to the assumed radius; the sum will be the required radius.

Example.-The first driving-pulley of a stepped cone has a radius of 12 inches, and the radius of the corresponding driven pulley is 4 inches; the distance between centres of pulleys is 3 feet, and there are three other pulleys on the driving-cone, having radii of 9, 6, and 3 inches respectively. It is required to find the radii of the corresponding pulleys on the driven cone. It will be necessary first to calculate the length of belt required, which is 10.334 feet, or about 10 feet 4 inches.

Next find whether the pulleys on the driving or driven cone are the largest. Half the length of belt is 5.167 feet. For the 9-inch pulley, 0.75 multiplied by 3.1416 is 2.356, and adding 3, the sum is 5.356, which is greater than 5.167, showing that the 9-inch pulley is larger than the pulley to be determined. For the 6-inch pulley: 0.5 multiplied by 3.1416 equals 1.571, and increased by 3 equals 4.571, and as this is less than 5.167, the 6-inch pulley is smaller than the pulley to be determined. Of course, then, the remaining 3-inch pulley is still smaller than the corresponding pulley in the driven cone.

To find the radius of the pulley corresponding to the one on the driving-cone whose radius is 9 inches: (1) 0.75 multiplied by 6.2832, 4.712. Subtracting 4.712 from 10.334, the remainder is 5.622; 5.622 divided by 3, 1.874. (2) 0.467 added to 1.874, 2.341. Square root of 2.341, 1.53. (3) Subtracting 1.53 from 1.571, the remainder is 0.041; 0.041 multiplied by 3, 0.123. (4) Subtracting 0.123 from 0.75, the remainder is 0.627 feet, or about 7-1/2 inches, radius of required pulley.

Pulley corresponding to driving-pulley whose radius is 6 inches: 0.5 multiplied by 6.2832 equals 3.142. This subtracted from 10.334 equals 7.192, which divided by 3 equals 2.397-(1). Adding 0.467, we have 2.864, the square root of which is 1.692 -(2). 1.692 diminished by 1.571 equals 0.121, which multiplied by 3 equals 0.363-(3). Adding 0.5 gives 0.863 feet, or about 10-11/32 inches, required radius.

Pulley corresponding to driving-pulley whose radius is 3 inches: 0.25 multiplied by 6.2832 equals 1.571. This, subtracted from 10.334, equals 8.763. Dividing the last by 3 gives 2.921-(1); and by adding 0.467 we have 3.388. Of this the square root is 1.841-(2). Subtracting 1.571 gives0.27, which multiplied by 3 equals 0.81, and adding 0.25 gives 1.06 feet, or about 12-1/16 inches, required radius.

The radii of the several pulleys on the two cones then will be:

Driving-cone...................12, 9, 6, 3 inches.

Driven cone.....................4, 7-1/2, 10-11/32, 12-1/16 inches.

Set-Screws For Pulleys

These should be made of cast-steel with hollow points; the ends should then be beveled to an edge surrounding the hole, and tempered to a dark straw. When set up, these screws cut circular indentations on the shaft, and exert an enormous force of resistance.

Working Value Of Pulleys

Pulleys covered with leather, iron pulleys polished, and mahogany pulleys polished, rank for working value as 36, 24, and 25 per cent respectively, wood and iron uncovered being almost identical.