per minute, the same belt at 848 feet per minute will transmit 848 as much, or 33,000 X 0.848 = 27,984 foot-pounds. The width of belt necessary to transmit 59,400 foot-pounds per minute at this speed will then be 59,400 ÷ 27,984 = 2.1 inches. No account has so far been taken of the power necessary to drive the machine itself. To allow for this the belt should evidently be not less than 2 inches wide. A 3-inch belt would allow considerable of a margin of safety, and further calculation will show that such a belt would develop, during about one-third of a revolution of the crank, the amount of energy which the fly-wheel had lost, so that, as the cutting operation takes about one-sixth of a revolution, the fly-wheel would be running at full speed for about one-half of a revolution of the crank, previous to the beginning of the cut, provided that it had not suffered any greater reduction of velocity than the 10 per cent. planned for.

If the press was employed doing punching the same method of procedure would be employed in the calculations, the area in shear in such a case being equal to the circumference of the hole multiplied by the thickness of the plate. The end of a punch is usually made slightly conical or slightly beveling, the effect in either case being to increase the shearing action, and make the work of punching easier.

CHAPTER XIX

TRAINS OF MECHANISM

FOR obtaining high speeds without the use of unduly large driving pulleys or gears, for securing gain in power by sacrificing speed, for securing reversal of direction, or for obtaining some particular velocity ratio between the driver and some part of the mechanism, pulleys, gears, worm-gears, or the like, may be substituted for direct acting driving-mechanisms.

To Secure Increase of Speed.-Let a shaft making 100 revolutions per minute be required to drive the spindle of a machine at 2000 revolutions per minute, the pulley on the spindle being 3 inches in diameter. If a direct drive were to be used, the pulley on the shaft would have to be as many times greater than the pulley on the spindle as 2000 is greater than 100, or 20 times.

This would mean a pulley on the shaft 60 inches in diameter. Practical considerations, such as the weight of the pulley, size of hangers and the like, would make such a pulley out of the question.

By interposing an intermediate countershaft between the first shaft and the spindle of the machine, however, having pulleys of such size that the product of the ratio of the pulley on the first shaft and the one to which it is belted on the countershaft, multiplied by the ratio of the second pulley

on the countershaft and the pulley on the spindle to which it is belted is equal to the ratio which it is desired to have between the first shaft and the spindle, the same speed may be secured by the use of pulleys of convenient size. Thus, if the ratio between the pulley on the first shaft and the one on the countershaft is as 1 to 4, and the ratio between the driving pulley on the countershaft and the one on the spindle of the machine is as 1 to 5, the product of these two ratios, 1 to 4 and 1 to 5, is 1 to 20, and the arrangement will give the

FIG. 204.-Reversal of Direction Obtained by Crossed Belt.

required speed. The pulley on the spindle being 3 inches in diameter, the driving pulley on the countershaft will be 15 inches in diameter, and if the driven pulley on the countershaft is 4 inches in diameter the pulley on the first shaft to which it is belted will be 16 inches in diameter, instead of 60 inches, as would be required with direct belting.

If the spindle of the machine, instead of being driven were made the driver, as it would be if it were the armature shaft of a motor, then this arrangement would give gain in power with consequent loss of speed.

To Secure Reversal of Direction.-In cases where shafts are belted together, reversal of direction of

rotation is secured by simply using a crossed belt instead of an open one, as shown in Fig. 204. When gears are used, reversal of direction of rotation follows as a natural condition of their meshing together, as shown in Fig. 205. In order that the two gears A and B shall rotate in the same direction, it is necessary to separate them slightly, and interpose an intermediate gear, or idler, between

B

B

FIG. 205.-Relative Direction of Rotation in a Pair of Gears.

FIG. 206.-Influence of Idler on Direction of Rotation.

them as shown in Fig. 206. The rates of rotation of A and B with regard to each other is not affected by the idler gear, whether the idler be large or small. That this is so may be seen by direct examination. If A is the driver, its circumference will impart to the circumference of C its own rate of motion, and C will in turn impart to B the same rate of motion, which is the same as it would have if in direct connection with A.

If, now, another idler be interposed between A and B, making four gears in the train, A and B will again rotate in opposite directions. From this it will be seen that when a train is composed of an

even number of gears, the first and last members rotate in opposite directions; but when the train is composed of an odd number of gears, the first and last members rotate in the same direction.

In Fig. 207 is shown the mechanism used in engine lathes to secure either direct or reversed motion, by having the working train consist of either an even or an odd number of gears. In this

FIG. 207.-Principle of Tum- FIG. 208.-Principle of Combler Gear. pound Idler.

arrangement A is a gear on the head-stock spindle, and B is a gear on a stud below. Pivoted on the axis of B is a triangular piece of metal, or bracket, shown in dotted lines, which can be swung back and forth by the handle E. Mounted on this bracket are the idler gears C and D, C being constantly in mesh with B, and D being in mesh with C. When it is required that B shall rotate in the same direction as A, the handle E is lowered until C meshes with A. The working train then consists