Modern Machine-Shop Practice Part 182

Suppose, now, we take the four divisions on A and the three on B to consider their proportions, and we may say 4 is 1-1/3 times 3, or we may with equal propriety say 3 is 3/4 of 4, hence 4 is not in the same proportion to 3 that 3 is to 4. Let it now be supposed that a driven pulley B is 18 inches in diameter, and requires to be driven one quarter faster than the driver, what then must be the diameter of that driver?

As the revolutions require to be increased one-fourth the pulley diameter must be increased one-fourth. Thus one quarter of 18 = 4-1/2, and this added to 18 is 22-1/2, which is therefore the diameter of the driving pulley, as may be proved as follows: Suppose the circ.u.mferences instead of the pulley diameters to be 22-1/2 and 18 respectively, and that the largest pulley makes 100 revolutions, then it will pa.s.s 2,250 (22-1/2 100 = 2,250) inches of belt over its circ.u.mference, and every 18 inches of this belt will cause the small pulley to make one revolution; hence we divide 2,250 by 18, which gives us 125 as the revolutions made by the small pulley, while the large one makes 100.

Thus it appears that we obtain the same result whether we take the circ.u.mferences or the diameters of the pulleys, because it is their relative proportions or relative revolutions that we are considering, and their actual diameters do not affect their proportions one to the other. Thus, if a 10-inch pulley drives a 30-inch one, the proportions being three to one, the revolutions will be three to one, and the driven being three times the largest, will make one revolution to every three of the driver. If the driver was 3 inches in diameter and the driven 9, the revolutions would be precisely the same as before, but with equal revolutions the velocities would be different, because in each revolution of the driver it will move a length of belt equal to its circ.u.mference; hence, the greater the circ.u.mference the greater length of belt it will move per revolution. To take the velocity into account, we must take into consideration the number of revolutions made in a given time by the driver. Suppose, for example, that the driver being 3 inches in diameter makes one revolution in a minute, then it will move in that minute a length of belt equal to its circ.u.mference, so that the circ.u.mference of the driver, multiplied by the number of its revolutions per minute, gives its velocity per minute. Thus, if a pulley has a circ.u.mference of 50 inches, and makes 120 revolutions per minute, then its velocity will be 6,000 inches per minute, because 50 120 = 6,000.

The velocity of the belt, and therefore that of the driven wheel, will also be 6,000 inches per minute, as has already been shown. From this train of reasoning the following rules will be obvious:--

To find the diameter of the driving pulley when the diameter of the driven pulley and the revolutions per minute of each are given:

_Rule._--Multiply the diameter of the driven by the number of its revolutions, and divide the product by the number of revolutions of the driver, and the quotient will be the diameter of the driver.

The diameter and revolutions of the driver in a given time being known, to find the diameter of a driven wheel that shall make a given number of revolutions in the same time:

_Rule._--Multiply the diameter of the driver by its number of revolutions, and divide the product by the number of revolutions of the driven. The quotient will be the diameter of the driven.

To find the number of revolutions of a driven pulley in a given time, its diameter and the diameter and revolutions of the driver being given:

_Rule._--Multiply the diameter of the driver by the number of its revolutions in the given time, and divide by the diameter of the driven, and the quotient will be the number of revolutions of the driven in the given time.

Suppose, however, that the speed of the shaft only is given, and we require to find the diameter of both pulleys, as, for example, suppose a shaft makes 150 revolutions per minute, and we require to drive the pulley on a machine 600 revolutions per minute. Here we have two considerations: first, the relative diameters of the two pulleys, and secondly, the diameter of pulley and width of belt necessary to transmit the amount of power necessary to drive the machine at the speed required. Leaving the second to be discussed hereafter in connection with the driving power of belts, we may proceed to determine the first as follows: The pulley on the machine must be as much smaller than that on the main shaft, as the speed of the pulley on the machine requires to run faster than does the main shaft, hence we divide the 600 by 150 and get four, which is the number of times smaller than the driver that the driven pulley must be. Suppose then the driver is made a 24-inch pulley, then the driven must be a 6-inch one, because 24 4 = 6; or we may make the driver 36, and the driven 9, because 36 4 = 9; or the driver being 48 inches in diameter, the driven must be 12, because 48 4 = 12. To reverse the case, suppose the shaft to make 200 revolutions per minute, and the machine pulley to make 50, then since 200 50 = 4, the driven (or machine pulley) must have a diameter four times that of the driver, and any two pulleys of which one is four times the diameter of the other may be used, as say: Pulley on line shaft 10 inches in diameter, pulley on machine 40 inches in diameter; or, pulley on line shaft 20 inches in diameter, pulley on machine 80 inches in diameter.

Now, in nearly all cases that are met with in practice, it would be inconvenient to have so large a pulley as 80 inches in diameter to drive a machine, and again in most cases a driving pulley of 10 inches in diameter would be too small. So likewise in cases where the machine pulley requires to run faster than the line shaft, a single pair of pulleys will be found to give, where great changes of revolution are required, too great a disproportion in the diameter of the pulleys; thus in the case of a shaft making 150, and the machine requiring to make 600, we may use the following pairs of pulleys:--

On Main Shaft. On Machine Shaft.

First 32 inch diameter 8 inch diameter.

Second 40 " " 10 " "

Third 48 " " 12 " "

Fourth 60 " " 15 " "

But the machine may require so much power to drive it, that with the width of belt it is desired to employ, a pulley larger than either of these is necessary, as, say, one 20 inches in diameter. Now, with a 20-inch driven pulley, the driver would require to be 80 inches in diameter, because 20 4 = 80. But there may not be room between the shaft and the ceiling for a pulley of so large a diameter, or such a large pulley may be too heavy to place on the shaft, or it may be too costly, and to avoid these evils, countershafts are used.

By the employment of a countershaft we simply obtain--with two pairs of pulleys and by means of small pulleys--that which could be obtained in a single pair, providing the great difference in their diameters (necessary to obtain great changes of rotation), were not objectionable; all that is necessary, therefore, is to accomplish part of the required change of rotation in one pair, and the remainder in the other. In doing this, however, while the velocity of each driver and driven will be equal (as was explained with reference to a single pair), notwithstanding the difference in their diameters, yet the velocity of one pair will necessarily differ from that of the other, so that the pulley on the machine will vary in its velocity as well as in its rotation from that of the first driver. The first driver is that on the main or driving shaft, and the pulley it drives is the first driven. The second driver is the second pulley on the countershaft, and the second driven is the one it drives or that on the machine. Suppose, then, a driving shaft makes 100 revolutions per minute, and the machine requires to make 600, then the speed of rotation requires to be increased six times. Now we may effect this change of six times in several ways; thus: Suppose we increase the rotations three times in the first pair, then the second pulley will make 300 rotations, or three times those of the main shaft, and all we have to do is to make the second driven one-half the diameter of the second driver, and its rotations will be double those of the second driver, which will give the required speed of 600 revolutions. Suppose, however, we change the speed four times in the first pair, and the 100 of the shaft becomes 400 on the countershaft, and to increase this to 600 on the second driven, all that is required is to make its diameter one-half less than that of the second driver, because 600 is one-half more than 400. From this it will be perceived that the number of changes or amount of increase or decrease of speed being given, the proportion of diameters for both pairs of pulleys will be represented by any two numbers which, multiplied together, will give a sum equal to the number of increased revolutions required. Having found the proportions for each pair, it remains to determine their actual diameters, and they will be found to vary under different conditions.

Suppose, for example, we have the following conditions: Main shaft runs 100; machine must run 600. The pulley on the line shaft is 36 inches in diameter; required, the diameters for the other three pulleys.

To make three changes in the first pair, the first driven must be 1/3 the diameter of the first driver, which is 12 inches. Now the second pair we may make any diameters that are two to one; and since the second driver is to be the smallest, we may select as small a pulley as will answer for the machine, and make its driver twice its diameter.

But suppose it is the diameter of the pulley on the machine that is fixed, and the diameter of the other three require to be found. Let the diameter of the second driven be 12; then its driver on the countershaft must be 24. The other two must have diameters 3 to 1 as before, any suitable wheels being selected.

Yet another condition may occur. Thus, suppose the countershaft is on hand, and that it has on it two pulleys, as a 12 and a 24-inch; then a 36 on the inner shaft will be three times as large as the 12, and a 12-inch on the machine will be twice as small; or, what is the same, one half as large as the 24.

When the principle is clearly understood the calculations can be performed mentally with ease so far as the required diameters to attain the necessary speed is concerned, but there are other considerations that claim attention.

Thus, for example, to multiply the rotations 6 times we may proportion the first pair as follows: Driver 48, driven 16; second pair, driver 30, driven 15 inches in diameter.

Or we may proportion them as follows: First pair: driver 36, driven 12; second pair: driver 28, driven 14 inches in diameter.

In the second arrangement of diameters the drivers are each 2 inches, and the driven each 1 inch less in diameter than in the first; hence their cost would be diminished, as would also be the wear of the journals, on account of the reduced weight of the pulleys; hence, if the driving capacity of each pulley is equal to the requirements the second arrangement would be preferable.

In considering this part of the subject, first let it be shown that although the horse-power transmitted by the two belts is equal whatever be the proportions of the pulleys (provided, of course, that the belts do not slip), yet the strain or wear and tear of the belts varies, and the requirements for one belt are therefore different from those for the other.

[Ill.u.s.tration: Fig. 2655.]

In Fig. 2655 let A represent a 36-inch pulley on the driving shaft, B a 12-inch, and C a 24-inch pulley on a countershaft, and D a 12-inch pulley on a machine shaft. Let the main shaft make 100 revolutions per minute, and the machine requires a force to move it equal to 50 pounds applied to the perimeter of D. Now the rotations of D will, with these pulleys, be six to one of the main shaft or A, which gives D 600 revolutions per minute, thus: 100 6 = 600. The circ.u.mference of D is about 37.69 inches, which, multiplied by 600 (the number of its revolutions), gives 22,614 inches, or 1,884.5 feet as its speed per minute. This multiplied by the 50 pounds it takes to move the machine at the perimeter of D, gives 94,225 as the foot pounds per minute required to drive the machine 600 revolutions per minute, and this, therefore, is the amount of power transmitted by each belt. On the second belt this is shown to be composed of 50 pounds moving 1,884-1/2 feet per minute, hence we may now find how it is composed on the first belt, as follows:--

The diameter of the first driver is 36 inches, and its circ.u.mference 113.09 inches, or 9.42 feet; this, multiplied by its revolutions per minute, will give its speed, thus: 9.42 100 = 942 feet per minute. To obtain the necessary amount of pull for this first belt, we must divide this speed into the number of foot pounds it takes to drive the machine, thus: 94,225 942 = 100.02. The duties of the two belts are therefore as follows:--

First belt, weight of pull 100.02 " speed per minute 942 feet.

Second belt, weight of pull 50.00 " speed per minute 1884.5 feet.

The duty in foot pounds being equal, as may be shown by multiplying the feet per minute by the force or weight of the pull, leaving out the fractions, thus:--

942 100 = 94,200.

1884 50 = 94,200.

The difference in the requirements is, then, that the first belt must have as much more weight or force of pull than the second as its speed is less than that of the second.

It is obvious that in determining the proportions of the pulleys this difference in the requirements should be considered, and the manner in which this should be done depends entirely upon the conditions.

Thus, in the case we have considered, the speed was increased, but the object of the countershaft may be to decrease the speed, and in that case the conditions would be reversed, inasmuch as though the foot pounds transmitted by both belts would still be equal, yet the speed would be greatest and the strain or pull the most on the second belt instead of on the first.

It is obvious, then, that the proportions of the pulleys being determined the actual diameters must be large enough to transmit the required amount of power without unduly straining the belt.

CHAPTER x.x.xII.--Leather Belting.

The names of the various parts of a hide of leather as known to commerce are as follows:--

[Ill.u.s.tration: Fig. 2656.]

In Fig. 2656 the oblong portion between the two belly parts marked G G is known as the "b.u.t.t," and when split down the ridge, as shown by the dotted line down the centre, the two pieces are known as "bends;" the two pieces marked Y are "belly offal;" D is known as "cheeks and faces."

The b.u.t.t within the dotted line may extend in length from A to B, or from A to C; if cut off between B and C that portion is called the "range" or the whole from B to X may be cut in one piece and termed a "shoulder."

Sometimes the range is cut off and the rest would be called a shoulder with "cheeks and faces" on; or, again, the range and shoulder may be in one nearly square piece. The manner of cutting this part depends upon the spread and size of the hide.

[Ill.u.s.tration: Fig. 2657.]

The part of the hide that is used to manufacture the best belting is shown in Fig. 2657, on which the characteristics of the various parts are marked. The piece enclosed by the dotted lines is that employed in the manufacture of the commonest belting, while that enclosed by the full lines B, C, D is that used for the best belting. The former includes the shoulder, which is more soft and spongy, while it contains numerous creases, as shown. These creases are plainly discernible in the belt when made up, and may be looked for near the belt points.

[Ill.u.s.tration: Fig. 2658.]

The centre of the length of the hide will stretch the least, and the outer edges on each side of the length of the hide the most. Hence it follows that the only strip of leather in the whole hide that will have an equal amount of stretch on each edge is that cut parallel to line A, and having that line as a centre of its width. All the remaining strips will have more stretch on one edge than on the other, and it follows that, to obtain the best results the leather should be stretched after it is cut into strips, and not as a whole in the hide, or in that part of it employed for the belt strips. It is found, indeed, that, even though stretched in strips, the leather is apt in time to curve. Thus a belt that is straight when rolled in the coil will, on being unrolled, be found to be curved. It is to be observed, also, that each time the width of the strips is reduced, this curving will subsequently take place; thus, if a belt 8 inches wide and quite straight, be cut into two belts of 4 inches wide, the latter will curve after a short time. The reason of this is almost obvious, because it is plain that the edge that was nearest the centre line of the hide offers the greatest resistance to stretching; hence, when the strip is stretched straight, and an equilibrium of tension is induced, reducing the width destroys to some extent this equilibrium, and the leather resumes, to some extent, its natural conformation. This, however, is not found to be of great practical importance, so long as the outer curve of one piece is on the same side as the outer curve of its neighbor, as shown on the left view in Fig. 2658, in which case the belt will run straight, notwithstanding its curve; but if the curves are reversed, as on the right in Fig. 2658, the belt will run crooked, wabbling from side to side on the pulley. To avoid this, small belts may be made continuous by cutting them from the hide, as shown in Fig. 2659; but in this case it is better that the belt be cut from the centre strip of the hide.

[Ill.u.s.tration: Fig. 2659.]

If the leather is stretched in strips after being cut from the hide, the amount of the stretch is about 6 inches in a length of 4-1/2 feet of a belt, say, 4 inches wide, but the stretch will be greater in proportion as the width of the strip is reduced. But if stretched as a whole, the amount of stretch will be about 1 inch per foot of length, the shoulder end stretching one-third more.

If the leather has been properly stretched in strips the length of the belt may be cut to the length of an ordinary tape line drawn tightly over the pulleys, which allows the same stretch for the belt as there is on the tape line, added to the degree of tension due to cutting the belt too short to an amount equalling its thickness (as would be the case if the belt is cut of the same length as the tape line); or if the belt is a double one, the belt thus cut to length would be too short to an amount equal to twice the thickness of the strips of leather of which it is composed.