Modern Machine-Shop Practice, Volumes I and II

Chapter VII: Details in Lathe Construction (1)

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Although in each class of lathe the requirements may be practically the same, yet there is a variety of different details of construction by means of which these requirements may be met or filled, and it may be profitable to enter somewhat into these requirements and the different constructions generally employed to meet them.

The cone spindle or live spindle of a lathe should be a close working fit to its boxes or bearings, so that it will not lift under a heavy cut, or lift and fall under a cut of varying pressure. This lifting and falling may occur even though the work be true, and the cut therefore of even depth all around the work, because of hard seams or spots in the metal.

It is obvious that the bearings should form a guide, compelling the live spindle to revolve in a true circle and in a fixed plane, the axis of revolution being in line with the centre line of the tail spindle and that means should be provided to maintain this alignment while preserving the fit, or in other words taking up the wear. The spindle journals must, to produce truly cylindrical work, be cylindrically true, or otherwise the axis of its revolution will change as it revolves, and this change will be communicated through the live centre to the work, or through the chuck plate to the work, as the case may be.

The construction of the bearings should be such, that end motion to the spindle is prevented in as short a length of the spindle as possible, the thrust in either direction being resisted by the mechanism contained in one bearing.

In Fig. 553 is a form of construction for the front bearing (as that nearest to the live centre is called), in which end motion to the spindle is prevented at the same time as the diametral fit is adjusted. The spindle is provided with a cone at C and is threaded at T to receive two nuts N which draw the spindle cone within the bearing. In this case the journal at the back end may be made parallel, so that if the spindle either expands or contracts more under variations of temperature than the frame or head carrying the bearings or bearing boxes, it will not bind endwise, nor will the fit be impaired save inasmuch as there may be an inequality of expansion in the length of the front journal and its box. In this case, however, the end pressure caused by holding the work between the lathe centres acts to force the spindle into its bearing and increase the tightness of its fit, hence it is not unusual to provide at the back bearing additional means to resist the thrust of the dead centre.

Fig. 554, which is taken from "Mechanics," represents Wohlemberg's patent lathe spindle, in which both journals are coned, fitting into bushes which can be replaced by new ones when worn; the end thrust is here taken by a steel screw, while the end fit is adjusted by means of a ring nut which binds the face of the large cone gear against the inside face of the front bearing and by the face of the gear that drives the change gears. It may be pointed out, however, that in this construction the spindle must be drawn within to adjust the fit of the front bearing, which can only be done by adjusting the pinion that drives the change gears, or by screwing up the nut that is inside the cone, and therefore cannot be got at. The back bearing can be adjusted by means of the ring nuts provided at each of its ends.

Fig. 555 represents another design of cone bearing, in which the spindle is threaded to receive the nuts A which draw it within the front bearing and thus adjust the fit, and at the same time prevent end motion. The back bearing is provided with a bush parallel outside, and furnished with a nut at B to adjust the fit of the end bearing. To prevent the end pressure of the dead centre from forcing the spindle cones too tightly within their bearings a cross piece P is employed (being supported by two studs provided in the head), and through P passes an adjusting screw D, having nuts N and C, one on each side of P. Between the end of D and of the lathe spindle a washer of leather or of raw hide is placed to prevent the end faces from abrading. A similar device for taking up the end thrust is often provided to lathes in which the journals are both parallel, fitting in ordinary boxes, a top view of the device being illustrated in Fig. 556, in which B is the back bearing box, S S two studs supporting cross-piece P, and N and C are adjusting nuts. G is the gear for driving the change wheels for screw cutting or for ordinary feeding as the case may be. In this design the gear wheel G remains fixed and the combinations of gears necessary to cut various pitches of thread must be made on the lead screw and on the swing frame, which must be long enough to permit the change gear stud to pass up to permit the smallest change wheel to gear with wheel G, and which is provided with two grooves E and F, Fig. 557, for two studs to carry two compounded pairs of change wheels. This compounding in two places on the swing frame enables gear G to be comparatively large, and thus saves the teeth from rapid wear, while it facilitates the cutting of left-hand threads, because it affords more convenience for putting in a gear to change the direction of feed screw revolution.

In many lathes of American design the journals are made parallel, and the end play is taken up at the back bearing, an example being given in Fig. 558, in which the back bearing boxes are made in two halves A and B, the latter having a set screw (with check nut) threaded through it and bearing against a washer that meets the end of the spindle.

A simple method of preventing end motion is shown in Fig. 559, a bracket B affording a support for a threaded adjusting screw, which is sometimes made pointed and at others flat. When pointed it acts to support the spindle, but on the other hand it also acts to prevent the journal from bedding fairly in the boxes. In some cases of small lathes the back bearing is dispensed with, and a similar pointed adjusting screw takes its place, which answers very well for very small work.

Since the strain of the cut carried by the cutting tool falls mainly upon the live centre end of the cone spindle, it is obvious that the bearing at that end has a greater tendency to wear.

In addition to this the weight of the cone itself is greatest at that end, and furthermore the weight of the face plate or chuck, and of the work, is carried mainly at that end. If, however, one journal and bearing wears more than the other, the spindle is thrown out of line with the lathe shears, and with the tail block spindle. The usual method of obviating this as far as possible is to give that end a larger journal-bearing area.

The direction in which this wear will take place depends in a great measure upon the kind of work done in the lathe; thus in a lathe running slowly and doing heavy work carried by chucks, or on the face plate, the wear would be downwards and towards the operator, the weight of the chuck, &c., causing the downward, and the resistance or work-lifting tendency of the cut causing the lateral wear. As a general rule the wear will be least in a lateral direction towards the back of the lathe, but the direction of wear is so variable that provision for its special prevention or adjustment is not usually made. In the S. W. Putnam lathe, provision is made that the bearing boxes may be rotated in the head, so that when the lathe is used on a class of work that caused the live spindle to wear the bearing boxes on one side more than on another, the boxes may be periodically partly rotated in the head so that further wear will correct the evil.

The coned hole to receive the live centre should run quite true, so that the live centre will run true without requiring, when inserted, to be placed in exactly the same position it occupied when being turned up at its conical point. But when this hole does not run true a centre punch dot is made on the end of the spindle, and another on the centre, so that by placing the two dots to coincide at all times, the centre will run true.

The taper given to lathe centres varies from 9/16 per foot to 1 inch per foot. In the practice of Pratt and Whitney a taper of 9/16 per foot is given to all lathes, the lengths of the tapers for different sizes of lathes being as follows:

Length of Taper Socket
Swing of Lathe. for Live Centre.

13 inches 5 inches.
16 " 3-3/4 "
18 and 19 inches 7-11/16 "
" " with hollow spindle 5 inches long
and 1-1/16 diameter at the small end.

The less the amount of taper the more firmly the centre is held, but the more difficult it becomes to remove the centre when necessary.

The principal methods of removing live centres are shown in Fig. 560, in which is shown at B a square part to receive a wrench, it being found that if not less than about 1/2-inch taper per foot of length be given to the live spindle socket, then revolving the centre with a wrench will cause it to release itself, enabling it to be removed by hand. Another method employed on small lathes is to drill a hole through the live spindle to receive a taper pin P, the live centre end being shown at C.

Another and excellent plan for large lathes, is to thread the centre and provide it with a nut M, which on being screwed against the end face of the live spindle will release the centre. The objection to the use of the pin P is that it is apt to become mislaid, and it is not advisable to use a hammer about the parts of the lathe, especially in such an awkward place as between the journal bearing and the cone, which is where the pin hole requires to be located. The square section is, therefore, the best method for small lathes, and the nut for large ones.

In cases where the live spindle is made hollow a bar may be passed through from the rear end to remove the centre; this also enables rods of iron to be passed through the spindle, leaving the end projecting through the chuck for any length necessary for the work to be turned out of its exposed end.

The dead centre may be extracted from the tail spindle by a pin and hole as in Fig. 560, or, what is better, by contact with the end of the tail screw as described when referring to the tail stock of the S. W. Putnam lathe.

The cone pulley should be perfectly balanced, otherwise at high speeds the lathe will shake or tremble from the unbalanced centrifugal motion, and the tremors will be produced to some extent on the work. The steps of the cone should be amply wide, so that it may have sufficient power, without overstraining the belt, to drive the heaviest cut the lathe is supposed to take without the aid of the back gear.

In some cases, as in spinning lathes, the order of the steps is reversed, the smallest step of the cone being nearest to the live centre, the object being to have the largest step on the left, and therefore more out of the way.

The steps of the cone should be so proportioned that the belt will shift from one to the other, and have the same degree of tension, while at the same time they should give a uniform graduation or variation of speed throughout, whether the lathe runs in single gear or with the back gear in. This is not usually quite the case although the graduation is sufficiently accurate for practical purposes. The variation in the diameter of the steps of a lathe cone varies from an inch for lathes of about 12-inch swing, up to 2 inches for lathes of about 30-inch swing, and 3 inches for lathes of 5 or more feet of swing.

To enable the graduation of speed of the cone to be uniform throughout, while the tension of the belt is maintained the same on whatever step the cone may be, the graduation of the steps may be varied, and this graduation may be so proportioned as to answer all practical purposes if the overhead or countershaft cone and that on the lathe are alike.

The following on this subject is from the pen of Professor D. E. Klein, of Yale College.

"The numbers given in the following tables are the differences between the diameters of the adjacent steps on either cone pulley, and are accurate within half a hundredth of an inch, which is a degree of accuracy sufficient for practical purposes.

By simply omitting a step at each end of the cone, the two tables given will be found equally well adapted for determining the diameters of cones having four and three steps respectively.

The following are examples in the use of the tables. Suppose the centres of a pair of pulley shafts to be 60 inches apart, and that the difference of diameter between the adjacent steps is to be as near to 2-1/2 inches as can be, to obtain a uniformity of speed graduation and belt tension, also that each cone is to have six steps, the smallest of which is to be of five inches diameter.

To find the diameters for the remaining steps, we look in Table I. (corresponding to cone pulleys with six steps), under 60 in. and opposite 2-1/2 in. and obtain the differences,

2.37 2.43 2.50 2.57 2.63

Each of these differences is _subtracted_ from the _larger_ diameter of the two adjacent steps to which it corresponds, thus:

17.50 = 1st step.
Difference of 1st and 2nd = 2.37
-----
15.13 = 2nd "
" 2nd " 3rd = 2.43
-----
12.70 = 3rd "
" 3rd " 4th = 2.50
-----
10.20 = 4th "
" 4th " 5th = 2.57
-----
7.63 = 5th "
" 5th " 6th = 2.63
-----
5.00 = 6th "

EXAMPLE 2. If we suppose the same conditions as in Example 1, with the exception that each cone is to have four steps instead of six, the largest diameter will, in this case, equal 12-1/2 in. and we may obtain the remaining diameters by omitting the end differences of the above example, and then subtracting the remaining differences as follows:

12.50 = 2nd step.
Difference of 2nd and 3rd = 2.43
-----
10.07 = 3rd "
" 3rd " 4th = 2.50
-----
7.57 = 4th "
" 4th " 5th = 2.57
-----
5.00 = 5th "

The 2nd, 3rd, 4th, and 5th steps of the table correspond respectively to the 1st, 2nd, 3rd, and 4th steps of the cone, having but four steps. If the smallest diameter had not been assumed equal to 5 in. we might have dropped a step at each end of the six-step cone of the preceding example, and employed the remaining four diameters, 15.13 in. 12.70 in. 10.20 in. and 7.63 in. for one four-step cone.

The present and the previous examples show that we can assume the size of the smallest step anything that we please, and, other things being equal, can make the required cones large or small.

I.--TABLE FOR FINDING CONE PULLEY DIAMETERS WHEN THE TWO PULLEYS ARE CONNECTED BY AN OPEN BELT, AND ARE EXACTLY ALIKE.

The numbers given in table are the differences between the diameters of the adjacent steps on either cone pulley, and can be employed when there are either six or four steps on a cone. When there are six steps, the largest is the first, and the smallest the sixth step of the table. When there are four steps, the largest is the second, and the smallest the fifth step of the table.

+-------------+-----------+------------------------------
| Average | Adjacent | DISTANCE BETWEEN THE CENTRES
| difference | steps, | OF CONE PULLEYS.
| between | whose +----+----+----+----+----+----+
| the | diffe- | | | | | | |
| adjacent | rence is | 10 | 20 | 30 | 40 | 50 | 60 |
| steps. | given in | i n c h e s. |
| | table. | | | | | | |
+-------------+-----------+----+----+----+----+----+----+
| |1st and 2nd|0.87|0.94|0.96|0.97|0.98|0.98|
| |2nd " 3rd|0.94|0.97|0.98|0.98|0.99|0.99|
| 1 inch |3rd " 4th|1.00|1.00|1.00|1.00|1.00|1.00|
| |4th " 5th|1.06|1.03|1.02|1.02|1.01|1.01|
| |5th " 6th|1.13|1.06|1.04|1.03|1.02|1.02|
+-------------+-----------+----+----+----+----+----+----+
| |1st and 2nd|1.21|1.36|1.40|1.43|1.44|1.45|
| |2nd " 3rd|1.36|1.43|1.45|1.46|1.47|1.48|
| 1-1/2 inch |3rd " 4th|1.50|1.50|1.50|1.50|1.50|1.50|
| |4th " 5th|1.64|1.57|1.55|1.54|1.53|1.52|
| |5th " 6th|1.79|1.64|1.60|1.57|1.56|1.55|
+-------------+-----------+----+----+----+----+----+----+
| |1st and 2nd|1.47|1.74|1.83|1.87|1.90|1.92|
| |2nd " 3rd|1.74|1.87|1.92|1.93|1.95|1.96|
| 2 inches |3rd " 4th|2.00|2.00|2.00|2.00|2.00|2.00|
| |4th " 5th|2.26|2.13|2.08|2.07|2.05|2.04|
| |5th " 6th|2.53|2.26|2.17|2.13|2.10|2.08|
+-------------+-----------+----+----+----+----+----+----+
| |1st and 2nd|1.66|2.10|2.23|2.30|2.34|2.37|
| |2nd " 3rd|2.10|2.30|2.37|2.40|2.42|2.43|
|2-1/2 inches |3rd " 4th|2.50|2.50|2.50|2.50|2.50|2.50|
| |4th " 5th|2.90|2.70|2.63|2.60|2.58|2.57|
| |5th " 6th|3.34|2.90|2.77|2.70|2.66|2.63|
+-------------+-----------+----+----+----+----+----+----+
| |1st and 2nd|1.76|2.42|2.62|2.71|2.77|2.81|
| |2nd " 3rd|2.42|2.71|2.81|2.86|2.88|2.90|
| 3 inches |3rd " 4th|3.00|3.00|3.00|3.00|3.00|3.00|
| |4th " 5th|3.58|3.29|3.19|3.14|3.12|3.10|
| |5th " 6th|4.24|3.58|3.38|3.29|3.23|3.19|
+-------------+-----------+----+----+----+----+----+----+
| |1st and 2nd| |3.95|3.31|3.49|3.59|3.66|
| |2nd " 3rd|2.94|3.49|3.66|3.75|3.80|3.83|
| 4 inches |3rd " 4th|4.00|4.00|4.00|4.00|4.00|4.00|
| |4th " 5th|5.06|4.51|4.34|4.25|4.20|4.17|
| |5th " 6th| |5.05|4.69|4.51|4.41|4.34|
+-------------+-----------+----+----+----+----+----+----+
| |1st and 2nd| |3.33|3.92|4.20|4.36|4.47|
| |2nd " 3rd|3.31|4.19|4.47|4.60|4.68|4.74|
| 5 inches |3rd " 4th|5.00|5.00|5.00|5.00|5.00|5.00|
| |4th " 5th|6.69|5.81|5.53|5.40|5.32|5.26|
| |5th " 6th| |6.67|6.09|5.80|5.64|5.53|
+-------------+-----------+----+----+----+----+----+----+
| |1st and 2nd| |3.52|4.42|4.83|5.08|5.23|
| |2nd " 3rd| |4.83|5.23|5.42|5.54|5.62|
| 6 inches |3rd " 4th| |6.00|6.00|6.00|6.00|6.00|
| |4th " 5th| |7.17|6.77|6.58|6.46|6.38|
| |5th " 6th| |8.48|7.58|7.17|6.92|6.77|
+-------------+-----------+----+----+----+----+----+----+

+-------------+-----------+-----------------------------+
| Average | Adjacent | DISTANCE BETWEEN THE CENTRES|
| difference | steps, | OF CONE PULLEYS. |
| between | whose +----+----+----+----+----+----+
| the | diffe- | | | | | | |
| adjacent | rence is | 70 | 80 | 90 | 100| 120| 240|
| steps. | given in | i n c h e s. |
| | table. | | | | | | |
+-------------+-----------+----+----+----+----+----+----+
| |1st and 2nd|0.98|0.98|0.99|0.99|0.99|1.00|
| |2nd " 3rd|0.99|0.99|0.99|0.99|1.00|1.00|
| 1 inch |3rd " 4th|1.00|1.00|1.00|1.00|1.00|1.00|
| |4th " 5th|1.01|1.01|1.01|1.01|1.00|1.00|
| |5th " 6th|1.02|1.02|1.01|1.01|1.01|1.00|
+-------------+-----------+----+----+----+----+----+----+
| |1st and 2nd|1.46|1.46|1.47|1.47|1.48|1.49|
| |2nd " 3rd|1.48|1.48|1.49|1.49|1.49|1.49|
| 1-1/2 inch |3rd " 4th|1.50|1.50|1.50|1.50|1.50|1.50|
| |4th " 5th|1.52|1.52|1.51|1.51|1.51|1.51|
| |5th " 6th|1.54|1.54|1.53|1.53|1.52|1.51|
+-------------+-----------+----+----+----+----+----+----+
| |1st and 2nd|1.93|1.93|1.94|1.95|1.96|1.98|
| |2nd " 3rd|1.96|1.97|1.97|1.97|1.98|1.99|
| 2 inches |3rd " 4th|2.00|2.00|2.00|2.00|2.00|2.00|
| |4th " 5th|2.04|2.03|2.03|2.03|2.02|2.01|
| |5th " 6th|2.07|2.07|2.06|2.05|2.04|2.02|
+-------------+-----------+----+----+----+----+----+----+
| |1st and 2nd|2.39|2.40|2.41|2.42|2.43|2.47|
| |2nd " 3rd|2.44|2.45|2.46|2.46|2.47|2.49|
| 2-1/2 inches|3rd " 4th|2.50|2.50|2.50|2.50|2.50|2.50|
| |4th " 5th|2.56|2.55|2.54|2.54|2.53|2.51|
| |5th " 6th|2.61|2.60|2.59|2.58|2.57|2.53|
+-------------+-----------+----+----+----+----+----+----+
| |1st and 2nd|2.84|2.86|2.87|2.88|2.90|2.95|
| |2nd " 3rd|2.92|2.93|2.94|2.94|2.95|2.98|
| 3 inches |3rd " 4th|3.00|3.00|3.00|3.00|3.00|3.00|
| |4th " 5th|3.08|3.07|3.06|2.06|3.05|3.02|
| |5th " 6th|3.16|3.14|3.13|3.12|3.10|3.05|
+-------------+-----------+----+----+----+----+----+----+
| |1st and 2nd|3.71|3.75|3.78|3.80|3.83|3.91|
| |2nd " 3rd|3.85|3.87|3.88|3.89|3.91|3.96|
| 4 inches |3rd " 4th|4.00|4.00|4.00|4.00|4.00|4.00|
| |4th " 5th|4.15|4.13|4.12|4.11|4.09|4.04|
| |5th " 6th|4.29|4.25|4.22|4.20|4.17|4.09|
+-------------+-----------+----+----+----+----+----+----+
| |1st and 2nd|4.55|4.60|4.64|4.68|4.74|4.87|
| |2nd " 3rd|4.77|4.80|4.82|4.84|4.86|4.93|
| 5 inches |3rd " 4th|5.00|5.00|5.00|5.00|5.00|5.00|
| |4th " 5th|5.23|5.20|5.18|5.16|5.14|5.07|
| |5th " 6th|5.45|5.40|5.36|5.32|5.26|5.13|
+-------------+-----------+----+----+----+----+----+----+
| |1st and 2nd|5.34|5.42|5.49|5.55|5.62|5.80|
| |2nd " 3rd|5.67|5.71|5.75|5.77|5.81|5.90|
| 6 inches |3rd " 4th|6.00|6.00|6.00|6.00|6.00|6.00|
| |4th " 5th|6.33|6.29|6.25|6.23|6.19|6.10|
| |5th " 6th|6.66|6.58|6.51|6.45|6.38|6.20|
+-------------+-----------+----+----+----+----+----+----+

EXAMPLE 3. Let distance apart of the centres = 30 in. the average difference between adjacent steps = 2 in. the diameter of the smallest step = 4 in., and the number of steps on each of the cones = 5. The largest step will then equal 12 in., and from Table II., under 30 in. and opposite 2 in., we obtain the differences

1.87 1.96 2.04 2.13

and then subtracting as before we get the required diameters

12 in. 10.30 in. 8.17 in. 6.13 in. 4 in.

EXAMPLE 4. Let the conditions be as in the preceding example, the cone pulley having, however, three steps instead of five, the largest diameter will then equal 8 in.; and by dropping the end differences and subtracting

8.00 = 2nd step.
Difference of 2nd and 3rd = 1.96
-----
6.04 = 3rd "
" 3rd " 4th = 2.04
-----
4.00 = 4th "

we get the diameters 8 in., 6.04, and 4 in., which correspond respectively to 2nd, 3rd, and 4th steps of the table, and to the 1st, 2nd, and 3rd steps of the three-step cone.

EXAMPLE 5. Let the distance apart of the centres be 60 in., the average difference between the adjacent steps be 2-1/8 in., the smallest step 7 in. and the number of steps = 5. The largest step will then be 7 in. + (4 × 2-1/8) = 15-1/2 inches.

Now an inspection of Table II. will show that it contains no horizontal lines corresponding to the average difference 2-1/8 inches, we cannot, therefore, as heretofore, obtain the required differences directly, but must interpolate as follows: since 2-1/8 inches is quarter way between 2 inches and 2-1/2 inches, the numbers corresponding to 2-1/8 inches (for any given distance apart of the centres), will be quarter way between the numbers of the table corresponding to 2 inches and 2-1/2 inches. Thus, in Table II., we have under 60 inches,

and opposite 2-1/2 in.: 2.40 2.47 2.53 2.60
" 2 1.93 1.98 2.02 2.07
---- ---- ---- ----
.47 .49 .51 .53

Dividing these differences by 4, we get:

.12 .12 .13 .13

to which we add,

1.93 1.98 2.02 2.07

and get for the differences corresponding to 2-1/8 inches

2.05 2.10 2.15 2.20

and subtracting as before,

15.5 1st step.
difference of 1st and 2nd = 2.05
-----
13.45 = 2nd "
" 2nd " 3rd = 2.10
-----
11.35 = 3rd "
" 3rd " 4th = 2.15
-----
9.20 = 4th "
" 4th " 5th = 2.20
-----
7.00 = 5th "

Thus far, however, we have considered only the case where the two cone pulleys were exactly alike. Now although this case occurs much more frequently than the case in which the cone pulleys are unlike, it is nevertheless true that unlike cone pulleys occur with sufficient frequency to make it desirable that convenient means be established for obtaining the diameters of their steps rapidly and accurately, and Table III. was calculated by the writer for this purpose; its accuracy is more than sufficient for the requirements of practice, the numbers in the table being correct to within a unit of the fourth decimal place (_i.e._ within .0001). It should be noticed that the tabular quantities are not the diameters of the steps, but these diameters divided by the distance between the centres of the cone pulleys; in other words, the tabular quantities are the effective diameters of the steps only when the centres of the pulleys are a unit's distance apart. By thus expressing the tabular quantities in terms of the distance apart of the axis, the table becomes applicable to all cone pulleys whatever their distance from each other, the effective diameters of the steps being obtained by multiplying the proper tabular quantities by the distance between the centres of the pulleys.

II.--TABLE FOR FINDING CONE PULLEY DIAMETERS WHEN THE TWO PULLEYS ARE CONNECTED BY AN OPEN BELT, AND ARE EXACTLY ALIKE.

The numbers given in table are the differences between the diameters of the adjacent steps on either cone pulley, and can be employed when there are either five or three steps on a cone.

+-------------+-----------+-----------------------------+
| Average | Adjacent | DISTANCE BETWEEN THE CENTRES|
| difference | steps, | OF CONE PULLEYS. |
| between | whose +----+----+----+----+----+----+
| the | diffe- | | | | | | |
| adjacent | rence is | 10 | 20 | 30 | 40 | 50 | 60 |
| steps. | given in | i n c h e s. |
| | table. | | | | | | |
+-------------+-----------+----+----+----+----+----+----+
| |1st and 2nd|0.90|0.95|0.97|0.98|0.98|0.98|
| |2nd " 3rd|0.97|0.98|0.99|0.99|0.99|0.99|
| 1 inch |3rd " 4th|1.03|1.02|1.01|1.01|1.01|1.01|
| |4th " 5th|1.10|1.05|1.03|1.02|1.02|1.02|
+-------------+-----------+----+----+----+----+----+----+
| |1st and 2nd|1.28|1.39|1.43|1.45|1.46|1.46|
| |2nd " 3rd|1.43|1.46|1.48|1.48|1.48|1.49|
| 1-1/2 inch |3rd " 4th|1.57|1.54|1.52|1.52|1.52|1.51|
| |4th " 5th|1.72|1.61|1.57|1.55|1.54|1.54|
+-------------+-----------+----+----+----+----+----+----+
| |1st and 2nd|1.61|1.81|1.87|1.90|1.92|1.93|
| |2nd " 3rd|1.87|1.94|1.96|1.97|1.97|1.98|
| 2 inches |3rd " 4th|2.13|2.06|2.04|2.03|2.03|2.02|
| |4th " 5th|2.39|2.19|2.13|2.10|2.08|2.07|
+-------------+-----------+----+----+----+----+----+----+
| |1st and 2nd|1.89|2.20|2.30|2.35|1.38|2.40|
| |2nd " 3rd|2.30|2.40|2.43|2.45|2.46|2.47|
| 2-1/2 inches|3rd " 4th|2.70|2.60|2.57|2.55|2.54|2.53|
| |4th " 5th|3.11|2.80|2.70|2.65|2.62|2.60|
+-------------+-----------+----+----+----+----+----+----+
| |1st and 2nd|2.10|2.57|2.71|2.78|2.83|2.86|
| |2nd " 3rd|2.71|2.86|2.90|2.93|2.94|2.95|
| 3 inches |3rd " 4th|3.29|3.14|3.10|3.07|3.06|3.05|
| |4th " 5th|3.90|3.43|3.29|3.22|3.17|3.14|
+-------------+-----------+----+----+----+----+----+----+
| |1st and 2nd| |3.22|3.49|3.62|3.69|3.75|
| |2nd " 3rd|3.48|3.74|3.83|3.87|3.90|3.91|
| 4 inches |3rd " 4th|4.52|4.26|4.17|4.13|4.10|4.09|
| |4th " 5th| |4.78|4.51|4.38|4.31|4.25|
+-------------+-----------+----+----+----+----+----+----+
| |1st and 2nd| |3.77|4.20|4.40|4.52|4.60|
| |2nd " 3rd|4.19|4.60|4.73|4.80|4.84|4.87|
| 5 inches |3rd " 4th|5.81|5.40|5.27|5.20|5.16|5.13|
| |4th " 5th| |6.23|5.80|5.60|5.48|5.40|
+-------------+-----------+----+----+----+----+----+----+
| |1st and 2nd| |4.21|4.83|5.13|5.31|5.42|
| |2nd " 3rd|4.82|5.42|5.62|5.71|5.77|5.81|
| 6 inches |3rd " 4th|7.18|6.58|6.38|6.29|6.23|6.19|
| |4th " 5th| |7.79|7.17|6.87|6.69|6.58|
+-------------+-----------+----+----+----+----+----+----+

+-------------+-----------+-----------------------------+
| Average | Adjacent | DISTANCE BETWEEN THE CENTRES|
| difference | steps, | OF CONE PULLEYS. |
| between | whose +----+----+----+----+----+----+
| the | diffe- | | | | | | |
| adjacent | rence is | 70 | 80 | 90 | 100| 120| 240|
| steps. | given in | i n c h e s. |
| | table. | | | | | | |
+-------------+-----------+----+----+----+----+----+----+
| |1st and 2nd|0.99|0.99|0.99|0.99|0.99|1.00|
| |2nd " 3rd|0.99|1.00|1.00|1.00|1.00|1.00|
| 1 inch |3rd " 4th|1.01|1.00|1.00|1.00|1.00|1.00|
| |4th " 5th|1.01|1.01|1.01|1.01|1.01|1.00|
+-------------+-----------+----+----+----+----+----+----+
| |1st and 2nd|1.47|1.47|1.48|1.48|1.48|1.49|
| |2nd " 3rd|1.49|1.49|1.49|1.49|1.49|1.49|
| 1-1/2 inch |3rd " 4th|1.51|1.51|1.51|1.51|1.51|1.51|
| |4th " 5th|1.53|1.53|1.52|1.52|1.52|1.51|
+-------------+-----------+----+----+----+----+----+----+
| |1st and 2nd|1.94|1.95|1.96|1.96|1.97|1.98|
| |2nd " 3rd|1.98|1.98|1.99|1.99|1.99|1.99|
| 2 inches |3rd " 4th|2.02|2.02|2.01|2.01|2.01|2.01|
| |4th " 5th|2.06|2.05|2.04|2.04|2.03|2.02|
+-------------+-----------+----+----+----+----+----+----+
| |1st and 2nd|2.41|2.42|2.43|2.44|2.45|2.47|
| |2nd " 3rd|2.47|2.47|2.48|2.48|2.48|2.49|
| 2-1/2 inches|3rd " 4th|2.53|2.53|2.52|2.52|2.52|2.51|
| |4th " 5th|2.59|2.58|2.57|2.56|2.55|2.53|
+-------------+-----------+----+----+----+----+----+----+
| |1st and 2nd|2.87|2.89|2.90|2.91|2.93|2.96|
| |2nd " 3rd|2.96|2.96|2.97|2.97|2.98|2.99|
| 3 inches |3rd " 4th|3.04|3.04|3.03|3.03|3.02|3.01|
| |4th " 5th|3.13|3.11|3.10|3.09|3.07|3.04|
+-------------+-----------+----+----+----+----+----+----+
| |1st and 2nd|3.78|3.81|3.83|3.84|3.87|3.94|
| |2nd " 3rd|3.92|3.94|3.94|3.95|3.96|3.98|
| 4 inches |3rd " 4th|4.08|4.06|4.06|4.05|4.04|4.02|
| |4th " 5th|4.22|4.19|4.17|4.16|4.13|4.06|
+-------------+-----------+----+----+----+----+----+----+
| |1st and 2nd|4.66|4.71|4.73|4.76|4.80|4.90|
| |2nd " 3rd|4.89|4.90|4.91|4.92|4.93|4.96|
| 5 inches |3rd " 4th|5.11|5.10|5.09|5.08|5.07|5.04|
| |4th " 5th|5.34|5.29|5.27|5.24|5.20|5.10|
+-------------+-----------+----+----+----+----+----+----+
| |1st and 2nd|5.51|5.57|5.62|5.66|5.71|5.86|
| |2nd " 3rd|5.83|5.86|5.87|5.88|5.90|5.95|
| 6 inches |3rd " 4th|6.17|6.14|6.13|6.12|6.10|6.05|
| |4th " 5th|6.49|6.43|6.38|6.34|6.29|6.14|
+-------------+-----------+----+----+----+----+----+----+

Before describing and applying the table, we will call attention to the term "effective" diameter. The effective radius--as is well known--extends from the centre of the pulley to the centre of the belt; the effective diameter, being twice this effective radius, must also equal the actual diameter plus thickness of belt.

The table is so arranged that the diameter (divided by distance between centres) of one step of a belted pair will always be found in the extreme right-hand column; while its companion step will be found on the same horizontal line, and in that vertical column of the table corresponding to the length of belt employed. For example, if column 14 of the table corresponded to the length of belt employed, some of the possible pairs of diameters would be as follows:

.7118 .5813 .42 .2164 .0474
.06 .24 .42 .60 .72

The upper row of this series of pairs being taken from column 14, and the lower row from the extreme right-hand column, the numbers in each pair being on the same horizontal line. If the distance between the centers of the pulleys were 60 ins. the effective diameters of the steps corresponding to the above pairs would be:

42.71 34.88 25.2 12.98 2.84 ins.
3.6 14.4 25.2 36.0 43.20

being obtained by multiplying the first series of pairs by 60; the length of belt which would be equally tight on each of these pairs would be 3.3195 × 60 ins. = 199.17 ins.

III.--TABLE FOR FINDING THE EFFECTIVE DIAMETERS OF THE STEPS OF CONE PULLEYS, WHEN THE PULLEYS ARE CONNECTED BY AN OPEN BELT AND ARE UNLIKE.

Each vertical cone of the table corresponds to a given length of belt, and the numbers in these columns are the required effective diameters of the steps when the centres of the pulleys are a Unit's distance apart.

+--------------------------------------------------------------+------+
| LENGTH OF BELT WHEN THE CENTRES OF THE CONE PULLEYS ARE A | |
| UNIT'S DISTANCE APART. | |
+------+------+------+------+------+------+------+------+------+ |
|2.9425|3.0367|3.1310|3.2252|3.3195|3.4137|3.5080|3.6022|3.6965| |
|------+------+------+------+------+------+------+------+------+ [A] |
| =10= | =11= | =12= | =13= | =14= | =15= | =16= | =17= | =18= | |
+------+------+------+------+------+------+------+------+------+------+
| .5514| .6020| .6518| .7010| .7495| .7974| .8447| .8913| .9373| 0.00 |
| .5300| .5814| .6320| .6818| .7310| .7795| .8274| .8747| .9213| 0.03 |
| .5079| .5600| .6114| .6620| .7118| .7610| .8095| .8574| .9047| 0.06 |
| .4850| .5379| .5900| .6414| .6920| .7418| .7910| .8395| .8874| 0.09 |
| .4613| .5150| .5679| .6200| .6714| .7220| .7718| .8210| .8695| 0.12 |
| .4367| .4913| .5450| .5979| .6500| .7014| .7520| .8018| .8510| 0.15 |
| .4113| .4667| .5213| .5750| .6279| .6800| .7314| .7820| .8318| 0.18 |
| .3850| .4413| .4967| .5513| .6050| .6579| .7100| .7614| .8120| 0.21 |
| .3577| .4150| .4713| .5267| .5813| .6350| .6879| .7400| .7914| 0.24 |
| .3294| .3877| .4450| .5013| .5567| .6113| .6650| .7179| .7700| 0.27 |
| .30 | .3594| .4177| .4750| .5313| .5867| .6413| .6950| .7479| 0.30 |
| .2694| .33 | .3894| .4477| .5050| .5613| .6167| .6713| .7250| 0.33 |
| .2375| .2994| .36 | .4194| .4777| .5350| .5913| .6467| .7013| 0.36 |
| .2044| .2675| .3294| .39 | .4494| .5077| .5650| .6213| .6767| 0.39 |
| .1700| .2344| .2975| .3594| .42 | .4794| .5377| .5950| .6513| 0.42 |
| .1340| .2000| .2644| .3275| .3894| .45 | .5094| .5677| .6250| 0.45 |
| .0964| .1640| .2300| .2944| .3575| .4194| .48 | .5394| .5977| 0.48 |
| .0571| .1264| .1940| .2600| .3244| .3875| .4494| .51 | .5694| 0.51 |
| .0160| .0871| .1564| .2240| .2900| .3544| .4175| .4794| .54 | 0.54 |
| | .0460| .1171| .1864| .2540| .3200| .3844| .4475| .5094| 0.57 |
| | .0029| .0760| .1471| .2164| .2840| .3500| .4144| .4775| 0.60 |
| | | .0329| .1060| .1771| .2464| .3140| .3800| .4444| 0.63 |
| | | | .0629| .1360| .2071| .2764| .3440| .4100| 0.66 |
| | | | .0174| .0929| .1660| .2371| .3064| .3740| 0.69 |
| | | | | .0474| .1229| .1960| .2671| .3364| 0.72 |
| | | | | | .0774| .1529| .2260| .2971| 0.75 |
| | | | | | .0292| .1074| .1829| .2560| 0.78 |
| | | | | | | .0592| .1374| .2129| 0.81 |
| | | | | | | .0081| .0892| .1674| 0.84 |
| | | | | | | | .0381| .1192| 0.87 |
| | | | | | | | | .0681| 0.90 |
| | | | | | | | | .0138| 0.93 |
+------+------+------+------+------+------+------+------+------+------+

+--------------------------------------------------------------+------+
| LENGTH OF BELT WHEN THE CENTRES OF THE CONE PULLEYS ARE A | |
| UNIT'S DISTANCE APART. | |
+------+------+------+------+------+------+------+------+------+ |
|3.7907|3.8850|3.9792|4.0735|4.1677|4.2620|4.3562|4.4504|4.5447| |
|------+------+------+------+------+------+------+------+------+ [A] |
| =19= | =20= | =21= | =22= | =23= | =24= | =25= | =26= | =27= | |
+------+------+------+------+------+------+------+------+------+------+
| .9828|1.0277|1.0721|1.1159|1.1593|1.2021|1.2444|1.2861|1.3274| 0.00 |
| .9673|1.0128|1.0577|1.1021|1.1459|1.1893|1.2321|1.2744|1.3161| 0.03 |
| .9513| .9973|1.0428|1.0877|1.1321|1.1759|1.2193|1.2621|1.3044| 0.06 |
| .9347| .9813|1.0273|1.0728|1.1177|1.1621|1.2059|1.2493|1.2921| 0.09 |
| .9174| .9647|1.0113|1.0573|1.1028|1.1477|1.1921|1.2359|1.2793| 0.12 |
| .8995| .9474| .9947|1.0413|1.0873|1.1328|1.1777|1.2221|1.2659| 0.15 |
| .8810| .9295| .9774|1.0247|1.0713|1.1173|1.1628|1.2077|1.2521| 0.18 |
| .8618| .9110| .9595|1.0074|1.0547|1.1013|1.1473|1.1928|1.2377| 0.21 |
| .8420| .8918| .9410| .9895|1.0374|1.0847|1.1313|1.1773|1.2228| 0.24 |
| .8214| .8720| .9218| .9710|1.0195|1.0674|1.1147|1.1613|1.2073| 0.27 |
| .8000| .8514| .9020| .9518|1.0010|1.0495|1.0974|1.1447|1.1913| 0.30 |
| .7779| .8300| .8814| .9320| .9818|1.0310|1.0795|1.1274|1.1747| 0.33 |
| .7550| .8079| .8600| .9114| .9620|1.0118|1.0610|1.1095|1.1574| 0.36 |
| .7313| .7850| .8379| .8900| .9414| .9920|1.0418|1.0910|1.1395| 0.39 |
| .7067| .7613| .8150| .8679| .9200| .9714|1.0220|1.0718|1.1210| 0.42 |
| .6813| .7367| .7913| .8450| .8979| .9500|1.0014|1.0520|1.1018| 0.45 |
| .6550| .7113| .7667| .8213| .8750| .9279| .9800|1.0314|1.0820| 0.48 |
| .6277| .6850| .7413| .7967| .8513| .9050| .9579|1.0100|1.0614| 0.51 |
| .5994| .6577| .7150| .7713| .8267| .8813| .9350| .9879|1.0400| 0.54 |
| .57 | .6294| .6877| .7450| .8013| .8567| .9113| .9650|1.0179| 0.57 |
| .5394| .60 | .6594| .7177| .7750| .8313| .8867| .9413| .9950| 0.60 |
| .5075| .5694| .63 | .6894| .7477| .8050| .8613| .9167| .9713| 0.63 |
| .4744| .5375| .5994| .66 | .7194| .7777| .8350| .8913| .9467| 0.66 |
| .4400| .5044| .5675| .6294| .69 | .7494| .8077| .8650| .9213| 0.69 |
| .4040| .4700| .5344| .5975| .6594| .72 | .7794| .8377| .8950| 0.72 |
| .3664| .4340| .5000| .5644| .6275| .6894| .75 | .8094| .8677| 0.75 |
| .3271| .3964| .4640| .5300| .5944| .6575| .7194| .78 | .8394| 0.78 |
| .2860| .3571| .4264| .4940| .5600| .6244| .6875| .7494| .81 | 0.81 |
| .2429| .3160| .3871| .4564| .5240| .5900| .6544| .7175| .7794| 0.84 |
| .1974| .2729| .3460| .4171| .4864| .5540| .6200| .6844| .7475| 0.87 |
| .1492| .2274| .3029| .3760| .4471| .5164| .5840| .6500| .7144| 0.90 |
| .0981| .1792| .2574| .3329| .4060| .4771| .5464| .6140| .6800| 0.93 |
| .0438| .1281| .2092| .2874| .3629| .4360| .5071| .5764| .6440| 0.96 |
| | .0738| .1581| .2392| .3174| .3929| .4660| .5371| .6064| 0.99 |
| | .0157| .1038| .1881| .2692| .3474| .4229| .4960| .5671| 1.02 |
| | | .0457| .1338| .2181| .2992| .3774| .4529| .5260| 1.05 |
| | | | .0757| .1638| .2481| .3292| .4074| .4829| 1.08 |
| | | | .0131| .1057| .1938| .2781| .3592| .4374| 1.11 |
| | | | | .0431| .1357| .2238| .3081| .3892| 1.14 |
| | | | | | .0731| .1657| .2538| .3381| 1.17 |
| | | | | | .0050| .1031| .1957| .2838| 1.20 |
| | | | | | | .0350| .1331| .2257| 1.23 |
| | | | | | | | .0650| .1631| 1.26 |
| | | | | | | | | .0950| 1.29 |
| | | | | | | | | .0200| 1.32 |
+------+------+------+------+------+------+------+------+------+------+

+-----------------------------------------+------+
| LENGTH OF BELT WHEN THE CENTRES OF THE | |
|CONE PULLEYS ARE A UNIT'S DISTANCE APART.| |
+------+------+------+------+------+------+ |
|4.6389|4.7332|4.8274|4.9217|5.0159|5.1102| |
|------+------+------+------+------+------+ [A] |
| =28= | =29= | =30= | =31= | =32= | =33= | |
+------+------+------+------+------+------+------+
|1.3682|1.4085|1.4484|1.4877|1.5266|1.5650| 0.00 |
|1.3574|1.3982|1.4385|1.4784|1.5177|1.5566| 0.03 |
|1.3461|1.3874|1.4282|1.4685|1.5084|1.5477| 0.06 |
|1.3344|1.3761|1.4174|1.4582|1.4985|1.5384| 0.09 |
|1.3221|1.3644|1.4061|1.4474|1.4882|1.5285| 0.12 |
|1.3093|1.3521|1.3944|1.4361|1.4774|1.5182| 0.15 |
|1.2959|1.3393|1.3821|1.4244|1.4661|1.5074| 0.18 |
|1.2821|1.3259|1.3693|1.4121|1.4544|1.4961| 0.21 |
|1.2677|1.3121|1.3559|1.3993|1.4421|1.4844| 0.24 |
|1.2528|1.2977|1.3421|1.3859|1.4293|1.4721| 0.27 |
|1.2373|1.2828|1.3277|1.3721|1.4159|1.4593| 0.30 |
|1.2213|1.2673|1.3128|1.3577|1.4021|1.4459| 0.33 |
|1.2047|1.2513|1.2973|1.3428|1.3877|1.4321| 0.36 |
|1.1874|1.2347|1.2813|1.3273|1.3728|1.4177| 0.39 |
|1.1695|1.2174|1.2647|1.3113|1.3573|1.4028| 0.42 |
|1.1510|1.1995|1.2474|1.2947|1.3413|1.3873| 0.45 |
|1.1318|1.1810|1.2295|1.2774|1.3247|1.3713| 0.48 |
|1.1120|1.1618|1.2110|1.2595|1.3074|1.3547| 0.51 |
|1.0914|1.1420|1.1918|1.2410|1.2895|1.3374| 0.54 |
|1.0700|1.1214|1.1720|1.2218|1.2710|1.3195| 0.57 |
|1.0479|1.1000|1.1514|1.2020|1.2518|1.3010| 0.60 |
|1.0250|1.0779|1.1300|1.1814|1.2320|1.2818| 0.63 |
|1.0013|1.0550|1.1079|1.1600|1.2114|1.2620| 0.66 |
| .9767|1.0313|1.0850|1.1379|1.1900|1.2414| 0.69 |
| .9513|1.0067|1.0613|1.1150|1.1679|1.2200| 0.72 |
| .9250| .9813|1.0367|1.0913|1.1450|1.1979| 0.75 |
| .8977| .9550|1.0113|1.0667|1.1213|1.1750| 0.78 |
| .8694| .9277| .9850|1.0413|1.0967|1.1513| 0.81 |
| .84 | .8994| .9577|1.0150|1.0713|1.1267| 0.84 |
| .8094| .87 | .9294| .9877|1.0450|1.1013| 0.87 |
| .7775| .8394| .90 | .9594|1.0177|1.0750| 0.90 |
| .7444| .8075| .8694| .93 | .9894|1.0477| 0.93 |
| .7100| .7744| .8375| .8994| .96 |1.0194| 0.96 |
| .6740| .7400| .8044| .8675| .9294| .99 | 0.99 |
| .6364| .7040| .7700| .8344| .8975| .9594| 1.02 |
| .5971| .6664| .7340| .8000| .8644| .9275| 1.05 |
| .5560| .6271| .6964| .7640| .8300| .8944| 1.08 |
| .5129| .5860| .6571| .7264| .7940| .8600| 1.11 |
| .4674| .5429| .6160| .6871| .7564| .8240| 1.14 |
| .4192| .4974| .5729| .6460| .7171| .7864| 1.17 |
| .3681| .4492| .5274| .6029| .6760| .7471| 1.20 |
| .3138| .3981| .4792| .5574| .6329| .7060| 1.23 |
| .2557| .3438| .4281| .5092| .5874| .6629| 1.26 |
| .1931| .2857| .3738| .4581| .5392| .6174| 1.29 |
| .1250| .2231| .3157| .4038| .4881| .5692| 1.32 |
| .0500| .1550| .2531| .3457| .4338| .5181| 1.35 |
| | .0800| .1850| .2831| .3757| .4638| 1.38 |
| | | .1100| .2150| .3131| .4057| 1.41 |
| | | .0255| .1400| .2450| .3431| 1.44 |
| | | | .0555| .1700| .2750| 1.47 |
| | | | | .0855| .2000| 1.50 |
+------+------+------+------+------+------+------+

Legend: [A] = Assumed diameter of steps, divided by distance between the centres of Cone Pulleys.

To get the actual diameters of these steps when thickness of belt = 7/32 = 0.22 in., we have simply to subtract 0.22 in. from the effective diameters just given, thus:

42.49 34.66 24.98 12.76 2.62 in.
3.38 14.18 24.98 35.78 42.98

would be the series of pairs of actual diameters.

In solving problems relating to the diameters of cone pulleys by means of the accompanying table, we must have, besides the distance between centres, sufficient data to determine the column representing the length of belt. The length of belt is seldom known because it is of small practical importance to know its exact length; but it may be estimated approximately, and then the determination of suitable diameters of the steps becomes an extremely simple matter, as may be seen from what has already preceded. When the length of the belt is not known, and has not been assumed, we indirectly prescribe the length of belt by assuming the effective diameters of the two steps of a belted pair; thus, in the following Figure (561), the length of belt is prescribed when the distance A B, and any one of the pairs of steps D_{1}_d__{1}, D_{2}_d__{2}, D_{3}_d__{3} and D_{4}_d__{4} are given. We will show in the following examples how the length of belt and its corresponding column of diameter may be found when a pair of steps (like D_{1}_d__{1}), are given.

EXAMPLE 1. Given the effective diameters

4.5 in. 9 in. 15 in. 21 in. on cone A,
-- -- 15 in. -- " B,

and the distance between centres equal to 50 inches.

Required the remaining diameters on cone B.

Since in this example the steps of the given pair are equal, we look for 15/50 = 0.30, in the extreme right-hand column of table; we will find it in the 11th line from the top; now looking along this line for the diameter of the other step, = 15/50 = 0.30, we will find it in column 10; consequently the numbers of this column may be taken as the diameters of the steps which are the companions or partners of those in the extreme right-hand column.

We can now easily determine the remaining members of the pairs to which 4.5 in., 9 in., and 21 in. steps respectively belong. To find the partner of the 4.5 step, we find 4.5/50 = 0.09 in the right-hand column, and look along the horizontal line on which 0.09 is placed till we come to column 10, in which we will find the number 0.4850; 0.4850 × 50 in. = 24.25 in. will be the effective diameter of the companion to the 4.5 in. step.

To find the partner to the 9 in. step, we proceed as before, looking for 9/50 = 0.18 in the right-hand column, and then along the horizontal line of 0.18 to column 10, then will 0.4113 × 50 in. = 20.57 in. be the required companion to the 9 in. step of cone A.

In like manner for the partner of the 21 in. step we get 0.1700 × 50 in. = 8.5 in. The effective diameter therefore will be,

4.5 in. 9 in. 15 in. 21 in. on cone A,
24.25 20.57 15 in. 8.5 " B.

If the thickness of belt employed were 0.25 in. the _actual_ diameters of steps would be,

4.25 8.75 14.75 20.75 on cone A,
24.00 20.32 14.75 8.25 " B,

and the length of belt would be 2.9425 × 50 = 147.125 in.

EXAMPLE 2. Given the effective diameters

6 in. 12 in. 18 in. 24 in. on cone A,
30 in. -- -- -- " B,

and the distance between centres = 40 in.

Required the unknown diameters on cone B.

We must, as before, first find the vertical column corresponding to the length of belt which joins the pair of steps 6 in/30 in. We find the number 6/40 = .15 in the right-hand column, and then look along its horizontal line for its partner 30/40 = 0.75. Since we do not find any number exactly equal to .7500, we must interpolate. For the benefit of those not familiar with the method of interpolation we will give in detail the method of finding intermediate columns of the table. On the aforesaid horizontal line we find in column 16 a number 0.7520, larger than the required 0.7500, and in column 15 a number 0.7014, smaller than 0.7500; evidently the intermediate column, containing the required 0.7500, must lie between columns 16 and 15. To find how far the required column is from column 16, we subtract as follows:

0.7520 0.7520
0.7500 0.7014
------ ------
.0020 0.0506

then the fraction .0020/.00506 = 0.04 nearly will represent the position of the required intermediate column; namely, that its distance from column 16 is about 4/100 of the distance between the adjacent columns, 15 and 16.

To find other numbers in this intermediate column we have only to multiply the difference between the adjacent numbers of columns 16 and 15 by 0.04, and subtract the product from the number in column 16. But it is not necessary to find as many numbers of the intermediate columns as are contained in either of the adjacent columns; it is only necessary to find as many numbers as there are steps in each of the cone pulleys. We will now illustrate what has preceded, by finding the partner to the 12 in. step of cone A. Find, as before, the horizontal line corresponding to 12/40 = 0.30, then take the difference between the numbers 0.6413 and 0.5867 of columns 16 and 15, and multiply this difference, 0.0546, by 0.04; this product = 0.0022 subtracted from 0.6413, will give 0.6391, a number of the intermediate columns corresponding to the length of belt of the present problem. Multiplying by the distance between the axes = 40 in. we get 0.6391 × 40 = 25.56, for the diameter of the step of cone B which is partner to the 12 in. step of cone A.

To find the companion to the 18 in. step, we proceed in the same manner, looking for the horizontal line 18/40 = 0.45, and interpolating as follows:

0.5094 - (0.5094 - 0.4500) × 0.04 = 0.5070.

Consequently, 0.5070 × 40 in. = 20.28 in. will be the required partner of the 18 in. step.

In like manner, for the 24 in. step, we have

0.3500 - (0.3500 - 0.2840) × 0.04 = 0.3474, and 0.3474 × 40 = 13.90.

The effective diameters are therefore

6 in. 12 in. 18 in. 24 in. on cone A.
30 25.56 20.28 13.9 " B.

The actual diameters, when thickness of belt = 0.20 in., are:

5.8 11.8 17.8 23.8 on cone A.
29.8 25.36 20.08 13.7 " B.

And the length of belt will be:

[3.5080 - (3.5080 - 3.4137) × 0.04] × 40 in. = 140.17 in.

EXAMPLE 3. Given the effective diameters:

12 in. 18 in. 24 in. 30 in. on cone A,
33 in. -- -- -- " B,

and the distance between the centres = 60 in.

Required the remaining diameters on cone B.

The horizontal corresponding to 12/60 = 0.20 lies 2/3rd way between the horizontal line, corresponding to 0.18 and 0.21; the number 33/60 = 0.5500, corresponding to the companion of the 12 in. step, will therefore lie 2/3rd way between the horizontal lines 0.18 and 0.21. We have now to find two numbers on this 2/3rd line, of which one will be less and the other greater than 0.5500. An inspection of the table will show that these greater and less numbers must lie in columns 13 and 12. The numbers on the 2/3rd line itself may now be found as follows:

In column 13, 0.5750 - 2/3(0.5750 - 0.5513) = 0.5592.

In column 12, 0.5213 - 2/3(0.5213 - 0.4967) = 0.5049.

0.5592 will be the number on the 2/3rd line, which is greater than 0.5500, and 0.5049 will be the one which is less than 0.5500. The position of the intermediate column, corresponding to the length of belt of the present example, may now be found, as before, briefly. It is:

0.5592 - 0.5500 = 0.0092
= 0.17.
0.5592 - 0.5049 = 0.0543

Consequently the required column lies nearest column 13, 17/100th way between columns 13 and 12. To find any other number in the required column, we have only to multiply the difference between two adjacent numbers of columns 13 and 12 by 17/100, and subtract the product from the number in column 13. For example, to find the diameter of the partner to the 18 in. step of cone A, we find the numbers 0.4750 and 0.4177 of columns 13 and 12, which lie on the horizontal line corresponding to 18/60 = 0.30; the difference, 0.0573, between the two numbers is multiplied by 0.17, and the product, 0.0573 × 0.17 = 0.0097, subtracted from 0.4750. This last difference will equal 0.4653, and will be the number sought. If we now multiply by 60, we will get 27.92 in. as the effective diameter of that step on cone B which is the partner to the 18 in. step of cone A.

To find the companion of the 24 in. step, we proceed after the same fashion; the horizontal line 24/60 = 0.40 lies 1/3rd way between 0.39 and 0.42; hence,

In column 13, 0.3900 - 1/3(0.3900 - 0.3594) = 0.3798;

In column 12, 0.3294 - 1/3(0.3294 - 0.2975) = 0.3188;

And 0.3798 - (0.3798 - 0.3188) × 0.17 = 0.3694.

The required effective diameter of the step, which is partner to the 24 in. step, will therefore be 0.3694 × 60 = 22.16 in.

In like manner we obtain partner for the 30 in. step, thus:

In column 13, 0.2944 - 2/3(0.2944 - 0.2600) = 0.2715.

In column 12, 0.2300 - 2/3(0.2300 - 0.1940) = 0.2060.

Also 0.2715 - (0.2715 - 0.2060) × 0.17 = 0.2604, and 0.2604 × 60 in. = 15.62 in. = diam. of step belonging to the same belted pair as the 30 in. step of cone A.

The effective diameters will be:

12 in. 18 in. 24 in. 30 in. on cone A,
33 27.92 22.16 15.62 " B,

and the actual diameters when belt is 0.22" thick:

11.78 17.78 23.78 29.78 in.
32.78 27.70 21.94 15.40

and the length of belt is found to be:

[3.2252 - (3.2252 - 3.1310) × 0.17] × 60 in. = 192.55 in.

In all the preceding problems it should be noticed that we arbitrarily assumed _all_ the steps on one cone, and _one_ of the steps on the other cone. It will be found that all of the practical problems relating to cone-pulley diameters can finally be reduced to this form, and can consequently be solved according to the methods just given.

For those who find difficulty in interpolating, the following procedure will be found convenient: Estimate approximately the necessary length of belt, then divide this length by the distance between the centres of the cone pulleys; now find which one of the 33 lengths of belt (per unit's distance apart of the centres) given in the table is most nearly equal to the quotient just obtained, and then take the vertical column, at the head of which it stands, for the companion to the right-hand column. Those numbers of these companion columns which are on the same horizontal line will be the companion steps of a belted pair. The table is so large, that in the great majority of cases not only exact, but otherwise satisfactory values can be obtained by this method, without any interpolation whatever."

The teeth of the back gear should be accurately cut so that there is no lost motion between the teeth of one wheel, and the spaces of the other, because on account of the work being of large diameter or of hard metal (so as to require the slow speed), the strain of the cut is nearly always heavy when the back gear is in use, and the strain on the teeth is correspondingly great, causing a certain amount of spring or deflection in the live spindle and back gear spindle. Suppose then, that at certain parts of the work there is no cut, then when the tool again meets the cut the work will meet the tool and stand still until the lost motion in the gear teeth and the spring of the spindles is taken up, when the cut will proceed with a jump that will leave a mark on the work and very often break the tool. When the cut again leaves the tool a second jump also leaving a mark on the work will be made. If the teeth of the gears are cut at an angle to the axial line of the spindle, as is sometimes the case, this jumping from the play between the teeth will be magnified on account of a given amount of play, affording more back lash in such gears.

The teeth of the wheels should always be of involute and not of epicycloidal form, for the following reasons. The transmission of motion by epicycloidal teeth is exactly uniform only when their pitch circles exactly coincide, and this may not be the case in time because of wear in the parts as in the live spindle journals and the bearings, and the back gear spindle and its bearings, and _every variation of speed_ in the cut, however slight it may be, produces a corresponding mark upon the work. In involute teeth the motion transmitted will be smooth and equal whether the pitch lines of the wheels coincide or not, hence the wear of the journals and bearings does not impair their action.

The object of cutting the teeth at an angle is to have the point of contact move or roll as it were from one end to the other of the teeth, and thus preserve a more conterminous contact on the line of centres of the two wheels, the supposition being that this would remove the marks on the work produced by the tremor of the back gear. But such tremor is due to errors in the form of the teeth, and also in the case of epicycloidal teeth from the pitch lines of the teeth not exactly coinciding when in gear.

The pitch of the teeth should be as fine as the requisite strength, with the usual allowance of margin for wear and safety will allow, so as to have as many teeth in continuous contact as possible.

Various methods of moving the back gear into and out of gear with the cone spindle gears are employed. The object is to place the back gears into gear to the exact proper depth to hold them securely in position, and to enable the operator to operate the gears without passing to the back of the lathe. Sometimes a sliding bearing box, such as shown in Fig. 562, is employed; _a_ is the back gear spindle, _b_ its bearing box, and _d_ a pin which when on the side shown holds _b_ in position, when the back gear is in action. To throw it out of action _d_ is removed, _b_ pushed back, and _d_ inserted in a hole on the right hand of _b_; the objection is that there is no means of taking up the wear of _b_, and it is necessary to pass to the back of the lathe to operate the device.

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Modern Machine-Shop Practice, Volumes I and IIChapter VII: Details in Lathe Construction (1)

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