Modern Machine-Shop Practice Part 41

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 a.s.sumed 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 a.s.sume 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 quant.i.ties 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 quant.i.ties 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 quant.i.ties 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 quant.i.ties 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.