Fig. 538

Fig. 538.

Fig. 538 represents a 50-inch swing lathe by the New Haven Manufacturing Company of New Haven, Connecticut. The compound rest is here provided with automatic feed so that it may be set at an angle to bore tapers with a uniform feed. The tailstock is provided with a bracket, carrying a pinion in gear with the hand-feed rack, so as to move the tailstock along the bed by means of the pinion. The feed screw is splined to give an independent feed, and the swing frame is operated by a worm as shown.

Gap Lathe or Break Lathe.

Fig. 539

Fig. 539.

The gap lathe is one in which the bed is provided with a gap beneath the face plate, so as to enable that plate or the chucks to swing work of larger diameter, an example being given in Fig. 539.

It is obvious, however, that the existence of the gap deprives the slide rest of support on one side, when it is used close to the face plate. This is obviated in some forms of gap lathes by fitting into the gap a short piece of bed that may be taken out when the use of the gap is required.

Fig. 540

Fig. 540.

The gap lathe has not found favor in the United States, the same result being more frequently obtained by means of the extension lathe, which possesses the advantages of the gap lathe, while at the same time enabling the width of the gap to be varied to suit the length of the work. Fig. 540 represents an extension lathe by Edwin Harrington and Son, of Philadelphia. There are two beds a and b, the former sliding upon the latter when operated by the hand-wheel e, which is upon the end of a screw that passes between the two beds, has journal bearing in the upper bed, and engages a nut in the lower one, so that as the screw is operated the wheel moves longitudinally with the upper bed. c is the feed rod which communicates motion to the feeding screw d, which has journal bearing on the upper bed and therefore travels with it when it is moved or adjusted longitudinally. The cross slide has sufficient length to enable the slide rest to face work of the full diameter that will swing in the gap, and to support the slide rest when moved outwards to the full limit, it is provided with a piece f, which slides at its base upon the guideway or slide g.

Fig. 541

Fig. 541.

Fig. 541 represents a double face plate lathe such as is used for turning the wheels for locomotives. The circumference of both the face plates are provided with spur teeth, so that both are driven by pinions, which by being capable of moving endways into or out of gear, enable either face plate to be used singly, if required, as for boring purposes.

The slide rests are operated by ratchet arms for the self feed, these arms being operated by an overhead shaft, with arms and chains.

Fig. 542

Fig. 542.

Fig. 542 represents a chucking lathe adapted more especially for boring purposes. Thus the cone pulley is of small diameter and the parts are light, so that the lathe is more handy than would be the case with a heavier built lathe, while at the same time it is sufficiently rigid for large work that is comparatively light.

The compound rest is upon a pedestal that can be bolted in any required position on the lower cross slide, and is made self-acting for the feed traverse by the change wheels and feed screw, while the self-acting cross feed is operated by a ratchet handle, actuated by a chain from an overhead reciprocating lever; the latter being actuated from the crank pin at a, which is adjustable in a slot in the crank disk b. A lathe of this kind is very suitable for brass work of unusually large diameter, because in such work the cuts and feeds are light, and the cutting speed is quick, hence a heavy construction is not essential.

Fig. 543

Fig. 543.

Fig. 544

Fig. 544.

Figs. 543 and 544 represent a large lathe built by Thomas Shanks and Co., of Johnstone, near Glasgow, Scotland; all the figures of this lathe being from The American Machinist.

Fig. 543 shows the headstock and two of the slide rests, while Fig. 544 represents the remainder of the bed, the tailstock, and two of the slide rests.

It will be seen from the figures that there are a compound rest and a column or pillar rest both at the front and at the back of the lathe, and that there is an additional rest on the front end of the tailstock which may be used for facing the ends of the work.

Fig. 545

Fig. 545.

Fig. 545 represents a section through, and a partial plan of the headstock, and it will be seen that the live spindle is free from the cone pulley and from the gearing, the chuck plate being driven from a pinion engaging an internal gear at the back of the chuck plate. By this construction the balancing of such work as crank shafts is facilitated, because the chuck plate is not affected by the friction of the driving gears, and may therefore be easily revolved to test the balance of the work.

Fig. 546

Fig. 546.

Fig. 546 represents a cross section through the bed, and through one of the compound rests, and one of the pillar rests, the latter rests being made thin so that they may pass between the cheeks of crank shafts, to turn their faces and the crank journals.

Fig. 547

Fig. 547.

Fig. 548

Fig. 548.

Fig. 547 represents a view from the back end of the headstock, and Fig. 548 a view of the lathe from the tailstock end.

Fig. 549

Fig. 549.

Fig. 550

Fig. 550.

Figs. 549 and 550 represent a plan and a side view of the headstock and the two slide rests nearest to it. The lathe being shown at work on the crank shaft of the steamship service, which is shown in dotted lines, and it will be seen that for turning the stem of the shaft all the rests can be used at once, those at the back of the lathe having their cutting tools turned upside down (as will be more clearly seen in the cross-sectional view of the rests in Fig. 546).

Fig. 551

Fig. 551.

Fig. 552

Fig. 552.

Figs. 551 and 552 represent a plan and a side view of the other half of the lathe in operation upon the same crank shaft, which is again shown in dotted lines.

Referring now to the general construction of the lathe, the headstock or live spindle has a front journal bearing 18 inches diameter and 24 inches long, and a back bearing 12 inches diameter and 15 inches long, the bearings being parallel. The driving cone has five changes of speed for a 6-inch belt, and is carried on an independent spindle. The cone is turned inside as well as outside, so as to be in balance at high speeds.

The face plate is 12 feet diameter, cast with internal gear at the back. It is provided with T-slots and square holes for fixing work. It is bolted to a large flange in one piece with the spindle, and fitted with four steel expanding gripping jaws worked with screws and toothed blocks. These are for doing chuck work, or for gripping work to be driven, as the collars of propeller or crank shafts, or work of a similar character. By the system of gearing adopted, when desired, the face plate can be revolved almost free, which facilitates balancing for turning crank shafts, as well as other operations. The thrust against the live spindle is taken by an adjustable steel tail piece.

The beds are double, 10 feet in width over all, the sections being joined together by massive ground plates and bolts. They are made with square lips to resist the upward strain of cutting. The front bed is fitted with two saddles, each carrying a compound slide rest having the following movements: First, screw-cutting, by means of a leading screw, situated inside the bed, with a sliding disengaging nut and reversing motion for right or left-hand threads, or for instantaneously stopping the longitudinal movement of the saddle. This is accomplished by a set of clutch mitres placed inside the bed at headstock end, and actuated by a lever in front: Second, a self-acting surfacing motion to slide rest by means of a longitudinal shaft at the front of the bed, and clutch mitres for reversing the saddle screw.

Third, power motion for moving the saddles quickly to position along the bed. This is done through the fast and loose pulleys at the headstock end of lathe.

Fourth, hand rack motion to saddle. The back bed is fitted with two saddles, each carrying a pillar rest, fitted for all movements in plain turning like the front rests, and also with swiveling motion for corner turning.

The tailstock has a spindle 9 inches diameter. It is fitted in Vs on the bed, and held down by three T-head bolts on each side. The top section is adjustable for turning tapers. It is moved along the ways by engaging a nut with the main screw. An end-cutting rest is fitted to the tailstock, which is adapted for operating on flanged couplings and similar work.

There is a separate set of change wheels for each saddle, so arranged as to cut standard pitches up to 3-inch pitch, and for self-acting feeds down to 50 per inch. By this means, when both tools are in operation on a piece of work, one tool may be used with coarse feed for roughing out, while the other may be taking a fine or finishing cut either on the same or a different part of the piece; or one tool may be cutting towards and the other from the face plate, always maintaining the balance of a front and back cut.

Complete counter driving motion, consisting of wall brackets, shaft, cone, and sets of fast and loose pulleys for quick reversing motion in screw cutting, also belt bar shipping motion, and full set of case-hardened wrenches are provided.


Chapter VII.—DETAILS IN LATHE CONSTRUCTION.

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.

Fig. 553

Fig. 553.

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

Fig. 554.

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

Fig. 555.

Fig. 556

Fig. 556.

Fig. 557

Fig. 557.

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.

Fig. 558

Fig. 558.

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.

Fig. 559

Fig. 559.

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 916 per foot to 1 inch per foot. In the practice of Pratt and Whitney a taper of 916 per foot is given to all lathes, the lengths of the tapers for different sizes of lathes being as follows:

Swing of Lathe.   Length of Taper Socket
for Live Centre.
13  inches    5    inches. 
16   3 34
18  and 19  inches    7 1116
with hollow spindle 5 inches long
  and 1116 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.

Fig. 560

Fig. 560.

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 12-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 212 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.

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
difference
between
the
adjacent
steps.
Adjacent
steps,
whose
difference
is given in
table.
Distance between the Centres of Cone Pulleys.
10
inches.
20
inches.
30
inches.
40
inches.
50
inches.
60
inches.
70
inches.
80
inches.
90
inches.
100
inches.
120
inches.
240
inches.
1 inch 1st  and  2nd 0.87 0.94 0.96 0.97 0.98 0.98 0.98 0.98 0.99 0.99 0.99 1.00
2nd 3rd 0.94 0.97 0.98 0.98 0.99 0.99 0.99 0.99 0.99 0.99 1.00 1.00
3rd 4th 1.00 1.00 1.00 1.00 1.00 1.00 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 1.01 1.01 1.01 1.01 1.00 1.00
5th 6th 1.13 1.06 1.04 1.03 1.02 1.02 1.02 1.02 1.01 1.01 1.01 1.00
112 inch 1st  and  2nd 1.21 1.36 1.40 1.43 1.44 1.45 1.46 1.46 1.47 1.47 1.48 1.49
2nd 3rd 1.36 1.43 1.45 1.46 1.47 1.48 1.48 1.48 1.49 1.49 1.49 1.49
3rd 4th 1.50 1.50 1.50 1.50 1.50 1.50 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 1.52 1.52 1.51 1.51 1.51 1.51
5th 6th 1.79 1.64 1.60 1.57 1.56 1.55 1.54 1.54 1.53 1.53 1.52 1.51
2 inches 1st  and  2nd 1.47 1.74 1.83 1.87 1.90 1.92 1.93 1.93 1.94 1.95 1.96 1.98
2nd 3rd 1.74 1.87 1.92 1.93 1.95 1.96 1.96 1.97 1.97 1.97 1.98 1.99
3rd 4th 2.00 2.00 2.00 2.00 2.00 2.00 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 2.04 2.03 2.03 2.03 2.02 2.01
5th 6th 2.53 2.26 2.17 2.13 2.10 2.08 2.07 2.07 2.06 2.05 2.04 2.02
212 inches 1st  and  2nd 1.66 2.10 2.23 2.30 2.34 2.37 2.39 2.40 2.41 2.42 2.43 2.47
2nd 3rd 2.10 2.30 2.37 2.40 2.42 2.43 2.44 2.45 2.46 2.46 2.47 2.49
3rd 4th 2.50 2.50 2.50 2.50 2.50 2.50 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 2.56 2.55 2.54 2.54 2.53 2.51
5th 6th 3.34 2.90 2.77 2.70 2.66 2.63 2.61 2.60 2.59 2.58 2.57 2.53
3 inches 1st  and  2nd 1.76 2.42 2.62 2.71 2.77 2.81 2.84 2.86 2.87 2.88 2.90 2.95
2nd 3rd 2.42 2.71 2.81 2.86 2.88 2.90 2.92 2.93 2.94 2.94 2.95 2.98
3rd 4th 3.00 3.00 3.00 3.00 3.00 3.00 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 3.08 3.07 3.06 2.06 3.05 3.02
5th 6th 4.24 3.58 3.38 3.29 3.23 3.19 3.16 3.14 3.13 3.12 3.10 3.05
4 inches 1st  and  2nd   3.95 3.31 3.49 3.59 3.66 3.71 3.75 3.78 3.80 3.83 3.91
2nd 3rd 2.94 3.49 3.66 3.75 3.80 3.83 3.85 3.87 3.88 3.89 3.91 3.96
3rd 4th 4.00 4.00 4.00 4.00 4.00 4.00 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 4.15 4.13 4.12 4.11 4.09 4.04
5th 6th   5.05 4.69 4.51 4.41 4.34 4.29 4.25 4.22 4.20 4.17 4.09
5 inches 1st  and  2nd   3.33 3.92 4.20 4.36 4.47 4.55 4.60 4.64 4.68 4.74 4.87
2nd 3rd 3.31 4.19 4.47 4.60 4.68 4.74 4.77 4.80 4.82 4.84 4.86 4.93
3rd 4th 5.00 5.00 5.00 5.00 5.00 5.00 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 5.23 5.20 5.18 5.16 5.14 5.07
5th 6th   6.67 6.09 5.80 5.64 5.53 5.45 5.40 5.36 5.32 5.26 5.13
6 inches 1st  and  2nd   3.52 4.42 4.83 5.08 5.23 5.34 5.42 5.49 5.55 5.62 5.80
2nd 3rd   4.83 5.23 5.42 5.54 5.62 5.67 5.71 5.75 5.77 5.81 5.90
3rd 4th   6.00 6.00 6.00 6.00 6.00 6.00 6.00 6.00 6.00 6.00 6.00
4th 5th   7.17 6.77 6.58 6.46 6.38 6.33 6.29 6.25 6.23 6.19 6.10
5th 6th   8.48 7.58 7.17 6.92 6.77 6.66 6.58 6.51 6.45 6.38 6.20

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 212 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 1212 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.

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
difference
between
the
adjacent
steps.
Adjacent
steps,
whose
difference
is given in
table.
Distance between the Centres of Cone Pulleys.
10
inches.
20
inches.
30
inches.
40
inches.
50
inches.
60
inches.
70
inches.
80
inches.
90
inches.
100
inches.
120
inches.
240
inches.
1 inch 1st  and  2nd 0.90 0.95 0.97 0.98 0.98 0.98 0.99 0.99 0.99 0.99 0.99 1.00
2nd 3rd 0.97 0.98 0.99 0.99 0.99 0.99 0.99 1.00 1.00 1.00 1.00 1.00
3rd 4th 1.03 1.02 1.01 1.01 1.01 1.01 1.01 1.00 1.00 1.00 1.00 1.00
4th 5th 1.10 1.05 1.03 1.02 1.02 1.02 1.01 1.01 1.01 1.01 1.01 1.00
112 inch 1st  and  2nd 1.28 1.39 1.43 1.45 1.46 1.46 1.47 1.47 1.48 1.48 1.48 1.49
2nd 3rd 1.43 1.46 1.48 1.48 1.48 1.49 1.49 1.49 1.49 1.49 1.49 1.49
3rd 4th 1.57 1.54 1.52 1.52 1.52 1.51 1.51 1.51 1.51 1.51 1.51 1.51
4th 5th 1.72 1.61 1.57 1.55 1.54 1.54 1.53 1.53 1.52 1.52 1.52 1.51
2 inches 1st  and  2nd 1.61 1.81 1.87 1.90 1.92 1.93 1.94 1.95 1.96 1.96 1.97 1.98
2nd 3rd 1.87 1.94 1.96 1.97 1.97 1.98 1.98 1.98 1.99 1.99 1.99 1.99
3rd 4th 2.13 2.06 2.04 2.03 2.03 2.02 2.02 2.02 2.01 2.01 2.01 2.01
4th 5th 2.39 2.19 2.13 2.10 2.08 2.07 2.06 2.05 2.04 2.04 2.03 2.02
212 inch 1st  and  2nd 1.89 2.20 2.30 2.35 1.38 2.40 2.41 2.42 2.43 2.44 2.45 2.47
2nd 3rd 2.30 2.40 2.43 2.45 2.46 2.47 2.47 2.47 2.48 2.48 2.48 2.49
3rd 4th 2.70 2.60 2.57 2.55 2.54 2.53 2.53 2.53 2.52 2.52 2.52 2.51
4th 5th 3.11 2.80 2.70 2.65 2.62 2.60 2.59 2.58 2.57 2.56 2.55 2.53
3 inches 1st  and  2nd 2.10 2.57 2.71 2.78 2.83 2.86 2.87 2.89 2.90 2.91 2.93 2.96
2nd 3rd 2.71 2.86 2.90 2.93 2.94 2.95 2.96 2.96 2.97 2.97 2.98 2.99
3rd 4th 3.29 3.14 3.10 3.07 3.06 3.05 3.04 3.04 3.03 3.03 3.02 3.01
4th 5th 3.90 3.43 3.29 3.22 3.17 3.14 3.13 3.11 3.10 3.09 3.07 3.04
4 inches 1st  and  2nd   3.22 3.49 3.62 3.69 3.75 3.78 3.81 3.83 3.84 3.87 3.94
2nd 3rd 3.48 3.74 3.83 3.87 3.90 3.91 3.92 3.94 3.94 3.95 3.96 3.98
3rd 4th 4.52 4.26 4.17 4.13 4.10 4.09 4.08 4.06 4.06 4.05 4.04 4.02
4th 5th   4.78 4.51 4.38 4.31 4.25 4.22 4.19 4.17 4.16 4.13 4.06
5 inches 1st  and  2nd   3.77 4.20 4.40 4.52 4.60 4.66 4.71 4.73 4.76 4.80 4.90
2nd 3rd 4.19 4.60 4.73 4.80 4.84 4.87 4.89 4.90 4.91 4.92 4.93 4.96
3rd 4th 5.81 5.40 5.27 5.20 5.16 5.13 5.11 5.10 5.09 5.08 5.07 5.04
4th 5th   6.23 5.80 5.60 5.48 5.40 5.34 5.29 5.27 5.24 5.20 5.10
6 inches 1st  and  2nd   4.21 4.83 5.13 5.31 5.42 5.51 5.57 5.62 5.66 5.71 5.86
2nd 3rd 4.82 5.42 5.62 5.71 5.77 5.81 5.83 5.86 5.87 5.88 5.90 5.95
3rd 4th 7.18 6.58 6.38 6.29 6.23 6.19 6.17 6.14 6.13 6.12 6.10 6.05
4th 5th   7.79 7.17 6.87 6.69 6.58 6.49 6.43 6.38 6.34 6.29 6.14

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