When it is required to polish and to keep the work as true and parallel as possible, these ends may be simultaneously obtained by means of clamps, such as shown in Fig. 1210, which represents a form of grinding and polishing clamp used by the Pratt and Whitney Company for grinding their standard cylindrical gauges. A cast-iron cylindrical body a is split partly through at b and entirely through at c, being closed by the screw d to take up the wear. The split b not only weakens the body a and enables its easy closure, but it affords ingress to the grinding material. It may be noted that cast iron is the best metal that can be used for this purpose, not only on account of the dead smooth surface it will take, but also because its porosity enables it to carry the oil better than a closer grained metal. For work of larger diameter, as, say, 2 or 3 inches, the form of lap shown in Fig. 1211 is used for external grinding, there being a hinge b c instead of a split, and handles are added to permit the holding and moving of the lap. The bore of this clamp is sometimes recessed and filled with lead. It is then reamed out to fit the work and used with emery and oil, the lathe running at about 300 feet per minute.
For grinding and polishing the bores of pieces, many different forms of expanding grinding mandrels have been devised, in most of which the mandrel has been given a slight degree of curvature in its length; or in other words, the diameter is slightly increased as the middle of the mandrel length is approached from either end. But with this curvature of outline, as small as it may be, it rather increases the difficulty of grinding a bore parallel instead of diminishing it. When expanding mandrels are caused to expand by a wedge acting upon split sections of the mandrel, they rarely expand evenly and do not maintain a true cylindrical form.
Fig. 1212 represents a superior form of expanding mandrel for this purpose. The length a is taper and contains a flute c. The lead is cast on and turned upon the mandrel, the metal in the flute c driving the lead. The diameter of the lap is increased by driving the taper mandrel through it, and the lead is therefore maintained cylindrically true.
While these appliances are supplied with the flour emery and oil, their action is to grind rather than to polish, but as they are used without the addition of emery, the action becomes more a polishing one.
Fig. 1213 represents at a a a wooden clamp for rough polishing with emery and oil. It consists of two arms hinged by leather at b and having circular recesses, as c, d, to receive the work. At j j is represented a similar grinding and polishing clamp for more accurate work. g and h are screws passing through the top arm and threaded into the lower, while e, f are threaded into the lower arm, and abut at their ends against the face of the upper arm. It is obvious that by means of these screws the clamp may be set to size, adjusted to give the required degree of pressure, and held firmly together. Lead bushes may be inserted in the bores as grinding laps. As this clamp is used by hand, it must be moved along the work at an exactly even speed of traverse, or else it will operate on the work for a longer period of time at some parts than at others; hence the greatest care is necessary in its use.
The best method of polishing cylindrical work to be operated on entirely in the lathe, the primary object being the polish, is by means of emery paper, and as follows:—
In all polishing the lathe should run at a fast speed; hence special high speeded lathes, termed speed lathes, are provided for polishing purposes only.
The emery paper or cloth should be of a fine grade, which is all that is necessary if the work has been properly filed, if cylindrical, and scraped if radial or of curved outline.
In determining whether emery paper or cloth should be used, the following is pertinent:—
The same grade of emery cuts more freely on cloth than on paper, because the surface of the cloth is more uneven; hence the emery grains project in places, causing them to cut more freely until worn down. If, then, the surface is narrow, so that there is no opportunity to move the emery cloth endways on the work, emery paper should be used. It should be wrapped closely (with not more than one, or at least two folds) around a smooth file, and not a coarse one, whose teeth would press the emery to the work at the points of its coarse teeth only. The file should be given short, rapid, light strokes.
For work of curved outline emery cloth should be used, because it will bend without cracking, and the cloth should be moved quickly backwards and forwards across, and not round, the curve; and when the work is long enough to permit it, the emery paper or cloth should be moved rapidly backwards and forwards along the work so that its marks cross and recross at an obtuse angle.
Now, suppose the grade of emery paper first used to be flour emery, and the final polish is to be of the highest order, then 0000 French emery paper will be required to finish, and it is to be observed that nothing will polish a metal so exquisitely as an impalpable powder of the metal itself: hence, while performing the earlier stages of polishing, it is well to prepare the final finishing piece, so as to give it a glaze of metal from the work surface.
When, therefore, all the file marks are removed by the use of the flour emery cloth, the surface of the work should be slightly oiled and then wiped, so as not to appear oily and yet not quite dry, with a piece of rag or waste, then the piece of 0000 emery paper, or, what is equally as good, a piece of crocus cloth, to be used for the final finishing should be applied to the work, and the slightly oily surface will cause the cuttings to clog and fill the crocus cloth. The cloth should be frequently changed in position so as to bring all parts of its surface in contact with the work and wear down all projections on the cloth as well as filling it with fine cuttings from the work. Then a finer grade, as, say, No. 0 French emery paper, must be used, moving it rapidly endwise of the work, as before, and using it until all the marks left by the flour emery have been removed.
One, or at most two drops of lard oil should then be put on the work, and spread over as far as it will extend with the palm of the hand, when the finishing crocus may again be applied and reversed as before in every direction; 00 emery paper may then be used until all the marks of the 0 are removed, and with the work left quite dry the crocus for final finishing may again be applied; 000 emery paper may then be used to efface all the marks left by the 00. This 000 emery paper should be used until it is very much worn, the final finish being laid with the glazed crocus.
If this crocus has been properly prepared, its whole surface will be covered with a film of fine particles of metal, so that if the metal be brass the crocus surface will appear like gold leaf. If cast iron, the crocus surface will appear as though polished with plumbago or blacklead, while in any case the crocus surface will be polished and quite dry. The crocus should be pressed lightly to the work, so that its polishing marks will not be visible to the naked eye.
If emery paper be applied to work finished to exact diameter it should be borne in mind that the process reduces to some extent the size of the work, and that the amount under proper conditions though small is yet of importance, where preciseness of diameter is a requisite.
In the practice, however, of some of the best machine shops of the United States, the lathe alone is not relied upon to produce the best of polish. Thus, in the engine works of Charles H. Brown, of Fitchburg, Massachusetts, whose engines are unsurpassed for finish and polish, and which the majority of mechanics would suppose were finely silver plated, the following is the process adopted for polishing connecting rods.
The rod is carefully tool-finished with a fine feed. The tool marks are then erased with a fine smooth file, and these file marks by a dead-smooth file, the work rotating at a quick speed, little metal being left, so as to file as little as possible. Next comes fine emery cloth to smooth down and remove the file marks. The lathe is then stopped and the rod stoned lengthwise with Hindostan stone and benzine, removing all streaks. The Scotch stone used with water follows, until the surface is without scratches or marks, as near perfect as possible. The next process is, for the finest work, the burnisher used by hand. But if not quite so exquisite a polish is required, the rod is finished by the use of three grades of emery cloth, the last being very fine.
Sometimes, however, the streaks made by polishing with emery paper used before the application of the stones are too difficult to remove by them. In this case, for a very fine finish, the lathe is stopped and draw-filing with the finest of files is performed, removing all streaks; and the stones then follow the draw-filing. All stoning is done by hand with the work at rest, as is also all burnishing.
After the burnisher comes fine imported crocus cloth, well worn, which makes the surface more even and dead than that left by the burnisher. The crocus is used with the lathe at its quickest speed, and is moved as slowly and as evenly as possible, the slower and more even the crocus movement along the rod, the more even the finish. If the rod has filleted corners, such corners are in all cases draw-filed before the stoning.
The method of polishing a cylinder cover at the Brown Engine Works is as follows.
The finishing cut is taken with a feed of 32 lathe-revolutions per inch of tool traverse, and at as quick a cutting speed as the hardness of the iron will permit. This is necessary in order to have the tool-edge cut the metal without breaking it out as a coarse one would do. With the fine feed and quick speed the pores of the iron do not show; with a coarse feed the pores show very plainly and are exposed for quite a depth.
After the lathe-tool comes a well oil-stoned hand-scraper, with a piece of leather between it and the tool rest to prevent the scraper from chattering. The scraper not only smooths the surface, but it cuts without opening the pores. It is used at a quick speed, as quick indeed as it will stand, which varies with the hardness of the metal, but is always greater than is possible with a slide-rest tool.
After the scraper the cover is removed from the lathe, and all flat surfaces are filed as level as possible with a second-cut file, and then stoned with soft Hindostan stone, used with benzine or turpentine, so as to wash away the cuttings and prevent them from clogging the stone or forming scratches. In using all stones the direction of motion is frequently reversed so as to level the surface. Next comes stoning with Scotch stone (Water of Ayr), used with water; in this part of the operation great care must be taken, otherwise the cuttings will induce scratches. When the Scotch stone marks have removed all those left by the Hindostan stone, and left the surface as smooth as possible, the cover is again put in the lathe and the grain is laid and finished with very fine emery cloth and oil. The emery cloth is pressed lightly to the work and allowed to become well worn so as to obtain a fine lustre without leaving any streaks.
It will be noticed here that the use of the emery stick and oil is entirely dispensed with; but for a less fine polish it may be used, providing it be kept in quick motion radially on the work. The objection to its use is that if there be any speck on the work it is apt to cut a streak or groove following the spot like a comet’s tail.
Turning Tapers.—There are five methods of turning outside tapers; 1st, by setting over the tailstock of the lathe; 2nd, by the use of a former or taper turning attachment such as was shown in Fig. 508; 3rd, by the use of a compound slide rest; 4th, by means of a lathe in which the head and tailstock are upon a bed that can be set at an angle to the lathe shears on which the lathe carriage slides; and 5th, by causing the cross-feed screw to operate simultaneously with the feed traverse.
Referring to the first method, it is objectionable, inasmuch as that the work axis is thrown at an angle to the axis of the lathe centres, which causes the work centres to wear rapidly, and this often induces them to move their positions and throw the work out of true. Furthermore, the tailstock has to be moved back in line with the live spindle axis for turning parallel again, and this is a troublesome matter, especially when the work is long.
Fig. 1214 shows the manner in which the lathe centres and the work centres have contact, l being the live and b the dead centre; hence c c is the axis of the live spindle which is parallel to the lathe shear slides, which are represented by g; obviously a is the work axis. The wear is greatest at the dead centre end of the work, but there is some wear at the live centre end, because there is at that end also a certain amount of motion of the work centre upon the live centre. Thus, in Fig. 1215, let c represent the live centre axis, a the work axis, d the lathe face plate, and e f the plane of the driver or dog upon the work, and it is obvious that the tail of the driver will when at one part of the lathe revolution stand at e, while when diametrically opposite it will stand at f; hence, during each work revolution the driver moves, first towards and then away from the face plate d, and care must be taken in adjusting the position of the driver to see that it has liberty to move in this direction, for if obstructed in its motion it will spring or bend the work.
To determine how much the tailstock of a lathe must be set over to turn a given taper, the construction shown in Fig. 1216 may be employed. Draw the outline of the work and mark its axis d, draw line c parallel to one side of the taper end, and the distance a between this line and the work axis is the amount the tailstock requires to be set over. This construction is proved in Fig. 1217, in which the piece of work is shown set over, c representing the line of the lathe ways, with which the side f of the taper must be parallel. d is the line of the live spindle, and e that of the work, and the distance b will be found the same as distance a in Fig. 1216.
It may be remarked, however, that in setting the tailstock over it is the point of the dead centre when set adjusted to the work length that must be measured, and not the tailblock itself.
Other methods of setting tailstocks for taper turning are as follows: If a new piece is to be made from an old one, or a duplicate of a piece of work is to be turned, the one already turned, or the old piece as the case may be, may be put in the lathe and we may put a tool in the tool post and set the tailstock over until the tool traversed along the work (the latter remaining stationary) will touch the taper surface from end to end.
If, however, the taper is given as so much per foot, the distance to set the tailstock over can be readily calculated.
Thus, suppose a piece of work has a taper part, having a taper of an inch per foot, the work being three feet long, then there would be three inches of taper in the whole length of the piece and the tailstock requires to be set over one-half of the three inches, or 11⁄2 inches. It will not matter how long the taper part of the work is, nor in what part of the work it is, the rule will be found correct so long as the tailstock is set over one-half the amount obtained by multiplying the full length of the work per foot by the amount of taper per foot.
If we have no pattern we may turn at each end of the part that is to be taper a short parallel place, truing it up and leaving it larger to the same amount at each end than the finished size, and taking care that the parallel part at the small end will all turn out in the finishing. We then fasten a tool in the lathe tool post, place it so that it will clear the metal of the part requiring to be turned taper, and placing it at one extreme end of said part, we take a wedge, or a piece of metal sufficiently thick, and place it to just contact with the turned part of the work and the tool point (adjusting the tool with the cross-feed screw), we then wind the rest to the other end of the required taper part, and inserting same wedge or piece of iron, gauge the distance from the tool point to the work, it being obvious that when the tool point wound along is found to stand at an equal distance from each end of the turned part, the lathe is set to the requisite taper.
Figs. 1218 and 1219 illustrate this method of setting. a represents a piece of work requiring to be turned taper from b to c, and turned down to within 1⁄32 inch of the required size at e and f. If then we place the tool point h first at one end and then at the other, and insert the piece i and adjust the lathe so that the piece of metal i will just fit between the tool point and the work at each extreme end of the required taper part, the lathe will be set to the requisite taper as near as practicable without trying the work to the taper hole. The parallel part at the small end of the work should be turned as true as possible, or the marks may not be obliterated in finishing the work.
Fig. 1220 (from The American Machinist) represents a gauge for setting the tailstock over for a taper. A groove is cut as at e and d, these diameters corresponding to the required taper; a holder a is then put in the tool point, and to this holder is pivoted the gauge b. The tailstock is set over until the point of b will just touch the bottom of the groove at each end of the work.
To try a taper into its place, we either make a chalked stripe along it from end to end, smoothing the chalked surface with the finger, or else apply red marking to it, and then while pressing it firmly into its place, revolve it backwards and forwards, holding it the while firmly to its seat in the hole; we move the longest outwardly projecting end up and down and sideways, carefully noting at which end of the taper there is the most movement. The amount of such movement will denote how far the taper is from fitting the hole, while the end having the least movement will require to have the most taken off it, because the fulcrum off which the movement takes place is the highest part, and hence requires the greatest amount of metal to be taken off.
Having fitted a taper as nearly as possible with the lathe tool, that is to say, so nearly that we cannot find any movement or unequal movement at the ends of the taper (for there is sure to be movement if the tapers do not agree, or if the surfaces do not touch at more than one part of their lengths), we must finish it with a fine smooth file as follows: After marking the inside of the hole with a very light coat of red marking, taking care that there is no dirt or grit in it, we press the taper into the hole firmly, forcing it to its seat while revolving it backwards and forwards.
By advancing it gradually on the forward stroke, the movement will be a reciprocating and yet a revolving one. The work must then be run in the lathe at a high speed, and a smooth file used to ease off the mark visible on the taper, applying the file the most to parts or marks having the darkest appearance, since the darker the marks the harder the bearing has been. Too much care in trying the taper to its hole cannot be taken, because it is apt to mark itself in the hole as though it were a correct fit when at the same time it is not; it is necessary therefore at each insertion to minutely examine the fit by the lateral and vertical movement of projecting part of the taper, as before directed.
A taper or cone should be fitted to great exactitude before it is attempted to grind it, the latter process being merely intended to make the surfaces even.
For wrought-iron, cast-iron, or steel work, oil and emery may be used as the grinding materials (for brass, burnt sand and water are the best). The oil and emery should be spread evenly with the finger over the surfaces of the hole and the taper; the latter should then be placed carefully in its place and pressed firmly to its seat while it is being revolved backwards and forwards, and slowly rotated forward by moving it farther during the forward than during the backward movement of the reciprocating motion.
After about every dozen strokes the taper should be carefully removed from the hole and the emery again spread evenly over the surfaces with the finger, and at and during about every fourth one of the back strokes of the reciprocating movement the taper should be slightly lifted from its bed in the hole, being pressed lightly home again on the return stroke, which procedure acts to spread the grinding material and to make the grinding smooth and even. The emery used should be about number 60 to 70 for large work, about 80 to 100 for small, and flour emery for very fine work.
Any attempt to grind work by revolving it steady in one direction will cause it to cut rings and destroy the surface.
Referring to the second method, all that is necessary in setting a former or taper attachment bar is to set it out of line with the lathe shears to half the amount of taper that is to be turned, the bar being measured along a length equal to that of the work. Turning tapers with a bar or taper-turning attachment possesses the advantage that the tailstock not being set over, the work centres are not thrown out of line with the live centres, and the latter are not subject to the wear explained with reference to Fig. 1214. Furthermore, the tailstock being kept set to turn parallel, the operator may readily change from turning taper to turning parallel, and may, therefore, rough out all parts before finishing any of them, and thus keep the work more true, whereas in turning tapers by setting the tailstock over we are confronted by the following considerations:—
If we turn up and finish the plain part first, the removing of the skin and the wear of the centres during the operation of turning the taper part will cause the work to run out of true, and hence it will not, when finished, be true; or if, on the other hand, we turn up the taper part first, the same effects will be experienced in afterwards turning the plain part. We may, it is true, first rough out the plain part, then rough out the taper part, and finish first the one and then the other; to do this, however, we shall require to set the lathe twice for the taper and once for the parallel part.
It is found in practice that the work will be more true by turning the taper part the last, because the work will alter less upon the lathe centres when changed from parallel to taper turning than when changed from the latter to the former. In cases, however, in which the parts fitting the taper part require turning, it is better to finish the parallel part last, and to then turn up the work fastened upon the taper part while it is fast upon its place: thus, in the case of a piston rod and piston, were we to turn up the parallel part of the rod first and the taper last, and the centres altered during the last operation, when the piston head was placed upon the rod, and the latter was placed in the lathe, the plain part or stem would not run true, and we should require to true the centres to make the rod run true before turning up the piston head. If, however, we first rough out the plain part or stem of the rod, and then rough out and finish the taper part, we may then fasten the head to its place on the rod, and turn the two together; that is to say, rough out the piston head and finish its taper hole; then rough out the parallel part of the rod, but finish its taper end. The rod may then be put together and finished at one operation; thus the head will be true with the rod whether the taper is true with the parallel part of the rod or not. With a taper-turning attachment the rod may be finished separately, which is a great advantage.
If, however, one part of the length of a taper turning attachment is much more used than another, it is apt to wear more, which impairs the use of the bar for longer work, as it affects its straightness and causes the slide to be loose in the part most used, and on account of the wear of the sliding block it is proper to wind the tool out from its cut on the back traverse, or otherwise the tool may cut deeper on the back than on the forward traverse, and thus leave a mark on the work surface.
Referring to the third method, a compound slide rest provides an excellent method of turning tapers whose lengths are within the capacity of the upper slide of the compound rest, because that slide may be used to turn the taper, while the ordinary carriage feed may be used for the parallel parts of the work, and as the tailstock does not require to set over, the work centres are not subject to undue wear.
If the seat for the upper slide of the rest is circular, and the taper is given in degrees of angle, a mark may be made on the seat, and the base of the upper slide may be marked in degrees of a circle, as shown in Fig. 1221, which will facilitate the setting; or the following construction, which is extracted from Mechanics, may be employed. Measure the diameter of the slide rest seat, and scribe on a flat surface a circle of corresponding diameter. Mark its centre, as a in Fig. 1222, and mark the line a b. From the centre a mark the point b, whose radius is that of the small end of the hole to be bored. Mark the length of the taper to be turned on the line a g and draw the line g d distant from a b equal to the diameter of the large end of the hole to be bored. Draw the line b d. Then the distance e f is the amount the rest must be swiveled to turn the required taper.
It is obvious that the same method may also be used for setting the rest.
Referring to the fourth method, by having an upper bed or base plate for the head and tailstock, so that the line of lathe centres may be set at the required angle to the Vs or slides on which the carriage traverses, it affords an excellent means of turning tapers, since it avoids the disadvantages mentioned with regard to other systems, while at the same time it enables the turning of tapers of the full length of the carriage traverse, but it is obvious that the head and tailstock are less rigidly supported than when they are bolted direct to the lathe shears.
In turning tapers it is essential that the tool point be set to the exact height of the work axis, or, in other words, level with the line of centres. If this is not the case the taper will have a curved outline along its length. Furthermore, it may be shown that if a straight taper be turned and the tool be afterwards either raised or lowered, the amount of taper will be diminished as well as the length being turned to a curve.
Figs. 1223 and 1224 demonstrate that the amount of taper will be changed by any alteration in the height of the tool. In Fig. 1223, a b represents the line of centres of the spindle of a lathe, or, in other words, the axis of the work w, when the lathe is set to turn parallel; a c represents the axis of the work or cone when the lathe tailstock is set over to turn the taper or cone; hence the length of the line c b represents the amount the tailstock is set over. Referring now to Fig. 1224, the cone is supposed to stand level, as it will do in the end view, because the lathe centres remain at an equal height from the lathe bed or Vs, notwithstanding that the tailstock is set over. The tool therefore travels at the same height throughout its whole length of feed; hence, if it is set, as at t, level with the line of centres, its line of feed while travelling from end to end of the cone is shone by the line a b. The length of the line a b is equal to the length of the line b c Fig. 1223. Hence, the line a b, Fig. 1224, represents two things: first, the line of motion of the point of tool t as it feeds along the cone, and second its length represents the amount the work axis is out of parallel with the line of lathe centres. Now, suppose that the tool be lowered to the position shown at i; its line of motion as it feeds will be the line c d, which is equal in length to the line a b. It is obvious, therefore, that though the tool is set to the diameter of the small end, it will turn at the large end a diameter represented by the dotted circle h. The result is precisely the same if the taper is turned by a taper-turning attachment instead of setting the tailstock out of line.
The demonstration is more readily understood when made with reference to such an attachment as the one just mentioned, because the line a b represents the line of tool feed along the work, and its length represents the amount the attachment causes the tool to recede from the work axis. Now as this amount depends upon the set-over of the attachment it will be governed by the degree of that set over, and is, therefore, with any given degree, the same whatever the length of the tool travel may be. All that is required, then, to find the result of placing the tool in any particular position, as at i in the end view, is to draw from the tool point a line parallel to a b and equal in length to it, as c d. The two ends of that line will represent in their distances from the work axis the radius the work will be turned to at each end with the tool in that position. Thus, at one end of the line c d is the circle k, representing the diameter the tool i would turn the cone at the small end, while at the other end the dotted circle h gives the diameter at the large end that the tool would turn to when at the end of its traverse. But if the tool be placed as at t, it will turn the same diameter k at the small end, and the diameter of the circle p at the large end.
We have here taken account of the diameters at the ends only of the work, without reference to the result given at any intermediate point along the cone surface, but this we may now proceed to do, in order to prove that a curved instead of a straight taper is produced if the tool be placed either above or below the line of lathe centres.
In Fig. 1225, d e f c represents the complete outline of a straight taper, whose diameter at the ends is represented in the end view by the outer and inner circles. Now, a line from a to b will represent the axis of the work, and also the line of tool point motion or traverse, if that point is set level with the axis. The line i k in the end view corresponds to the line a b in the side view, in so far that it represents the line of tool traverse when the tool point is set level with the line of centres. Now, suppose the tool point to be raised to stand level with the line g h, instead of at i k, and its line of feed traverse be along the line g h, whose length is equal to that of i k. If we divide the length of g h into six equal divisions, as marked from 1 to 6, and also divide the length of the work in the side view into six equal divisions (a to f), we shall have the length of line g h in the first division in the end view (that is, the length from h to g), representing the same amount or length of tool traverse as from the end b of the cone to the line a in the side view. Now, suppose the tool point has arrived at 1; the diameter of work it will turn when in that position is evidently given by the arc or half-circle h, which meets the point 1 on g h. To mark that diameter on the side view, we first draw a horizontal line, as h p, just touching the top of h; a perpendicular dropped from it cutting the line a b, gives the radius of work transferred from the end view to the side view. When the tool point has arrived at 2 on g h in the end view, its position will be shown in the side view at the line b, and the diameter of work it will turn is shown in the end view by the half-circle k. To transfer this diameter to the side view we draw the line k g, and where it cuts the line b in the side view is the radius of the work diameter when the tool has arrived at the point b in the side view. Continuing this process, we mark half-circles, as l, m, n, o, and the lines l r, m s, n t, o u, by means of which we find in the side view the work radius when the tool has arrived at c, d, e, and f respectively. All that remains to be done is to draw on the side view a line, as u e, that shall pass through the points. This line will represent the outline of the work turned by the tool when its height is that denoted by g h. Now, the line u e is shown to be a curve, hence it is proved that with the tool at the height g h a curved, and not a straight, taper will be turned.
It may now be proved that if the tool point is placed level with the line of centres, a straight taper will be turned. Thus its line of traverse will be denoted by a b in the side view and the line i k in the end view; hence we may divide i k into six equal divisions, and a b into six equal divisions (as a, b, c, &c.). From the points of division i k, we may draw half-circles as before, and from these half-circles horizontal lines, and where the lines meet the lines of division in the side view will be points in the outline of the work, as before. Through these points we draw a line, as before, and this line c f, being straight, it is proven that with the tool point level with the work axis, it will turn a straight taper.
It may now be shown that it is possible to turn a piece of work to a curve of equal curvature on each side of the middle of the work length. Suppose, for example, that the cutting tool stands on top of the work, as in the end view in Fig. 1226, and that while the tool is feeding along the work it also has a certain amount of motion in a direction at right angles to the work axis, so that its line of motion is denoted by the line b b in the top view. The outline of the work turned will be a curve, as is shown in Fig. 1227, in which the line of tool traverse is the line c d. Now the amount of tool motion that occurs during this traverse in a direction at right angles to the work axis is represented by the line f e, because the upper end is opposite to the upper end of c d, while the lower end is opposite the lower end of c d. We may then divide one-half of the length of f e into the divisions marked from 1 to 6. Now, as we have taken half the length of f e, we must also take half the length of the work and divide it into six equal divisions marked from a to f. Now, suppose the tool point to stand in the line f s in the end view, its position in the top view will be at c. When it is at 1 on the end view it will have arrived at p in the top view. The radius of work it will then turn is shown in the end view by the length of line running from 1 to the work centre. Take this length, and from a in the work axis set it off on the line a h, and make the length equal the height of 1 s. In like manner, when the tool point has arrived at 2, the radius it will cut the work is shown by the length of line i; hence from 2 on the work axis we may set off the length of 2 s, making 2 s and b i of equal length. Continuing this process, we make the length of c k equal that of 3 s, the length of d l equal 4 s, and so on. All that remains then is to draw a line, o g, that shall meet the tops of these lines. This line will show the curve to which that half of the work length will be turned to. The other half of the work length will obviously be turned to the same curvature.
It is obvious that the curvature of the work outline will be determined by the proportion existing between the length of the work and the amount of tool motion in a direction at right angles to the work axis, or, in other words, between the length of the work and that of the line f e. It is evident, also, that with a given amount of tool motion across the work, the curvature of outline turned will be less in proportion as the work length is greater. Now, suppose that the smaller and the larger diameter of the work, together with its length, are given, and it is required to find how much curvature the tool must have, we may find this and work out the curve it will cut by the construction shown in Fig. 1228, in which the circle k is the smallest and the circle p the largest diameter. The line m c is drawn to just touch the perimeter of k, and this at once gives the amount of cross-motion for the tool. Hence, we may draw the line m b and c b, and from their extremities draw the line b b representing the path of traverse of the tool point. We may then obtain the full curve on one side of the work by dividing one-half the length of m c into six equal divisions and proceeding as before, except that we have here added the lines of division in the second half as from f to l. It will be observed that the centre of the curve is at the point where the tool point crosses the axis of the work; hence, by giving to the tool more traverse on one side than on the other of the work axis, the location of the smallest point of work diameter may be made to fall on one side of the middle of the work length.
In either turning or boring tapers that are to drive or force in or together, the amount to be allowed for the fit may be ascertained, so that the work may be made correct without driving each piece to its place to try its fit.
Suppose, for example, that the pieces are turned, and the holes are to be reamed, then the first hole reamed may be made to correct diameter by fit and trial, and a collar may be put on the reamer to permit it to enter the holes so far and no farther.