Fig. 1224.—End View.
[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.
Side View. Fig. 1225. End View.
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.