In making patterns wherefrom gear-wheels may be cast in a mould, the true curves are frequently represented by arcs of circles struck from the requisite centres and of the most desirable radius with compasses, and this will be treated after explaining the pattern maker’s method of obtaining true curves by rolling segments by hand. If, then, the wheels are of small diameter, as say, less than 12 inches in diameter, and precision is required, it is best to turn in the lathe wooden disks representing in their diameters the base and generating circles. But otherwise, wooden segments to answer the same purpose may be made as from a piece of soft wood, such as pine or cedar, about three-eighths inch thick, make two pieces a and b, in Fig. 115, and trim the edges c and d to the circle of the pitch line of the required wheel. If the diameter of the pitch circle is marked on a drawing, the pieces may be laid on the drawing and sighted for curvature by the eye. In the absence of a drawing, strike a portion of the pitch circle with a pair of sharp-pointed compasses on a piece of zinc, which will show a very fine line quite clear. After the pieces are filed to the circle, try them together by laying them flat on a piece of board, bringing the curves in contact and sweeping a against b, and the places of contact will plainly show, and may be filed until continuous contact along the curves is obtained. Take another similar piece of wood and form it as shown in Fig. 116, the edge e representing a portion of the rolling circle. In preparing these segments it is an excellent plan to file the convex edges, as shown in Fig. 117, in which p is a piece of iron or wood having its surface s trued; f is a file held firmly to s, while its surface stands vertical, and t is the template laid flat on s, while swept against the file. This insures that the edge shall be square across or at least at the same angle all around, which is all that is absolutely necessary. It is better, however, that the edges be square. So likewise in fitting a and b (Fig. 115) together, they should be laid flat on a piece of board. This will insure that they will have contact clear across the edge, which will give more grip and make slip less likely when using the segments. Now take a piece of stiff drawing paper or of sheet zinc, lay segment a upon it, and mark a line coincident with the curved edge. Place the segment representing the generating circle flat on the paper or zinc, hold its edge against segment a, and roll it around a sufficient distance to give as much of the curve as may be required; the operation being illustrated in Fig. 118, in which a is the segment representing the pitch or base circle, e is the segment representing the generating circle, p is the paper, c the curve struck by the tracing point or pencil o.

This tracing point should be, if paper be used to trace on, a piece of the hardest pencil obtainable, and should be filed so that its edge, if flat, shall stand as near as may be in the line of motion when rolled, thus marking a fine line. If sheet zinc be used instead of paper a needle makes an excellent tracing point. Several of the curves, c, should be struck, moving the position of the generating segment a little each time.

On removing the segments from the paper, there will appear the lines shown in Fig. 119; a representing the pitch circle, and o o o the curves struck by the tracing point.

Cut out a piece of sheet zinc so that its edge will coincide with the curve a and the epicycloid o, trying it with all four of the epicycloids to see that no slip has occurred when marking them; shape a template as shown in Fig. 120. Cutting the notches at a b, acts to let the file clear well when filing the template, and to allow the scriber to go clear into the corner. Now take the segment a in Fig. 118, and use it as a guide to carry the pitch circle across the template as at p, in Fig. 120. A zinc template for the flank curve is made after the same manner, using the rolling segment in conjunction with the segment b in Fig. 115.

But the form of template for the flank should be such as shown in Fig. 121, the curve p representing, and being of the same radius as the pitch circle, and the curve f being that of the hypocycloid. Both these templates are set to the pitch circles and to coincide with the marks made on the wheel teeth to denote the thickness, and with a hardened steel point a line is traced on the tooth showing the correct curve for the same.

An experienced hand will find no difficulty in producing true templates by this method, but to avoid all possibility of the segments slipping on coarse pitches, and with large segments, the segments may be connected, as shown in Fig. 122, in which o represents a strip of steel fastened at one end into one segment and at the other end to the other segment. Sometimes, indeed, where great accuracy is requisite, two pieces of steel are thus employed, the second one being shown at p p, in the figure. The surfaces of these pieces should exactly coincide with the edge of the segments.

The curve templates thus produced being shaped to apply to the pitch circle may be correctly applied to that circle independently of its concentricity to the wheel axis or of the points of the teeth, but if the points of the teeth are turned in the lathe so as to be true (that is, concentric to the wheel axis) the form of the template may be such as shown in Fig. 123, the radius of the arc a a equalling that of the addendum circle or circumference at the points of the teeth, and the width at b (the pitch circle) equaling the width of a space instead of the thickness of a tooth. The curves on each side of the template may in this case be filed for the full side of a tooth on each side of the template so that it will completely fill the finished space, or the sides of two contiguous teeth may be marked at one operation. This template may be set to the marks made on the teeth at the pitch circle to denote their requisite thickness, or for greater accuracy, a similar template made double so as to fill two finished tooth spaces may be employed, the advantage being that in this case the template also serves to mark or test the thickness of the teeth. Since, however, a double template is difficult to make, a more simple method is to provide for the thickness of a tooth, the template shown in Fig. 124, the width from a to b being either the thickness of tooth required or twice the thickness of a tooth plus the width of a space, so that it may be applied to the outsides of two contiguous teeth. The arc c may be made both in its radius and distance from the pitch circle d d to equal that of the addendum circle, so as to serve as a gauge for the tooth points, if the latter are not turned true in the lathe, or to rest on the addendum circle (if the teeth points are turned true), and adjust the pitch circle d d to the pitch circle on the wheel.

The curves for the template must be very carefully filed to the lines produced by the rolling segments, because any error in the template is copied on every tooth marked from it. Furthermore, instead of drawing the pitch circle only, the addendum circle and circle for the roots of the teeth or spaces should also be drawn, so that the template may be first filed to them, and then adjusted to them while filing the edges to the curves.

Another form of template much used is shown in Fig. 125. The curves a and b are filed to the curve produced by rolling segments as before, and the holes c, d, e, are for fastening the template to an arm, such as shown in Fig. 126, which represents a section of a wheel w, with a plug p, fitting tightly into the hub h of the wheel. This plug carries at its centre a cylindrical pin on which pivots the arm a. The template t is fastened to the arm by screws, and set so that its pitch circle coincides with the pitch circle p on the wheel, when the curves for one side of all the teeth may be marked. The template must then be turned over to mark the other side of the teeth.

The objection to this form of template is that the length of arc representing the pitch circle is too short, for it is absolutely essential that the pitch line on the template (or line representing the arc of the addendum if that be used) be greater than the width of a single tooth, because an error of the thickness of a line (in the thickness of a tooth), in the coincidence of the pitch line of the template with that of the tooth, would throw the tooth curves out to an extent altogether inadmissible where true work is essential.

To overcome this objection the template may be made to equal half the thickness of a tooth and its edge filed to represent a radial line on the wheel. But there are other objections, as, for example, that the template can only be applied to the wheel when adjusted on the arm shown in Fig. 126, unless, indeed, a radial line be struck on every tooth of the wheel. Again, to produce the template a radial line representing the radius of the wheel must be produced, which is difficult where segments only are used to produce the curves. It is better, therefore, to form the template as shown in Fig. 127, the projections at a b having their edges filed to coincide with the pitch circle p, so that they may be applied to a length of one arc of pitch circle at least equal to the pitch of the teeth.

The templates for the tooth curves being obtained, the wheel must be divided off on the pitch circle for the thickness of the teeth and the width of the spaces, and the templates applied to the marks or points of division to serve as guides to mark the tooth curves. Since, however, as already stated, the tooth curves are as often struck by arcs of circles as by templates, the application of such arcs and their suitability may be discussed.

Marking the Curves by Hand.

In the employment of arcs of circles several methods of finding the necessary radius are found in practice.

In the best practice the true curve is marked by the rolling segments already described, and the compass points are set by trial to that radius which gives an arc nearest approaching to the true face and flank curves respectively. The degree of curve error thus induced is sufficient that the form of tooth produced cannot with propriety be termed epicycloidal teeth, except in the case of fine pitches in which the arc of a circle may be employed to so nearly approach the true curve as to be permissible as a substitute. But in coarse pitches the error is of much importance. Thus in Fig. 128 is shown the curve of the former or template attachment used on the celebrated Corliss Bevel Gear Cutting Machine, to cut the teeth on the bevel-wheels employed upon the line shafting at the Centennial Exhibition. These gears, it may be remarked, were marvels of smooth and noiseless running, and attracted wide attention both at home and abroad. The engraving is made from a drawing marked direct from the former itself, and kindly furnished me by Mr. George H. Corliss. a a is the face and b b the flank of the tooth, c c is the arc of a circle nearest approaching to the face curve, and d d the arc of a circle nearest approaching the flank curve. In the face curve, there are but two points where the circle coincides with the true curve, while in the flank there are three such points; a circle of smaller radius than c c would increase the error at b, but decrease it at a; one of a greater radius would decrease it at b, and increase it at a. Again, a circle larger in radius than d d would decrease the error at e and increase it at f; while one smaller would increase it at e and decrease it at f. Only the working part of the tooth is given in the illustration, and it will be noted that the error is greatest in the flank, although the circle has three points of coincidence.

In this case the depth of the former tooth is about three and three-quarter times greater than the depth of tooth cut on the bevel-wheels; hence, in the figure the actual error is magnified three and three-quarter times. It demonstrates, however, the impropriety of calling coarsely pitched teeth that are found by arcs of circles “epicycloidal” teeth.

When, however, the pitches of the teeth are fine as, say an inch or less, the coincidence of an arc of a circle with the true curve is sufficiently near for nearly all practical purposes, and in the case of cast gear the amount of variation in a pitch of 2 inches would be practically inappreciable.

To obtain the necessary set of the compasses to mark the curves, the following methods may be employed.

First by rolling the true curves with segments as already described, and the setting the compass points (by trial) to that radius which gives an arc nearest approaching the true curves. In this operation it is not found that the location for the centre from which the curve must be struck always falls on the pitch circle, and since that location will for every tooth curve lie at the same radius from the wheel centre it is obvious that after the proper location for one of the curves, as for the first tooth face or tooth flank as the case may be, is found, a circle may be struck denoting the radius of the location for all the teeth. In Fig. 129, for example, p p represents the pitch circle, a b the radius that will produce an arc nearest approaching the true curve produced by rolling segments, and a the location of the centre from which the face arc b should be struck. The point a being found by trial with the compasses applied to the curve b, the circle a c may be struck, and the location for the centres from which the face arcs of each tooth must be struck will also fall on this circle, and all that is necessary is to rest one point of the compasses on the side of the tooth as, say at e, and mark on the second circle a c the point c, which is the location wherefrom to mark the face arc d.

If the teeth flanks are not radial, the locations of the centre wherefrom to strike the flank curves are found in like manner by trial of the compasses with the true curves, and a third circle, as i in Fig. 130, is struck to intersect the first point found, as at g in the figure. Thus there will be upon the wheel face three circles, p p the pitch circle, j j wherefrom to mark the face curves, and i wherefrom to mark the flank curves.

When this method is pursued a little time may be saved, when dividing off the wheel, by dividing it into as many divisions as there are teeth in the wheel, and then find the locations for the curves as in Fig. 131, in which 1, 2, 3 are points of divisions on the pitch circle p p, while a, b, struck from point 2, are centres wherefrom to strike the arcs e, f; c, d, struck also from point 2 are centres wherefrom to strike the flank curves g, h.

It will be noted that all the points serving as centres for the face curves, in Fig. 130, fall within a space; hence if the teeth were rudely cast in the wheel, and were to be subsequently cut or trimmed to the lines, some provision would have to be made to receive the compass points.

To obviate the necessity of finding the necessary radius from rolling segments various forms of construction are sometimes employed.

Thus Rankine gives that shown in Fig. 132, which is obtained as follows. Draw the generating circle d, and a d the line of centres. From the point of contact at c, mark on circle d, a point distance from c one-half the amount of the pitch, as at p, and draw the line p c of indefinite length beyond c. Draw a line from p, passing through the line of centres at e, which is equidistant between c and a. Then multiply the length from p to c by the distance from a to d, and divide by the distance between d and e. Take the length and radius so found, and mark it upon p c, as at f, and the latter will be the location of centre for compasses to strike the face curve.

Another method of finding the face curve, with compasses, is as follows: In Fig. 133, let p p represent the pitch circle of the wheel to be marked, and b c the path of the centre of the generating or describing circle as it rolls outside of p p. Let the point b represent the centre of the generating circle when that circle is in contact with the pitch circle at a. Then from b, mark off on b c any number of equidistant points, as d, e, f, g, h, and from a, mark on the pitch circle, points of division, as 1, 2, 3, 4, 5, at the intersection of radial lines from d, e, f, g, and h. With the radius of the generating circle, that is, a b, from b, as a centre, mark the arc i, from d the arc j, from e the arc k, &c., to m, marking as many arcs as there are points of division on b c. With the compasses set to the radius of divisions 1, 2, step off on arc m the five divisions, n, o, s, t, v, and v will be a point in the epicycloidal curves. From point of division 4, step off on l four points of division, as a, b, c, d, and d will be another point in the epicycloidal curve. From point 3 set off three divisions on k, from point 2 two dimensions on l, and so on, and through the points so obtained, draw by hand or with a scroll the curve represented in the cut by curve a v.

Hypocycloids for the flanks of the teeth may be traced in a similar manner. Thus in Fig. 134 p p is the pitch circle, and b c the line of motion of the centre of the generating circle to be rolled within p p, and r a radial line. From 1 to 6 are points of equal division on the pitch circle, and d to i are arc locations for the centre of the generating circle. Starting from a, which represents the supposed location for the centre of the generating circle, the point of contact between the generating and base circles will be at b. Then from 1 to 6 are points of equal division on the pitch circle, and from d to i are the corresponding locations for the centres of the generating circle. From these centres the arcs j, k, l, m, n, o, are struck. From 6 mark the six points of division from a to f, and f is a point in the curve. Five divisions on n, four on m, and so on, give respectively points in the curve which is marked in the figure from a to f.

There is this, however, to be noted concerning the constructions of the last two figures. Since the circle described by the centre of the generating circle is of different arc or curve to that of the pitch circle, the chord of an arc having an equal length on each will be different. The amount is so small as to be practically correct. The direction of the error is to give to the curves a less curvature, as though they had been produced by a generating circle of larger diameter. Suppose, for example, that the difference between the arc n 5 (Fig. 133) and its chord is .1, and that the difference between the arc 4 5, and its chord is .01, then the error in one step is .09, and, as the point v is formed in 5 steps, it will contain this error multiplied five times. Point d would contain it multiplied four times, because it has 4 steps, and so on.

The error will increase in proportion as the diameter of the generating is less than that of the pitch circle, and though in large wheels, working with large wheels (so that the difference between the radius of the generating circle and that of the smallest wheel is not excessive), it is so small as to be practically inappreciable, yet in small wheels, working with large ones, it may form a sensible error.

Fig. 135.

An instrument much employed in the best practice to find the radius which will strike an arc of a circle approximating the true epicycloidal curve, and for finding at the same time the location of the centre wherefrom that curve should be struck, is found in the Willis’ odontograph. This is, in reality, a scale of centres or radii for different and various diameters of wheels and generating circles. It consists of a scale, shown in Fig. 135, and is formed of a piece of sheet metal, one edge of which is marked or graduated in divisions of one-twentieth of an inch. The edge meeting the graduated edge at o is at angle of 75° to the graduated edge.

On one side of the odontograph is a table (as shown in the cut), for the flanks of the teeth, while on the other is the following table for the faces of the teeth:

TABLE SHOWING THE PLACE OF THE CENTRES UPON THE SCALE.

CENTRES FOR THE FACES OF THE TEETH.

Pitch in Inches and Parts.