Chapter III: THE TEETH OF GEAR-WHEELS (continued) (3)

It is obvious, then, that it would be more in accordance with the requirements to proportion the amount of clearance to the diameter of the wheel, so as to keep the clearance as small as possible. This will possess the advantage that the teeth will be stronger, it being obvious that the teeth are weakened both from the loss of thickness and the increase of height due to the clearance.

It is usual in epicycloidal teeth to fill in the corner at the root of the tooth with a fillet, as at C, D, in Fig. 143, to strengthen it. This is not requisite when the diameter of the generating circle is so small in proportion to the base circle as to produce teeth that are spread at the roots; but it is especially advantageous when the teeth have radial flanks, in which case the fillets may extend farther up the flanks than when they are spread; because, as shown in Fig. 47, the length of operative flank is a minimum in teeth having radial flanks, and as the smallest pinion in the set is that with radial flanks, and further as it has the least number of teeth in contact, it is the weakest, and requires all the strengthening that the fillets in the corners will give, and sometimes the addition of the flanges on the sides of the pinion, such gears being termed "shrouded."

The proportion of the teeth to the pitch as found in ordinary practice is given by Professor Willis as follows:--

Depth to pitch line 3/10 of the pitch.
Working depth 6/10 " "
Whole depth 7/10 " "
Thickness of tooth 5/11 " "
Breadth of space 6/11 " "

The depth to pitch line is, of course, the same thing as the height of the addendum, and is measured through the centre of the tooth from the point to the pitch line in the direction of a radial line and not following the curve of tooth face.

Referring to the working depth, it was shown in Figs. 42 and 44 that the height of the addendum remaining constant, it varies with the diameter of the generating circle.

From these proportions or such others as may be selected, in which the proportions bear a fixed relation to the pitch, a scale may be made and used as a gauge, to set the compasses by, and in marking off the teeth for any pitch within the capacity of the scale. A vertical line A B in Fig. 144, is drawn and marked off in inches and parts of an inch, to represent the pitches of the teeth; at a right angle to A B, the line B C is drawn, its length equalling the whole depth of tooth, which since the coarsest pitch in the scale is 4 inches will be 7/10 of 4 inches. From the end of line C we draw a diagonal line to A, and this gives us the whole depth of tooth for any pitch up to 4 inches: thus the whole depth for a 4-inch pitch is the full length of the horizontal line B C; the whole depth for a 3-inch pitch will be the length of the horizontal line running from the 3 on line A B, to line A C on the right hand of the figure; similarly for the full depth of tooth for a 2-inch pitch is the length of the horizontal line running from 2 to A C. The working depth of tooth being 6/10 of the pitch a diagonal is drawn from A meeting line C at a distance from B of 6/10 of 4 inches and we get the working depth for any other pitch by measuring (along the horizontal line corresponding to that pitch), from the line of pitches to the diagonal line for working depth of tooth. The thickness of tooth is 5/11 of the pitch and its diagonal is distant 5/11 of 4 (from B) on line B C, the thickness for other pitches being obtained on the horizontal line corresponding to those pitches as before.

The construction of a pattern wherefrom to make a foundry mould, in which to cast a spur gear-wheel, is as shown in section, and in plan of Fig. 145. The method of constructing these patterns depends somewhat on their size. Large patterns are constructed with the teeth separate, and the body of the wheel is built of separate pieces, forming the arms, the hub, the rim, and the teeth respectively. Pinion patterns, of six inches and less in diameter, are usually made out of a solid piece, in which case the grain of the wood must lie in the direction of the teeth height. The chuck or face plate of the lathe, for turning the piece, must be of smaller diameter than the pinion, so that it will permit access to a tool applied on both sides, so as to strike the pitch circle on both sides. A second circle is also struck for the roots or depths of the teeth, and also, if required, an extra circle for striking the curves of the teeth with compasses, as was described in Fig. 130. All these circles are to be struck on both sides of the pattern, and as the pattern is to be left slightly taper, to permit of its leaving the mould easily, they must be made of smaller diameter on one side than on the other of the pattern; the reduction in diameter all being made on the same side of the pattern. The pinion body must then be divided off on the pitch line into as many equal divisions as there are to be teeth in it; the curves of the teeth are then marked by some one of the methods described in the remarks on curves of gear-teeth. The top of the face curves are then marked along the points of the teeth by means of a square and scribe, and from these lines the curves are marked in on the other side of the pinion, and the spaces cut out, leaving the teeth projecting. For a larger pinion, without arms, the hub or body is built up of courses of quadrants, the joints of the second course _breaking joint_ with those of the first.

The quadrants are glued together, and when the whole is formed and the glue dry, it is turned in the lathe to the diameter of the wheel at the roots of the teeth. Blocks of wood, to form the teeth, are then planed up, one face being a hollow curve to fit the circle of the wheel. The circumference of the wheel is divided, or pitched off, as it is termed, into as many points of equal division as there are to be teeth, and at these points lines are drawn, using a square, having its back held firmly against the radial face of the pinion, while the blade is brought coincidal with the point of division, so as to act as a guide in converting that point into a line running exactly true with the pinion. All the points of division being thus carried into lines, the blocks for the teeth are glued to the body of the pinion, as denoted by A, in Fig. 145. Another method is to dovetail the teeth into the pinion, as in Fig. 145 at B. After the teeth blocks are set, the process is, as already described, for a solid pinion.

The construction of a wheel, such as shown in Fig. 145, is as follows: The rim R must be built up in segments, but when the courses of segments are high enough to reach the flat sides of the arms they should be turned in the lathe to the diameter on the inside, and the arms should be let in, as shown in the figure at O. The rest of the courses of segments should then be added. The arms are then put in, and the inside of the segments last added may then be turned up, and the outside of the rim turned. The hub should then be added, one-half on each side of the arms, as in the figure. The ribs C of the arms are then added, and the body is completed (ready to receive the teeth), by filleting in the corners. An excellent method of getting out the teeth is as follows: Shape A piece of hard wood, as in Fig. 146, making it some five or six inches longer than the teeth, and about three inches deeper, the thickness being not less than the thickness of the required teeth at the pitch line. Parallel to the edge B C, mark the line A D, distant from B C to an amount equal to the required depth of tooth. Mark off, about midway of the piece, the lines A B and C D, distant from each other to an amount equal to the breadth of the wheel rim, and make two saw cuts to those lines. Take a piece of board an inch or two longer than the radius of the gear-wheel and insert a piece of wood (which is termed a box) tightly into the board, as shown in Fig. 147, E representing the box. Let the point F on the board represent the centre of the wheel, and draw a radial line R from F through the centre of the box. From the centre F, with a trammel, mark the addendum line G G, pitch line H I, and line J K for the depth of the teeth (and also a line wherefrom to strike the teeth curves, as shown in Fig. 129 if necessary). From the radial line R, as a centre, mark off on the pitch circle, points of division for several teeth, so as to be able to test the accuracy of the spacing across the several points, as well as from one point to the next, and mark the curves for the teeth on the end of the box, as shown. Turn the box end for end in the board, and mark out a tooth by the same method on the other end of the box. The box being removed from the board must now have its sides planed to the lines, when it will be ready to shape the teeth in. The teeth are got out for length, breadth, and thickness at the pitch line as follows: The lumber from which they are cut should be very straight grained, and should be first cut into strips of a width and thickness slightly greater than that of the teeth at the pitch line. These strips (which should be about two feet long) should then be planed down on the sides to very nearly the thickness of the tooth at the pitch line, and hollow on one edge to fit the curvature of the wheel rim. From these strips, pieces a trifle longer than the breadth of the wheel rim are cut, these forming the teeth. The pieces are then planed on the ends to the exact width of the wheel rim. To facilitate this planing a number of the pieces or blank teeth may be set in a frame, as in Figs. 148 and 149, in which A is a piece having the blocks B B affixed to it. C is a clamp secured by the screws at S S, and 1, 2, 3, 4, 5, 6 are the ends of the blank teeth. The clamp need not be as wide as the teeth, as in Fig. 148, but it is well to let the pieces A and B B equal the breadth of the wheel rim, so that they will act as a template to plane the blank teeth ends to. The ends of B B may be blackleaded, so as to show plainly if the plane blade happens to shave them, and hence to prevent planing B B with the teeth. The blank teeth may now be separately placed in the box (Fig. 146) and secured by a screw, as shown in that figure, in which S is the screw, and T the blank tooth. The sides of the tooth must be carefully planed down equal and level with the surface of the box. The rim of the wheel, having been divided off into as many divisions as there are to be teeth in the wheel, as shown in Fig. 150, at _a_, _a_, _a_, &c., the finished teeth are glued so that the same respective side of each tooth exactly meets one of the lines _a_. Only a few spots of glue should be applied, and these at the middle of the root thickness, so that the glue shall not exude and hide the line _a_, which would make it difficult to set the teeth true to the line. When the teeth are all dry they must be additionally secured to the rim by nails. Wheels sufficiently large to incur difficulty of transportation are composed of a number of sections, each usually consisting of an arm, with an equal length of the rim arc on each side of it, so that the joint where the rim segments are bolted together will be midway between the two arms.

This, however, is not absolutely necessary so long as the joints are so arranged as to occur in the middle of tooth spaces, and not in the thickness of the tooth. This sometimes necessitates that the rim sections have an unequal length of arc, in which event the pattern is made for the longest segment, and when these are cast the teeth superfluous for the shorter segments are stopped off by the foundry moulder. This saves cutting or altering the pattern, which, therefore, remains good for other wheels when required.

When the teeth of wheels are to be cut in a gear-cutting machine the accurate spacing of the teeth is determined by the index plate and gearing of the machine itself; but when the teeth are to be cast upon the wheel and a pattern is to be made, wherefrom to cast the wheel the points of division denoting the thickness of the teeth and the width of the spaces are usually marked by hand. This is often rendered necessary from the wheels being of too large a diameter to go into dividing machines of the sizes usually constructed.

To accurately divide off the pitch circle of a gear-wheel by hand, requires both patience and skilful manipulation, but it is time and trouble that well repays its cost, for in the accuracy of spaces lies the first requisite of a good gear-wheel.

It is a very difficult matter to set the compasses so that by commencing at any one point and stepping the compasses around the circle continuously in one direction, the compass point shall fall into the precise point from which it started, for if the compass point be set the 1-200th inch out, the last space will come an inch out in a circle having 200 points of divisions. It is, therefore, almost impossible and quite impracticable to accurately mark or divide off a circle having many points of division in this manner, not only on account of the fineness of the adjustment of the compass points, but because the frequent trials will leave so many marks upon the circle that the true ones will not be distinguishable from the false. Furthermore, the compass points are apt to spring and fall into the false marks when those marks come close to the true ones.

In Fig. 151 is shown a construction by means of which the compass points may be set more nearly than by dividing the circumference of the circle by the number of divisions it is required to be marked into and setting the compasses to the quotient, because such a calculation gives the length of the division measured around the arc of the circle, instead of the distance measured straight from point of division to point of division.

The construction of Fig. 151 is as follows: P P is a portion of the circle to be divided, and A B is a line at a tangent to the point C of the circle P P. The point D is set off distant from C, to an amount obtained by dividing the circumference of P P by the number of divisions it is to have. Take one-quarter of this distance C D, and mark it from C, giving the point E, set one point of the compass at E and the other at D, and draw the arc D F, and the distance from F to C, as denoted by G, is the distance to which to set the compasses to divide the circle properly. The compasses being set to this distance G, we may rest one compass point at C, and mark the arc F H, and the distance between arc H and arc D, measured on the line A B, is the difference between the points C, F when measured around the circle P P, and straight across, as at G.

A pair of compasses set even by this construction will not, however, be entirely accurate, because there will be some degree of error, even though it be in placing the compass points on the lines and on the points marked, hence it is necessary to step the compasses around the circle, and the best method of doing this is as follows: Commencing at A, Fig. 152, we mark off continuously one from the other, and taking care to be very exact to place the compass point exactly coincident with the line of the circle, the points B, C, D, &c., continuing until we have marked half as many divisions as the circle is to contain, and arriving at E, starting again at A, we mark off similar divisions (one half of the total number), F, G, H, arriving at I, and the centre K, between the two lines E, I, will be the true position of the point diametrally opposite to point A, whence we started. These points are all marked inside the circle to keep them distinct from those subsequently marked.

It will be, perhaps, observed by the reader that it would be more expeditious, and perhaps cause less variation, were we to set the compasses to the radius of the circle and mark off the point K, as shown in Fig. 153, commencing at the point A, and marking off on the one side the lines B, C, and D, and on the other side E, F, and G, the junction or centre, between G and D, at the circle being the true position of the point K. For circles struck upon flat surfaces, this plan may be advantageous; and in cases where there are not at hand compasses large enough, a pair of trammels may be used for the purpose; but our instructions are intended to apply also to marking off equidistant points on such circumferences as the faces of pulleys or on the outsides of small rings or cylinders, in which cases the use of compasses is impracticable. The experienced hand may, it is true, adjust the compasses as instructed, and mark off three or four of the marks B, C, &c., in Fig. 152, and then open out the compasses to the distance between the two extreme marks, and proceed as before to find the centre K, but as a rule, the time saved will scarcely repay the trouble; and all that can be done to save time in such cases is, if the holes come reasonably close together, to mark off, after the compasses are adjusted, three or four spaces, as shown in Fig. 154. Commencing at the point A, and marking off the points B, C, and D, we then set another pair of compasses to the distance between A and D, and then mark, from D on one side and from A on the other, the marks from F to L and from M to T, thus obtaining the point K. This method, however expeditious and correct for certain work, is not applicable to circumferential work of small diameter and in which the distance between two of the adjacent points is, at the most, 1/20 of the circumference of the circle; because the angle of the surface of the metal to the compass point causes the latter to spring wider open in consequence of the pressure necessary to cause the compass point to mark the metal. This will be readily perceived on reference to Fig. 155 in which A represents the stationary, and B the scribing or marking point of the compasses.

The error in the set of the compasses as shown by the distance apart of the two marks E and I on the circle in Fig. 152 is too fine to render it practicable to remedy it by moving the compass legs, hence we effect the adjustment by oilstoning the points on the outside, throwing them closer together as the figure shows is necessary.

Having found the point K, we mark (on the outside of the circle, so as to keep the marks distinct from those first marked) the division B, C, D, Fig. 156, &c., up to G, the number of divisions between B and G being one quarter of those in the whole circle. Then, beginning at K, we mark off also one quarter of the number of divisions arriving at M in the figure and producing the point 3. By a similar operation on the other side of the circle, we get the true position of point No. 4. If, in obtaining points 3 and 4, the compasses are not found to be set dead true, the necessary adjustment must be made; and it will be seen that, so far, we have obtained four true positions, and the process of obtaining each of them has served as a justification of the distance of the compass points. From these four points we may proceed in like manner to mark off the holes or points between them; and the whole will be as true as it is practicable to mark them off upon that size of circle. In cases, however, where mathematical precision is required upon flat and not circumferential surfaces, the marking off may be performed upon a circle of larger diameter, as shown in Fig. 157. If it is required to mark off the circle A, Fig. 157, into any even number of equidistant points, and if, in consequence of the closeness together of the points, it becomes difficult to mark them (as described) with the compasses, we mark a circle B B of larger diameter, and perform our marking upon it, carrying the marks across the smaller circle with a straightedge placed to intersect the centres of the circles and the points marked on each side of the diameter. Thus, in Fig. 157, the lines 1 and 2 on the smaller circle would be obtained from a line struck through 1 and 4 on the outer circle; and supposing the larger circle to be three times the size of the smaller, the deviation from truth in the latter will be only 1/3 of whatever it is in the former.

In this example we have supposed the number of divisions to be an even one, hence the point K, Fig. 152, falls diametrically opposite to A, whereas in an odd number of points of division this would not be the case, and we must proceed by either of the two following methods:--

In Fig. 158 is shown a circle requiring to be divided by 17 equidistant points. Starting from point 1 we mark on the outside of the circumference points 2, 3, 4, &c., up to point 9. Starting again from point 1 we mark points 10, 11, &c., up to 17. If, then, we try the compasses to 17 and 9 we shall find they come too close together, hence we take another pair of compasses (so as not to disturb the set of our first pair) and find the centre between 9 and 17 as shown by the point A. We then correct the set of our first pair of compasses, as near as the judgment dictates, and from point A, we mark with the second compasses (set to one half the new space of the first compasses) the points B, C. With the first pair of compasses, starting from B, we mark D, E, &c., to G; and from I, we mark divisions H, I, &c., to K, and if the compasses were set true, K and G would meet at the circle. We may, however, mark a point midway between K and G, as at 5. Starting again from points C and I, we mark the other side of the circle in a similar manner, producing the lines P and Q, midway between which (the compasses not being set quite correct as yet) is the true point for another division. After again correcting the compasses, we start from B and 5 respectively, and mark point 7, again correcting the compasses. Then from C and the point between P and Q, we may mark an intermediate point, and so on until all the points of division are made. This method is correct enough for most practical purposes, but the method shown in Fig. 159 is more correct for an odd number of points of division. Suppose that we have commenced at the point marked I, we mark off half the required number of holes on one side and arrive at the point 2; and then, commencing at the point I again, we mark off the other half of the required number of holes, arriving at the point 3. We then apply our compasses to the distance between the points 2 and 3; and if that distance is not exactly the same to which the compasses are set, we make the necessary adjustment, and try again and again until correct adjustment is secured.

It is highly necessary, in this case, to make the lines drawn at each trial all on the same side of the circle and of equal length, but of a different length to those marked on previous trials. For example, left the lines A, B, C, D, in Fig. 159 represent those made on the first trial, and E, F, G, H, those made on the second trial; and when the adjustment is complete, let the last trial be made upon the outside or other side of the circle, as shown by the lines I, J, K, L. Having obtained the three true points, marked 1, 2, 3, we proceed to mark the intermediate divisions, as described for an even number of divisions, save that there will be a space, 2 and 3, opposite point 1, instead of a point, as in case of a circle having an even number of divisions.

The equal points of division thus obtained may be taken for the centres of the tooth at the pitch circle or for one side of the teeth, as the method to be pursued to mark the tooth curves may render most desirable. If, for example, a template be used to mark off the tooth curves, the marks may be used to best advantage as representing the side of a tooth, and from them the thickness of the tooth may be marked or not as the kind of template used may require. Thus, if the template shown in Fig. 21 be used, no other marks will be used, because the sides of a tooth on each side of a space may be marked at one setting of the template to the lines or marks of division. If, however, a template, such as shown in Fig. 81 be used, a second set of lines marked distant from the first to a radius equal to the thickness of a tooth becomes necessary so that the template may be set to each line marked. If the Willis odontograph or the Robinson template odontograph be used the second set of lines will also be necessary. In using the Walker scale a radial line, as G in Fig. 142, will require to be marked through the points of equal division, and the thickness of the tooth at the points on the pitch circle and at the root must be marked as was shown in Fig. 142.

But if the arcs for the tooth curves are to be marked by compasses, the location for the centres wherefrom to strike these arcs may be marked from the points of division as was shown in Fig. 130.

To construct a pattern wherefrom to cast a bevel gear-wheel.--When a pair of bevel-wheels are in gear and upon their respective shafts all the teeth on each wheel incline, as has been shown, to a single point, hence the pattern maker draws upon a piece of board a sketch representing the conditions under which the wheels are to operate. A sketch of this kind is shown in Fig. 160, in which A, B, C, D, represent in section the body of a bevel pinion. F G is the point of a tooth on one side, and E the point of a tooth on the other side of the pinion, while H I are pitch lines for the two teeth. Thus, the cone surface, the points, the pitch lines and the bottom of the spaces, projected as denoted by the dotted lines, would all meet at X, which represents the point where the axes of the shafts would meet.

In making wooden patterns wherefrom to cast the wheels, it is usual, therefore, to mark these lines on a drawing-board, so that they may be referred to by the workman in obtaining the degree of cone necessary for the body A B C D, to which the teeth are to be affixed. Suppose, then, that the diameter of the pinion is sufficiently small to permit the body A B C D to be formed of one piece instead of being put together in segments, the operation is as follows: The face D C is turned off on the lathe, and the piece is reversed on the lathe chuck, and the face A B is turned, leaving a slight recess at the centre to receive and hold the cone point true with the wheel. A bevel gauge is then set to the angle A B C, and the cone of the body is turned to coincide in angle with the gauge and to the required diameter, its surface being made true and straight so that the teeth may bed well. While turning the face D C in the lathe a fine line circle should be struck around the circumference of the cone and near D C, on which line the spacing for the teeth may be stepped off with the compasses. After this circle or line is divided off into as many equidistant points as there are to be teeth on the wheel, the points of division require to be drawn into lines, running across the cone surface of the wheel, and as the ordinary square is inapplicable for the purpose, a suitable square is improvised as follows: In Fig. 161 let the outline in full lines denote the body of a pinion ready to receive the teeth, and A B the circle referred to as necessary for the spacing or dividing with the compasses. On A B take any point, as C, as a centre, and with a pair of compasses mark equidistant on each side of it two lines, as D, D. From D, D as respective centres mark two lines, crossing each other as at F, and draw a line, joining the intersection of the lines at F with C, and the last line, so produced, will be in the place in which the teeth are to lie; hence the wheel will require as many of these lines as it is to contain teeth, and the sides of the teeth, being set to these lines all around the pinion, will be in their proper positions, with the pitch lines pointing to X, in Fig. 160.

To avoid, however, the labor involved in producing these lines for each tooth, two other plans may be adopted. The first is to make a square, such as shown in Fig. 162, the face _f_ _f_ being fitted to the surface C, in Fig. 161, while the edges of its blade coincide with the line referred to; hence the edge of the blade may be placed coincident successively with each point of division, as D D, and the lines for the place of the length of each tooth be drawn. The second plan is to divide off the line A B before removing the body of the pinion from the lathe, and produce, as described, a line for one tooth. A piece of wood may then be placed so that when it lies on the surface of the hand-rest its upper surface will coincide with the line as shown in Fig. 163, in which W is the piece of wood, and A, B, C, &c., the lines referred to. If the teeth are to be glued and bradded to the body, they are first cut out in blocks, left a little larger every way than they are to be when finished, and the surfaces which are to bed on the cone are hollowed to fit it. Then blocks are glued to the body, one and the same relative side of each tooth being set fair to the lines. When the glue is dry, the pinion is again turned on the lathe, the gauge for the cone of the teeth being set in this case to the lines E, F, G in Fig. 160. The pitch circles must then be struck at the ends of the teeth. The turned wheel is then ready to have the curves of the teeth marked. The wheel must now again be divided off on the pitch circle at the large end of the cone into as many equidistant points as there are to be teeth on the wheel, and from these points, and on the same relative side of them, mark off a second series of points, distant from the points of division to an amount equal to the thickness the teeth are required to be. From these points draw in the outline of the teeth (upon the ends of the blocks to form the teeth) at the large end of the cone. Then, by use of the square, shown in Fig. 162, transfer the points of the teeth to the small end of the cone, and trace the outline of the teeth at the small end, taking centres and distances proportionate to the reduced diameter of the pitch circle at the small end, as shown in Fig. 160, where at J are three teeth so marked for the large end, and at K three for the small end, P P representing the pitch circle, and R R a circle for the compass points. The teeth for bevel pinions are sometimes put on by dovetails, as shown in Fig. 164, a plan which possesses points of advantage and disadvantage. Wood shrinks more across the grain than lengthwise with it, hence when the grain of the teeth crosses that of the body with every expansion or contraction of the wood (which always accompanies changes in the humidity of the atmosphere) there will be a movement between the two, because of the unequal expansion and contraction, causing the teeth to loosen or to move. In the employment of dovetails, however, a freedom of movement lengthways of the tooth is provided to accommodate the movement, while the teeth are detained in their proper positions. Again, if in making the founders' mould, one of the mould teeth should break or fall down when the pattern is withdrawn, a tooth may be removed from the pattern and used by the moulder to build up the damaged part of the mould again. And if the teeth of a bevel pinion are too much undercut on the flank curves to permit the whole pattern from being extracted from the mould without damaging it, dovetailed teeth may be drawn, leaving the body of the pattern to be extracted from the mould last. On the other hand, the dovetail is a costly construction if applied to large wheels. If the teeth are to be affixed by dovetails, the construction varies as follows: Cut out a wooden template of the dovetail, leaving it a little narrower than the thickness of the tooth at the root, and set the template on the cone at a distance from one of the lines A, B, C, Fig. 163, equal to the margin allowed between the edge of the dovetail and the side of the root of the tooth, and set it true by the employment of the square, shown in Fig. 162, and draw along the cone surface of the body lines representing the location of the dovetail grooves. The lines so drawn will give a taper toward X (Fig. 160), providing that, the template sides being parallel, each side is set to the square. While the body is in the lathe, a circle on each end may be struck for the depth of the dovetails, which should be cut out to gauge and to template, so that the teeth will interchange to any dovetail. The bottom of the dovetails need not be circular, but flat, which is easier to make. Dovetail pieces or strips are fitted to the grooves, being left to project slightly above the face of the cone or body. They are drawn in tight enough to enable them to keep their position while being turned in the lathe when the projecting points are turned down level with the cone of the body. The teeth may then be got out as described for glued teeth, and the dovetails added, each being marked to its place, and finally the teeth are cut to shape.

In wheels too large to have their cones tested by a bevel gauge, a wooden gauge may be made by nailing two pieces of wood to stand at the required angle as shown in Fig. 165, which is extracted from _The American Machinist_, or the dead centre C and a straightedge may be used as follows. In the figure the other wheel of the pair is shown dotted in at B, and the dead centre is set at the point where the axes of A and B would meet; hence if the largest diameter of the cone of A is turned to correct size, the cone will be correct when a straightedge applied as shown lies flat on the cone and meets the point of the dead centre E. The pinion B, however, is merely introduced to explain the principle, and obviously could not be so applied practically, the distance to set _e_, however, is the radius _a_.

Skew Bevel.[9]--When the axles of the shaft are inclined to each other instead of being in a straight line, and it is proposed to connect and communicate motion to the shafts by means of a single pair of bevel-gears, the teeth must be inclined to the base of the frustra to allow them to come into contact.

[9] From the "Engineer and Machinists' Assistant."

To find the line of contact upon a given frustrum of the tangent-cone; let the Fig. 166 be the plane of the frustrum; _a_ the centre. Set off _a_ _e_ equal to the shortest distance between the axes (called the _eccentricity_), and divide it in _c_, so that _a_ _c_ is to _e_ _c_ as the mean radius of the frustrum to the mean radius of that with which it is to work; draw _c_ _p_ perpendicular to _a_ _e_, and meeting the circumference of the conical surface at _m_; perform a similar operation on the base of the frustrum by drawing a line parallel to _c_ _m_ and at the same distance _a_ _c_ from the centre, meeting the circumference in _p_.

The line _p_ _c_ is then plainly the line of direction of the teeth. We are also at liberty to employ the equally inclined line _c_ _q_ in the opposite direction, observing only that, in laying out the two wheels, the pair of directions be taken, of which the inclinations correspond.

Fig. 167 renders this mode of laying off the outlines of the wheels at once obvious. In this figure the line _a_ _e_ corresponds to the line marked by the same letters in Fig. 166; and the division of it at _c_ is determined in the manner directed. The line _c_ _m_ being thus found in direction, it is drawn indefinitely to _d_. Parallel to this line and from the point _c_ draw _e_ to _e_, and in this line take the centre of the second wheel. The line _c_ _m_ _d_ gives the direction of the teeth; and if from the centre _a_ with radius at _c_ a circle be described, the direction of any tooth of the wheel will be a tangent to it, as at _c_, and similarly if a centre _e_ be taken in the line _e_ _d_, and with radius _e_ _d_, _c_ _e_ a circle be drawn, the direction of the teeth of the second wheel will be tangents to this last, as at _d_.

Having thus found the direction of the teeth, these outlines may be formed as in the case of ordinary bevel-wheels and with equal exactness and facility, all that is necessary being to find the curves for the teeth as described for bevel-wheels, and follow precisely the same construction, except that the square, Fig. 162, marking the lines across the cones, requires to be set to the angle for the tooth instead of at a right angle, and this angle may be found by the construction shown in Fig. 167, it being there represented by line _d_ _c_. It is obvious, however, that the bottoms of the blocks to form the teeth must be curved to bed on the cone along the line _d_ _c_, Fig. 167, and this may best be done by bedding two teeth, testing them by trial of the actual surfaces.

Then two teeth may be set in as No. 1 and No. 6 in the box shown in Fig. 148, the intermediate ones being dressed down to them.

Where a bevel-wheel pattern is too large to be constructed in one piece and requires to be built up in pieces, the construction is as in Fig. 168, in which on the left is shown the courses of segments 1, 2, 3, 4, 5, &c., of which the rim is built up (as described for spur wheels), and on the right is shown the finished rim with a tooth, _c_, in position.

The tooth proper is of the length of face of the wheel as denoted by _b b´_; now all the lines bounding the teeth must converge to the point X. Suppose, then, that the teeth are to be shaped for curve of face and flank in a box as described for spur-wheel teeth in Fig. 146, then in Fig. 168 let _a_, _a_ represent the bottom and _b b´_ the top of the box, and _c_ a tooth in the box, its ends filling the opening in the box at _b b´_ then the curve on the sides of the box at _b´_ must be of the form shown at F, and the curve on the sides of the box (at the point _b_ of its length) must be as shown at G, the teeth shown in profile at G and U representing the forms of the teeth at their ends, on the outside of the wheel rim at _b´_, and on the inside at _b_; having thus made a box of the correct form on its sides, the teeth may be placed in it and planed down to it, thus giving all the teeth the same curve.

The spacing for the teeth and their fixing may be done as described for the bevel pinion.

To construct a pattern wherefrom to cast an endless screw, worm, or tangent screw, which is to have the worm or thread cut in a lathe.--Take two pieces, each to form one longitudinal half of the pattern; peg and screw them together at the ends, an excess of stuff being allowed at each end for the accommodation of two screws to hold the two halves together while turning them in the lathe, or dogs, if the latter are more convenient, as they might be in a large pattern. Turn the piece down to the size over the top of the thread, after which the core prints are turned. The body thus formed will be ready to have the worm or thread cut, and for this purpose the tools shown in Figs. 169 and 140 are necessary.

That shown in Fig. 169 should be flat on the face similar to a parting tool for cast iron, but should have a great deal more bottom rake, as strength is not so much an object, and the tool is more easily sharpened. It has also in addition two little projections A B like the point of a penknife, formed by filing away the steel in the centre; these points are to cut the fibres of the wood, the severed portion being scraped away by the flat part of the tool.

The degree of side rake given to the tool must be sufficient to let the tool sides well clear the thread or worm, and will therefore vary with the pitch of the worm.

The width of the tool must be a shade narrower than the narrowest part of the space in the worm. Having suitably adjusted the change wheels of the lathe to cut the pitch required the parting tool is fed in until the extreme points reach the bottom of the spaces, and a square nosed parting tool without any points or spurs will finish the worm to the required depth. This will have left a square thread, and this we have now to cut to the required curves on the thread or worm sides, and as the cutting will be performed on the end grain of the wood, the top face of the tool must be made keen by piercing through the tool a slot A, Fig. 170, and filing up the bevel faces B, C and D, and then carefully oilstoning them. This tool should be made slightly narrower than the width of the worm space, so that it may not cut on both sides at once, as it would have too great a length of cutting edge.

Furthermore, if the pattern is very large, it will be necessary to have two tools for finishing, one to cut from the pitch line inwards and the other to complete the form from the pitch line outwards. It is advisable to use hard wood for the pattern.

If it is decided to cut the thread by hand instead of with these lathe tools, then, the pattern being turned as before, separate the two halves by taking out the screws at the ends; select the half that has not the pegs, as being a little more convenient for tracing lines across. Set out the sections of the thread, A, B, C, and D, Fig. 171, similar to a rack; through the centres of A, B, C, and D, square lines across the piece; these lines, where they intersect the pitch line, will give the centres of teeth on that side: or if we draw lines, as E, F, through the centres of the spaces, they will pass through the centres of the teeth (so to speak) on the other side; in this position complete the outline on that side. It will be found, in drawing these outlines, that the centres of some of the arcs will lie outside the pattern. To obtain support for the compasses, we must fit over the pattern a piece of board such as shown by dotted lines at G H.

It now remains to draw in the top of the thread upon the curved surface of the half pattern; for this purpose take a piece of stiff card or other flexible material, wrap it around the pattern and fix it temporarily by tacks, we then trim off the edges true to the pattern, and mark upon the edges of the card the position of the tops of the thread upon each side; we remove the card and spread it out on a flat surface, join the points marked on the edges by lines as in Fig. 172, replace the card exactly as before upon the pattern, and with a fine scriber we prick through the lines. The cutting out is commenced by sawing, keeping, of course, well within the lines; and it is facilitated by attaching a stop to the saw so as to insure cutting at all parts nearly to the exact depth. This stop is a simple strip of wood and may be clamped to the saw, though it is much more convenient to have a couple of holes in the saw blade for the passage of screws. For finishing, a pair of templates, P and Q, Fig. 173, right and left, will be found useful; and finally the work should be verified and slight imperfections corrected by the use of a form or template taking in three spaces, as shown at R in Fig. 173. In drawing the lines on the card, we must consider whether it is a right or left-handed worm that we desire. In the engraving the lines are those suitable for a right-handed thread. Having completed one half of the pattern, place the two halves together, and trace off the half that is uncut, using again the card template for drawing the lines on the curved surface. The cutting out will be the same as before.

As the teeth of cast wheels are, from their deviation from accuracy in the tooth curves and the concentricity of the teeth to the wheel centre, apt to create noise in running, it is not unusual to cast one or both wheels with mortises in the rim to receive wooden teeth. In this case the wheel is termed a mortise wheel, and the teeth are termed _cogs_. If only one of a pair of wheels is to be cogged, the largest of the pair is usually selected, because there are in that case more teeth to withstand the wear, it being obvious that the wear is greatest upon the wheel having the fewest teeth, and that the iron wheel or pinion can better withstand the wear than the mortise wheel. The woods most used for cogs are hickory, maple, hornbeam and locust. The blocks wherefrom the teeth are to be formed are usually cut out to nearly the required dimensions, and kept in stock, so as to be thoroughly well-seasoned when required for use, and, therefore less liable to come loose from shrinkage after being fitted to the mortise in the wheel. The length of the shanks is made sufficient to project through the wheel rim and receive a pin, as shown in Fig. 174, in which B is a blank tooth, and C a finished tooth inserted in the wheel, the pin referred to being at P. But, if a mortise should fall in an arm of the wheel, this pin-hole must pass through the rim, as shown in the mortise A. The wheel, however, should be designed so that the mortises will not terminate in the arms of the wheel.

Another method of securing the teeth in the mortises is to dovetail them at the small end and drive wedges between them, as shown in Fig. 175, in which C C are two contiguous teeth, R the wheel rim and W W two of the wedges. On account of the dovetailing the wedges exert A pressure pressing the teeth into the mortises. This plan is preferable to that shown in the Fig. 174 inasmuch as from the small bearing area of the pins they become loose quicker, and furthermore there is more elasticity to take up the wear in the case of the wedges.

The mortises are first dressed out to a uniform size and taper, using two templates to test them with, one of which is for the breadth and the other for the width of the mortise. The height above the wheel requires to be considerably more than that due to the depth of the teeth, so that the surface bruised by driving the cogs or when fitting them into the mortises may be cut off. To avoid this damage as much as possible, a broad-face hammer should be employed--a copper, lead, lignum vitæ, or a raw hide hammer being preferable, and the last the best. The teeth are got out in a box and two guides, such as shown in Figs. 176, 177, and 178, similar letters of reference denoting the same parts in all three illustrations.