Modern Machine-Shop Practice Part 14

Each of these parts is marked with the number of teeth the wheel is to contain, and with the pitch of the teeth as shown in Fig. 140, which represents part C full size. Now suppose it is required to find the thickness at the root, for a tooth of a wheel having 60 teeth of one inch pitch, the circles from the point A, pitch line B and root C being drawn, and a radial line representing the middle of the tooth being marked, as is shown in Fig. 142, the compa.s.s points are set to the distance F B, Fig. 140--F being at the junction of line 1 with line 60; the compa.s.ses are then rested at G, and the points H I are marked. Then, from the portion B, Fig. 139 of the diagram, which is shown full-size in Fig. 141, the compa.s.ses may be set to half the thickness at the pitch circle, as in this case (for ordinary teeth) from E to E, and the points J K, Fig. 142, are marked. By a reference to the portion D of the diagram, half the thickness of the tooth at the point is obtained, and marked as at L M in Fig. 142. It now remains to set compa.s.ses to the radius for the face and that for the flank curves, both of which may be obtained from the part A of the diagram. The locations of the centres, wherefrom to strike these curves, are obtained as in Fig. 142. The compa.s.ses set for the face curve are rested at H, and the arc N is struck; they are then rested at J and the arc O struck; and from the intersection of N O, as a centre, the face curve H J is marked. By a similar process, reference to the portion D of the diagram, half the thickness of the tooth at the point is obtained, and marked as at L M in Fig. 142. It now remains to set the compa.s.ses to the radius to strike the respective face and flank curves, and for this purpose the operator turns to the portion A, Fig. 139, of the diagram or scale, and sets the compa.s.ses from the marks on that portion to the required radii.

[Ill.u.s.tration: Fig. 141.]

It now remains to find the proper location from which to strike the curves.

[Ill.u.s.tration: Fig. 142.]

The face curve on the other side of the tooth is struck. The compa.s.ses set to the flank radius is then rested at M, and the arc P is marked and rested at K to mark the arc Q; and from the intersection of P Q, as a centre, the flank curve K M is marked: that on the other side of the tooth being marked in a similar manner.

Additional scales or diagrams, not shown in Fig. 139, give similar distances to set the compa.s.ses for the teeth of internal wheels and racks.

It now remains to explain the method whereby the author of the scale has obtained the various radii, which is as follows: A wheel of 200 teeth was given the form of tooth curve that would be obtained by rolling it upon another wheel, containing 200 teeth of the same pitch. It was next given the form of tooth that would be obtained by rolling upon it a wheel having 10 teeth of the same pitch, and a line intermediate between the two curves was taken as representing the proper curve for the large wheel. The wheel having 10 teeth was then given the form of tooth that would be obtained by rolling upon it another wheel of the same diameter of pitch circle and pitch of teeth. It was next given the form of tooth that would be given by rolling upon it a wheel having 200 teeth, and a curve intermediate between the two curves thus obtained was taken as representing the proper curve for the pinion of 10 teeth. By this means the inventor does not claim to produce wheels having an exactly equal velocity ratio, but he claims that he obtains a curve that is the nearest approximation to the proper epicycloidal curve. The radii for the curves for all other numbers of teeth (between 10 and 200) are obtained in precisely the same manner, the pinion for each pitch being supposed to contain 10 teeth. Thus the scale is intended for interchangeable cast gears.

The nature of the scale renders it necessary to a.s.sume a constant height of tooth for all wheels of the same pitch, and this Mr. Walker has a.s.sumed as .40 of the pitch, from the pitch line to the base, and .35 from the pitch line to the point.

The curves for the faces obtained by this method have rather more curvature than would be due to the true epicycloid, which causes the points to begin and leave contact more easily than would otherwise be the case.

For a pair of wheels Mr. Walker strikes the face curve by a point on the pitch rolling circle, and the flanks by a point on the addendum circle, fastening a piece of wood to the pitch circle to carry the tracing point. The flank of each wheel is struck with a tracing point, thus attached to the pitch circle of the other wheel.

The proportions of teeth and of the s.p.a.ces between them are usually given in turns of the pitch, so that all teeth of a given pitch shall have an equal thickness, height, and breadth, with an equal addendum and flank, and the same amount of clearance.

The term "clearance" as applied to gear-wheel teeth means the amount of s.p.a.ce left between the teeth of one wheel, and the s.p.a.ces in the other, or, in other words, the difference between the width of the teeth and that of the s.p.a.ces between the teeth.

This clearance exists at the sides of the teeth, as in Fig. 143, at A, and between the tops of the teeth and the bottoms or roots of the s.p.a.ces as at B. When, however, the simple term clearance is employed it implies the side clearance as at A, the clearance at B being usually designated as _top and bottom clearance_. Clearance is necessary for two purposes; first, in teeth cut in a machine to accurate form and dimensions, to prevent the teeth of one wheel from binding in the s.p.a.ces of the other, and second, in cast teeth, to allow for the imperfections in the teeth which are incidental to casting in a founder's mould. In machine-cut teeth the amount of clearance is a minimum.

In wheels which are cast with their teeth complete and on the pattern, the amount of clearance must be a maximum, because, in the first place, the teeth on the pattern must be made taper to enable the extraction of the pattern from the mould without damage to the teeth in the mould, and the amount of this taper must be greater than in machine-moulded teeth, because the pattern cannot be lifted so truly vertical by hand as to avoid, in all cases, damage to the mould; in which case the moulder repairs the mould either with his moulding tools and by the aid of the eye, or else with a tooth and a s.p.a.ce made on a piece of wood for the purpose. But even in this case the concentricity of the teeth is scarcely likely to be preserved.

It is obvious that by reason of this taper each wheel is larger in diameter on one side than on the other, hence to preserve the true curves to the teeth the pitch circle is made correspondingly smaller.

But if in keying the wheels to their shafts the two large diameters of a pair of wheels be placed to work together, the teeth of the pair would have contact on that side of the wheel only, and to avoid this and give the teeth contact across their full breadth the wheels are so placed on their shafts that the large diameter of one shall work with the small one of the other, the amount of taper being the same in each wheel irrespective of their relative diameters. This also serves to keep the clearance equal in amount both top, and bottom, and sideways.

A second imperfection is that in order to loosen the pattern in the sand or mould, and enable its extraction by hand from the mould, the pattern requires to be _rapped_ in the mould, the blows forcing back the sand of the mould and thus loosening the pattern. In ordinary practice the amount of this rapping is left entirely to the judgment of the moulder, who has nothing to guide him in securing an equal amount of pattern movement in each direction in the mould; hence, the finished mould may be of increased radius at the circ.u.mference in the direction in which the wheel moved most during the rapping. Again, the wood pattern is apt in time to shrink and become _out of round_, while even iron patterns are not entirely free from warping. Again, the cast metal is liable to contract in cooling more in one direction than in another. The amount of clearance usually allowed for pattern-moulded cast gearing is given by Professor Willis as follows:--Whole depth of tooth 7/10, of the pitch working depth 6/10; hence 1/10 of the pitch is allowed for top and bottom clearance, and this is the amount shown at B in Fig. 143. The amount of side clearance given by Willis as that ordinarily found in practice is as follows:--"Thickness of tooth 5/11 of the pitch; breadth of s.p.a.ce 6/11; hence, the side clearance equals 1/11 of the pitch, which in a 3-inch pitch equals .27 of an inch in each wheel." Calling this in round figures, which is near enough for our purpose, 1/4 inch, we have thickness of tooth 1-1/4, width of s.p.a.ce 1-3/4, or 1/2 inch of clearance in a 3-inch pitch, an amount which on wheels of coa.r.s.e pitch is evidently more than that necessary in view of the accuracy of modern moulding, however suitable it may have been for the less perfect practice of Professor Willis's time. It is to be observed that the rapping of the pattern in the founder's mould reduces the thickness of the teeth and increases the width of the s.p.a.ces somewhat, and to that extent augments the amount of side clearance allowed on the pattern, and the amount of clearance thus obtained would be nearly sufficient for a small wheel, as say of 2 inches diameter. It is further to be observed that the amount of rapping is not proportionate to the diameter of the wheel; thus, in a wheel of 2 inches diameter, the rapping would increase the size of the mould about 1/32 inch. But in the proportion of 1/32 inch to every 2 inches of diameter, the rapping on a 6-foot wheel would amount to 1-1/16 inches, whereas, in actual practice, a 6-foot wheel would not enlarge the mould more than at most 1/8 inch from the rapping.

[Ill.u.s.tration: Fig. 143.]

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 f.l.a.n.g.es 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 s.p.a.ce 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.

[Ill.u.s.tration: Fig. 144. Scale of Proportions given by Willis]

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 compa.s.ses 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 coa.r.s.est 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.

[Ill.u.s.tration: Fig. 145.]

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 compa.s.ses, 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 s.p.a.ces 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.

[Ill.u.s.tration: Fig. 146.]

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 circ.u.mference 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.

[Ill.u.s.tration: Fig. 147.]

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 s.p.a.cing 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.

[Ill.u.s.tration: Fig. 148.]

This, however, is not absolutely necessary so long as the joints are so arranged as to occur in the middle of tooth s.p.a.ces, 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.

[Ill.u.s.tration: Fig. 149.]

When the teeth of wheels are to be cut in a gear-cutting machine the accurate s.p.a.cing 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 s.p.a.ces 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 s.p.a.ces lies the first requisite of a good gear-wheel.

It is a very difficult matter to set the compa.s.ses so that by commencing at any one point and stepping the compa.s.ses around the circle continuously in one direction, the compa.s.s point shall fall into the precise point from which it started, for if the compa.s.s point be set the 1-200th inch out, the last s.p.a.ce 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 compa.s.s 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 compa.s.s points are apt to spring and fall into the false marks when those marks come close to the true ones.

[Ill.u.s.tration: Fig. 150.]

In Fig. 151 is shown a construction by means of which the compa.s.s points may be set more nearly than by dividing the circ.u.mference of the circle by the number of divisions it is required to be marked into and setting the compa.s.ses 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.

[Ill.u.s.tration: Fig. 151.]

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 circ.u.mference 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 compa.s.s 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 compa.s.ses to divide the circle properly. The compa.s.ses being set to this distance G, we may rest one compa.s.s 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.

[Ill.u.s.tration: Fig. 152.]

A pair of compa.s.ses 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 compa.s.s points on the lines and on the points marked, hence it is necessary to step the compa.s.ses 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 compa.s.s 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.

[Ill.u.s.tration: Fig. 153.]

It will be, perhaps, observed by the reader that it would be more expeditious, and perhaps cause less variation, were we to set the compa.s.ses 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 compa.s.ses 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 circ.u.mferences as the faces of pulleys or on the outsides of small rings or cylinders, in which cases the use of compa.s.ses is impracticable. The experienced hand may, it is true, adjust the compa.s.ses as instructed, and mark off three or four of the marks B, C, &c., in Fig. 152, and then open out the compa.s.ses 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 compa.s.ses are adjusted, three or four s.p.a.ces, 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 compa.s.ses 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 circ.u.mferential work of small diameter and in which the distance between two of the adjacent points is, at the most, 1/20 of the circ.u.mference of the circle; because the angle of the surface of the metal to the compa.s.s point causes the latter to spring wider open in consequence of the pressure necessary to cause the compa.s.s 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 compa.s.ses.