Modern Machine-Shop Practice Part 19

"But these circular arcs may be rectified and subdivided with great facility and accuracy by a very simple process, which we take from Prof.

Rankine's "Machinery and Mill Work," and is ill.u.s.trated in Fig. 195. Let O B be tangent at O to the arc O D, of which C is the centre. Draw the chord D O, bisect it in E, and produce it to A, making O A = O E; with centre A and radius A D describe an arc cutting the tangent in B; then O B will be very nearly equal in length to the arc O D, which, however, should not exceed about 60; if it be 60, the error is theoretically about 1/900 of the length of the arc, O B being so much too short; but this error varies with the fourth power of the angle subtended by the arc, so that for 30 it is reduced to 1/16 of that amount, that is, to 1/14400. Conversely, let O B be a tangent of given length; make O F = 1/4 O B; then with centre F and radius F B describe an arc cutting the circle O D G (tangent to O B at O) in the point D; then O D will be approximately equal to O B, the error being the same as in the other construction and following the same law.

[Ill.u.s.tration: Fig. 195.]

"The extreme simplicity of these two constructions and the facility with which they may be made with ordinary drawing instruments make them exceedingly convenient, and they should be more widely known than they are. Their application to the present problem is shown in Fig. 196, which represents a quadrant of an ellipse, the approximate arcs C D, D E, E F, F A having been determined by trial and error. In order to s.p.a.ce this off, for the positions of the teeth, a tangent is drawn at D, upon which is constructed the rectification of D C, which is D G, and also that of D E in the opposite direction, that is, D H, by the process just explained. Then, drawing the tangent at F, we set off in the same manner F I = F E, and F K = F A, and then measuring H L = I K, we have finally G L, equal to the whole quadrant of the ellipse.

[Ill.u.s.tration: Fig. 196.]

"Let it now be required to lay out 24 teeth upon this ellipse; that is, 6 in each quadrant; and for symmetry's sake we will suppose that the centre of one tooth is to be at A, and that of another at C, Fig. 196.

We therefore divide L G into six equal parts at the points 1, 2, 3, &c., which will be the centres of the teeth upon the rectified ellipse. It is practically necessary to make the s.p.a.ces a little greater than the teeth; but if the greatest attainable exactness in the operation of the wheel is aimed at, it is important to observe that backlash, in elliptical gearing, has an effect quite different from that resulting in the case of circular wheels. When the pitch-curves are circles, they are always in contact; and we may, if we choose, make the tooth only half the breadth of the s.p.a.ce, so long as its outline is correct. When the motion of the driver is reversed, the follower will stand still until the backlash is taken up, when the motion will go on with a perfectly constant velocity ratio as before. But in the ease of two elliptical wheels, if the follower stand still while the driver moves, which must happen when the motion is reversed if backlash exists, the pitch-curves are thrown out of contact, and, although the continuity of the motion will not be interrupted, the velocity ratio will be affected. If the motion is never to be reversed, the perfect law of the velocity ratio due to the elliptical pitch-curve may be preserved by reducing the thickness of the tooth, not equally on each side, as is done in circular wheels, but wholly on the side not in action. But if the machine must be capable of acting indifferently in both directions, the reduction must be made on both sides of the tooth: evidently the action will be slightly impaired, for which reason the backlash should be reduced to a minimum. Precisely what _is_ the minimum is not so easy to say, as it evidently depends much upon the excellence of the tools and the skill of the workmen. In many treatises on constructive mechanism it is variously stated that the backlash should be from one-fifteenth to one-eleventh of the pitch, which would seem to be an ample allowance in reasonably good castings not intended to be finished, and quite excessive if the teeth are to be cut; nor is it very obvious that its amount should depend upon the pitch any more than upon the precession of the equinoxes. On paper, at any rate, we may reduce it to zero, and make the teeth and s.p.a.ces equal in breadth, as shown in the figure, the teeth being indicated by the double lines. Those upon the portion L H are then laid off upon K I, after which these divisions are transferred to curves. And since under that condition the motion of this third line, relatively to each of the others, is the same as though it rolled along each of them separately while they remained fixed, the process of constructing the generated curves becomes comparatively simple. For the describing line, we naturally select a circle, which, in order to fulfil the condition, must be small enough to roll within the pitch ellipse; its diameter is determined by the consideration, that if it be equal to A P, the radius of the arc A F, the flanks of the teeth in that region will be radial.

We have, therefore, chosen a circle whose diameter, A B, is three-fourths of A P, as shown, so that the teeth, even at the ends of the wheels, will be broader at the base than on the pitch line. This circle ought strictly to roll upon the true elliptical curve, and a.s.suming as usual the tracing-point upon the circ.u.mference, the generated curves would vary slightly from true epicycloids, and no two of those used in the same quadrant of the ellipse would be exactly alike. Were it possible to divide the ellipse accurately, there would be no difficulty in laying out these curves; but having subst.i.tuted the circular arcs, we must now roll the generating circle upon these as bases, thus forming true epicycloidal teeth, of which those lying upon the same approximating arc will be exactly alike. Should the junction of two of these arcs fall within the breadth of a tooth, as at D, evidently both the face and the flank on one side of that tooth will be different from those on the other side; should the junction coincide with the edge of a tooth, which is very nearly the case at F, then the face on that side will be the epicycloid belonging to one of the arcs, its flank a hypocycloid belonging to the other; and it is possible that either the face or the flank on one side should be generated by the rolling of the describing circle partly on one arc, partly on the one adjacent, which, upon a large scale and where the best results are aimed at, may make a sensible change in the form of the curve.

[Ill.u.s.tration: Fig. 197.]

"The convenience of the constructions given in Fig. 194 is nowhere more apparent than in the drawing of the epicycloids, when, as in the case in hand, the base and generating circles may be of incommensurable diameters; for which reason we have, in Fig. 197, shown its application in connection with the most rapid and accurate mode yet known of describing those curves. Let C be the centre of the base circle; B that of the rolling one; A the point of contact. Divide the semi-circ.u.mference of B into six equal parts at 1, 2, 3, &c.; draw the common tangent at A, upon which rectify the arc A2 by process No. 1, then by process No. 2 set out an equal arc A2 on the base circle, and stepping it off three times to the right and left, bisect these s.p.a.ces, thus making subdivisions on the base circle equal in length to those on the rolling one. Take in succession as radii the chords A1, A2, A3, &c., of the describing circle, and with centres 1, 2, 3, &c., on the base circle, strike arcs either externally or internally, as shown respectively on the right and left; the curve tangent to the external arcs is the epicycloid, that tangent to the internal ones the hypocycloid, forming the face and flank of a tooth for the base circle.

"In the diagram, Fig. 196, we have shown a part of an ellipse whose length is 10 inches and breadth 6, the figure being half size. In order to give an idea of the actual appearance of the combination when complete, we show in Fig. 198 the pair in gear, on a scale of 3 inches to the foot. The excessive eccentricity was selected merely for the purpose of ill.u.s.tration. Fig. 198 will serve also to call attention to another serious circ.u.mstance, which is that although the ellipses are alike, the wheels are not; nor can they be made so if there be an even number of teeth, for the obvious reason that a tooth upon one wheel must fit into a s.p.a.ce on the other; and since in the first wheel, Fig. 196, we chose to place a tooth at the extremity of each axis, we must in the second one place there a s.p.a.ce instead; because at one time the major axes must coincide, at another the minor axis, as in Fig. 191. If then we use even numbers, the distribution and even the forms of the teeth are not the same in the two wheels of the pair. But this complication may be avoided by using an odd number of teeth, since, placing a tooth at one extremity of the major axis, a s.p.a.ce will come at the other.

"It is not, however, always necessary to cut teeth all round these wheels, as will be seen by an examination of Fig. 199, C and D being the fixed centres of the two ellipses in contact at P. Now P must be on the line C D, whence, considering the free foci, we see P B is equal to P C, and P A to P D; and the common tangent at P makes equal angles with C P and P A, as is also with P B and P D; therefore, C D being a straight line, A B is also a straight line and equal to C D. If then the wheels be overhung, that is, fixed on the ends of the shafts outside the bearings, leaving the outer faces free, the moving foci may be connected by a rigid link A B, as shown.

[Ill.u.s.tration: Fig. 198.]

"This link will then communicate the same motion that would result from the use of the complete elliptical wheels, and we may therefore dispense with most of the teeth, retaining only those near the extremities of the major axes which are necessary in order to a.s.sist and control the motion of the link at and near the dead-points. The arc of the pitch-curves through which the teeth must extend will vary with their eccentricity: but in many cases it would not be greater than that which in the approximation may be struck about one centre, so that, in fact, it would not be necessary to go through the process of rectifying and subdividing the quarter of the ellipse at all, as in this case it can make no possible difference whether the s.p.a.cing adopted for the teeth to be cut would "come out even" or not if carried around the curve. By this expedient, then, we may save not only the trouble of drawing, but a great deal of labor in making, the teeth round the whole ellipse. We might even omit the intermediate portions of the pitch ellipses themselves; but as they move in rolling contact their retention can do no harm, and in one part of the movement will be beneficial, as they will do part of the work; for if, when turning, as shown by the arrows, we consider the wheel whose axis is D as the driver, it will be noted that its radius of contact, C P, is on the increase; and so long as this is the case the other wheel will be compelled to move by contact of the pitch lines, although the link be omitted. And even if teeth be cut all round the wheels, this link is a comparatively inexpensive and a useful addition to the combination, especially if the eccentricity be considerable. Of course the wheels shown in Fig. 198 might also have been made alike, by placing a tooth at one end of the major axis and a s.p.a.ce at the other, as above suggested. In regard to the variation in the velocity ratio, it will be seen, by reference to Fig. 199, that if D be the axis of the driver, the follower will in the position there shown move faster, the ratio of the angular velocities being PD/PB; if the driver turn uniformly the velocity of the follower will diminish, until at the end of half a revolution, the velocity ratio will be PB/PD; in the other half of the revolution these changes will occur in a reverse order. But P D = L B; if then the centres B D are given in position, we know L P, the major axis; and in order to produce any a.s.sumed maximum or minimum velocity ratio, we have only to divide L P into segments whose ratio is equal to that a.s.sumed value, which will give the foci of the ellipse, whence the minor axis may be found and the curve described. For instance, in Fig. 198 the velocity ratio being nine to one at the maximum, the major axis is divided into two parts, of which one is nine times as long as the other; in Fig. 199 the ratio is as one to three, so that, the major axis being divided into four parts, the distance A C between the foci is equal to two of them, and the distance of either focus from the nearer extremity of the major axis equal to one, and from the more remote extremity equal to three of these parts."

[Ill.u.s.tration: Fig. 199.]

Another example of obtaining a variable motion is given in Fig. 200. The only condition necessary to the construction of wheels of this cla.s.s is that the sum of the radii of the pitch circles on the line of centres shall equal the distance between the axes of the two wheels. The pitch curves are to be considered the same as pitch circles, "so that," says Willis, "if any given circle or curve be a.s.sumed as a describing (or generating) curve, and if it be made to roll on the inside of one of these pitch curves and on the outside of the corresponding portion of the other pitch curve, then the motion communicated by the pressure and sliding contact of one of the curved teeth so traced upon the other will be exactly the same as that effected by the rolling contact (by friction) of the original pitch curves."

[Ill.u.s.tration: Fig. 200.]

It is obvious that on B the corner sections are formed of simple segments of a circle of which the centre is the axis of the shaft, and that the sections between them are simply racks. The corners of A are segments of a circle of which the axis of A is the centre, and the sections between the corners curves meeting the pitch circles of the rack at every point as it pa.s.ses the line of centres.

Intermittent motion may also be obtained by means of a worm-wheel constructed as in Fig. 201, the worm having its teeth at a right angle to its axis for a distance around the circ.u.mference proportioned to the required duration of the period of rest; or the motion may be made variable by giving the worm teeth different degrees of inclination (to the axis), on different portions of the circ.u.mference.

In addition to the simple operation of two or more wheels transmitting motion by rotating about their fixed centres and in fixed positions, the following examples of wheel motion may be given.

[Ill.u.s.tration: Fig. 201.]

[Ill.u.s.tration: Fig. 202.]

In Fig. 202 are two gear-wheels, A, which is fast upon its stationary shaft, and B, which is free to rotate upon its shaft, the link C affording journal bearing to the two shafts. Suppose that A has 40 teeth, while B has 20 teeth, and that the link C is rotated once around the axis of A, how many revolutions will B make? By reason of there being twice as many teeth in A as in B the latter will make two rotations, and in addition to this it will, by reason of its connection to the arm C, also make a revolution, these being two distinct motions, one a rotation of B about the axis of A, and the other two rotations of B upon its own axis.

A simple arrangement of gearing for reversing the direction of rotation of a shaft is shown in Fig. 203. I and F are fast and loose pulleys for the shaft D, A and C are gears free to rotate upon D, N is a clutch driven by D; hence if N be moved so as to engage with C the latter will act as a driver to rotate the shaft B, the wheel upon B rotating A in an opposite direction to the rotation of D. But if N be moved to engage with A the latter becomes the driving wheel, and B will be caused to rotate in the opposite direction. Since, however, the engagement of the clutch N with the clutch on the nut of the gear-wheels is accompanied with a violent shock and with noise, a preferable arrangement is shown in Fig. 204, in which the gears are all fast to their shafts, and the driving shaft for C pa.s.ses through the core or bore of that for A, which is a sleeve, so that when the driving belt acts upon pulley F the shaft B rotates in one direction, while when the belt acts upon E, B rotates in the opposite direction, I being a loose pulley.

[Ill.u.s.tration: Fig. 203.]

If the speed of rotation of B require to be greater in one direction than in the other, then the bevel-wheel on B is made a double one, that is to say, it has two annular toothed surfaces on its radial face, one of larger diameter than the other; A gearing with one of these toothed surfaces, and C with the other. It is obvious that the pinions A C, being of equal diameters, that gearing with the surface or gear of largest diameter will give to B the slowest speed of rotation.

[Ill.u.s.tration: Fig. 204.]

Fig. 205 represents Watt's sun-and-planet motion for converting reciprocating into rotary motion; B D is the working beam of the engine, whose centre of motion is at D. The gear A is so connected to the connecting rod that it cannot rotate, and is kept in gear with the wheel C on the fly-wheel shaft by means of the link shown. The wheel A being prevented from rotation on its axis causes rotary motion to the wheel C, which makes two revolutions for one orbit of A.

[Ill.u.s.tration: Fig. 205.]

An arrangement for the rapid increase of motion by means of gears is shown in Fig. 206, in which A is a stationary gear, B is free to rotate upon its shaft, and being pivoted upon the shaft of A, at D, is capable of rotation around A while remaining in gear with C. Suppose now that the wheel A were absent, then if B were rotated around C with D as a centre of motion, C and its shaft E would make a revolution even though B would have no rotation upon its axis. But A will cause B to rotate upon its axis and thus communicate a second degree of motion to C, with the result that one revolution of B causes two rotations of C.

The relation of motion between B and C is in this case constant (2 to 1), but this relation may be made variable by a construction such as shown in Fig. 207, in which the wheel B is carried in a gear-wheel H, which rides upon the shaft D. Suppose now that H remains stationary while A revolves, then motion will be transmitted through B to C, and this motion will be constant and in proportion to the relative diameters of A and C. But suppose by means of an independent pinion the wheel H be rotated upon its axis, then increased motion will be imparted to C, and the amount of the increase will be determined by the speed of rotation of H, which may be made variable by means of cone pulleys or other suitable mechanical devices.

[Ill.u.s.tration: Fig. 206.]

Fig. 208 represents an arrangement of gearing used upon steam fire-engines and traction engines to enable them to turn easily in a short radius, as in turning corners in narrow streets. The object is to enable the driving wheel on either side of the engine to increase or diminish its rotation to suit the conditions caused by the leading or front pair of steering wheels.

[Ill.u.s.tration: Fig. 207.]

In the figures A is a plate wheel having the lugs L, by means of which it may be rotated by a chain. A is a working fit on the shaft S, and carries three pinions E pivoted upon their axes P. F is a bevel-gear, a working fit on S, while C is a similar gear fast to S. The pinions B, D are to drive gears on the wheels of the engine, the wheels being a working fit on the axle. Let it now be noted that if S be rotated, C and F will rotate in opposite directions and A will remain stationary. But if A be rotated, then all the gears will rotate with it, but E will not rotate upon P unless there be an unequal resistance to the motion of pinions D and B. So soon, however, as there exists an inequality of resistance between D and B then pinions E operate. For example, let B have more resistance than D, and B will rotate more slowly, causing pinion E to rotate and move C faster than is due to the motion of the chain wheel A, thus causing the wheel on one side of the engine to r.e.t.a.r.d and the other to increase its motion, and thus enable the engine to turn easily. From its action this arrangement is termed the equalizing gear.

[Ill.u.s.tration: Fig. 208.]

In Figs. 209 to 214 are shown what are known as mangle-wheels from their having been first used in clothes mangling machines.

[Ill.u.s.tration: Fig. 209.]

[Ill.u.s.tration: Fig. 210.]

The mangle-wheel[10] in its simplest form is a revolving disc of metal with a centre of motion C (Fig. 209). Upon the face of the disc is fixed a projecting annulus _a_ _m_, the outer and inner edges of which are cut into teeth. This annulus is interrupted at _f_, and the teeth are continued round the edges of the interrupted portion so as to form a continued series pa.s.sing from the outer to the inner edge and back again.

[10] From Willis's "Principles of Mechanism."

A pinion B, whose teeth are of the same pitch as those of the wheel, is fixed to the end of an axis, and this axis is mounted so as to allow of a short travelling motion in the direction B C. This may be effected by supporting this end of it either in a swing-frame moving upon a centre as at D, or in a sliding piece, according to the nature of the train with which it is connected. A short pivot projects from the centre of the pinion, and this rests in and is guided by a groove B S _f_ _t_ _b_ _h_ K, which is cut in the surface of the disc, and made concentric to the pitch circles of the inner and outer rays of teeth, and at a normal distance from them equal to the pitch radius of the pinion.

Now when the pinion revolves it will, if it be on the outside, as in Fig. 209, act upon the spur teeth and turn the wheel in the opposite direction to its own, but when the interrupted portion _f_ of the teeth is thus brought to the pinion the groove will guide the pinion while it pa.s.ses from the outside to the inside, and thus bring its teeth into action with the annular or internal teeth. The wheel will then receive motion in the same direction as that of the pinion, and this will continue until the gap _f_ is again brought to the pinion, when the latter will be carried outwards and the motion again be reversed. The _velocity ratio_ in either direction will remain constant, but the ratio when the pinion is inside will differ slightly from the ratio when it is outside, because the pitch radius of the annular or internal teeth is necessarily somewhat less than that of the spur teeth. However, the change of direction is not instantaneous, for the form of the groove S _f_ _t_, which connects the inner and outer grooves, is a semicircle, and when the axis of the pinion reaches S the velocity of the mangle-wheel begins to diminish gradually until it is brought to rest at _f_, and is again gradually set in motion from _f_ to _t_, when the constant ratio begins; and this r.e.t.a.r.dation will be increased by increasing the difference between the radius of the inner and outer pitch circles.

The teeth of a mangle-wheel are, however, most commonly formed by pins projecting from the face of the disc as in Fig. 210. In this manner the pitch circles for the inner and outer wheels coincide, and therefore the velocity ratio is the same within and without, also the s.p.a.ce through which the pinion moves in shifting is reduced.

[Ill.u.s.tration: Fig. 211.]

[Ill.u.s.tration: Fig. 212.]

This s.p.a.ce may be still further reduced by arranging the teeth as in Fig. 211, that is, by placing the spur-wheel within the annular or internal one; but at the same time the difference of the two velocity ratios is increased.

If it be required that the velocity ratio vary, then the pitch lines of the mangle-wheel must no longer be concentric.

Thus in Fig. 212 the groove _k_ _l_ is directed to the centre of the mangle-wheel, and therefore the pinion will proceed during this portion of its path without giving any motion to the wheel, and in the other lines of teeth the pitch radius varies, hence the angular velocity ratio will vary.

In Figs. 209, 210, and 211 the curves of the teeth are readily obtained by employing the same describing circle for the whole of them. But when the form Fig. 212 is adopted, the shape of the teeth requires some consideration.

Every tooth of such a mangle-wheel may be considered as formed of two ordinary teeth set back to back, the pitch line pa.s.sing through the middle. The outer half, therefore, appropriated to the action of the pinion on the outside of the wheel, resembles that portion of an ordinary spur-wheel tooth that lies beyond its pitch line, and the inner half which receives the inside action of the pinion resembles the half of an annular wheel that lies within the pitch circle. But the consequence of this arrangement is, that in both positions the action of the driving teeth must be confined to the approach of its teeth to the line of centres, and consequently these teeth must be wholly within their pitch line.