Modern Machine-Shop Practice Part 4

[Ill.u.s.tration: Fig. 47.]

But if a pair or a particular train of gears are to be constructed, then a diameter of generating circle may be selected that is considered most suitable to the particular conditions; as, for example, it may be equal to the radius of the smallest wheel giving it radial flanks, or less than that radius giving parallel or spread flanks. But in any event, in order to transmit continuous motion, the diameter of generating circle must be such as to give arcs of action that are equal to the pitch, so that each pair of teeth will come into action before the preceding pair have gone out of action.

[Ill.u.s.tration: Fig. 48.]

It may now be pointed out that the degrees of angle that the teeth move through always exceeds the number of degrees of angle contained in the paths of contact, or, in other words, exceeds the degrees contained in the arcs of approach and recess combined.

[Ill.u.s.tration: Fig. 49.]

In Fig. 50, for example, are a wheel A and pinion B, the teeth on the wheel being extended to a point. Suppose that the wheel A is the driver, and contact will begin between the two teeth D and F on the dotted arc.

Now suppose tooth D to have moved to position C, and F will have been moved to position H. The degrees of angle the pinion has been moved through are therefore denoted by I, whereas the degrees of angle the arcs of contact contain are therefore denoted by J.

The degrees of angle that the wheel A has moved through are obviously denoted by E, because the point of tooth D has during the arcs of contact moved from position D to position C. The degrees of angle contained in its path of contact are denoted by K, and are less than E, hence, in the case of teeth terminating in a point as tooth D, the excess of angle of action over path of contact is as many degrees as are contained in one-half the thickness of the tooth, while when the points of the teeth are cut off, the excess is the number of degrees contained in the distance between the corner and the side of the tooth as marked on a tooth at P.

[Ill.u.s.tration: Fig. 50.]

With a given diameter of pitch circle and pitch diameter of wheel, the length of the arc of contact will be influenced by the height of the addendum from the pitch circle, because, as has been shown, the arcs of approach and of recess, respectively, begin and end on the addendum circle.

If the height of the addendum on the follower be reduced, the arc of approach will be reduced, while the arc of recess will not be altered; and if the follower have no addendum, contact between the teeth will occur on the arc of recess only, which gives a smoother motion, because the action of the driver is that of dragging rather than that of pushing the follower. In this case, however, the arc of recess must, to produce continuous motion, be at least equal to the pitch.

It is obvious, however, that the follower having no addendum would, if acting as a driver to a third wheel, as in a train of wheels, act on its follower, or the fourth wheel of the train, on the arc of approach only; hence it follows that the addendum might be reduced to diminish, or dispensed with to eliminate action, on the arc of approach in the follower of a pair of wheels only, and not in the case of a train of wheels.

To make this clear to the reader it may be necessary to refer again to Fig. 33 or 34, from which it will be seen that the action of the teeth of the driver on the follower during the arc of approach is produced by the flanks of the driver on the faces of the follower. But if there are no such faces there can be no such contact.

On the arc of recess, however, the faces of the driver act on the flanks of the follower, hence the absence of faces on the follower is of no import.

From these considerations it also appears that by giving to the driver an increase of addendum the arc of recess may be increased without affecting the arc of approach. But the height of addendum in machinists'

practice is made a constant proportion of the pitch, so that the wheel may be used indiscriminately, as circ.u.mstances may require, as either a driver or a follower, the arcs of approach and of recess being equal.

The height of addendum, however, is an element in determining the number of teeth in contact, and upon small pinions this is of importance.

[Ill.u.s.tration: Fig. 51.]

In Fig. 51, for example, is shown a section of two pinions of equal diameters, and it will be observed that if the full line A determined the height of the addendum there would be contact either at C or B only (according to the direction in which the motion took place).

With the addendum extended to the dotted circle, contact would be just avoided, while with the addendum extended to D there would be contact either at E or at F, according to which direction the wheel had motion.

This, by dividing the strain over two teeth instead of placing it all upon one tooth, not only doubles the strength for driving capacity, but decreases the wear by giving more area of bearing surface at each instant of time, although not increasing that area in proportion to the number of teeth contained in the wheel.

In wheels of larger diameter, short teeth are more permissible, because there are more teeth in contact, the number increasing with the diameters of the wheels. It is to be observed, however, that from having radial flanks, the smallest wheel is always the weakest, and that from making the most revolutions in a given time, it suffers the most from wear, and hence requires the greatest attainable number of teeth in constant contact at each period of time, as well as the largest possible area of bearing or wearing surface on the teeth.

It is true that increasing the "depth of tooth to pitch line" increases the whole length of tooth, and, therefore, weakens it; but this is far more than compensated for by distributing the strain over a greater number of teeth. This is in practice accomplished, _when circ.u.mstances will permit_, by making the pitch finer, giving to a wheel, of a given diameter, a greater number of teeth.

[Ill.u.s.tration: Fig. 52.]

When the wheels are required to transmit motion rather than power (as in the case of clock wheels), to move as frictionless as possible, and to place a minimum of thrust on the journals of the shafts of the wheels, the generating circle may be made nearly as large as the diameter of the pitch circle, producing teeth of the form shown in Fig. 52. But the minimum of friction is attained when the two flanks for the tooth are drawn into one common hypocycloid, as in Fig. 53. The difference between the form of tooth shown in Fig. 52 and that shown in Fig. 53, is merely due to an increase in the diameter of the generating circle for the latter. It will be observed that in these forms the acting length of flank diminishes in proportion as the diameter of the generating circle is increased, the ultimate diameter of generating circle being as large as the pitch circles.

[Ill.u.s.tration: Fig. 53.]

[1]This form is undesirable in that there is contact on one side only (on the arc of approach) of the line of centres, but the flanks of the teeth may be so modified as to give contact on the arc of recess also, by forming the flanks as shown in Fig. 54, the flanks, or rather the parts within the pitch circles, being nearly half circles, and the parts without with peculiarly formed faces, as shown in the figure. The pitch circles must still be regarded as the rolling circles rolling upon each other. Suppose _b_ a tracing point on B, then as B rolls on A it will describe the epicycloid _a_ _b_. A parallel line _c_ _d_ will work at a constant distance as at _c_ _d_ from _a_ _b_, and this distance may be the radius of that part of D that is within the pitch line, the same process being applied to the teeth on both wheels. Each tooth is thus composed of a spur based upon a half cylinder.

[1] From an article by Professor Robinson.

[Ill.u.s.tration: Fig. 54.]

Comparing Figs. 53 and 54, we see that the bases in 53 are flattest, and that the contact of faces upon them must range nearer the pitch line than in 54. Hence, 53 presents a more favorable obliquity of the line of direction of the pressures of tooth upon tooth. In seeking a still more favorable direction by going outside for the point of contact, we see by simply recalling the method of generating the tooth curves, that tooth contacts outside the pitch lines have no possible existence; and hence, Fig. 53 may be regarded as representing that form of toothed gear which will operate with less friction than any other known form.

[Ill.u.s.tration: Fig. 55.]

This statement is intended to cover fixed teeth only, and not that complicated form of the trundle wheel in which the cylinder teeth are friction rollers. No doubt such would run still easier, even with their necessary one-sided contacts. Also, the statement is supposed to be confined to such forms of teeth as have good practical contacts at and near the line of centres.

Bevel-gear wheels are employed to transmit motion from one shaft to another when the axis of one is at an angle to that of the other. Thus in Fig. 55 is shown a pair of bevel-wheels to transmit motion from shafts at a right angle. In bevel-wheels all the lines of the teeth, both at the tops or points of the teeth, at the bottoms of the s.p.a.ces, and on the sides of the teeth, radiate from the centre E, where the axes of the two shafts would meet if produced. Hence the depth, thickness, and height of the tooth decreases as the point E is approached from the diameter of the wheel, which is always measured on the pitch circle at the largest end of the cone, or in other words, at the largest pitch diameter.

The principles governing the practical construction of the curves for the teeth of the bevel-wheels may be explained as follows:--

In Fig. 56 let F and G represent two shafts, rotating about their respective axes; and having cones whose greatest diameters are at A and B, and whose points are at E. The diameter A being equal to that of B their circ.u.mferences will be equal, and the angular and velocity ratios will therefore be equal.

[Ill.u.s.tration: Fig. 56.]

Let C and D represent two circles about the respective cones, being equidistant from E, and therefore of equal diameters and circ.u.mferences, and it is obvious that at every point in the length of each cone the velocity will be equal to a point upon the other so long as both points are equidistant from the points of intersection of the axes of the two shafts; hence if one cone drive the other by frictional contact of surfaces, both shafts will be rotated at an equal speed of rotation, or if one cone be fixed and the other moved around it, the contact of the surfaces will be a rolling contact throughout. The line of contact between the two cones will be a straight line, radiating at all times from the point E. If such, however, is not the case, then the contact will no longer be a rolling one. Thus, in Fig. 57 the diameters or circ.u.mferences at A and B being equal, the surfaces would roll upon each other, but on account of the line of contact not radiating from E (which is the common centre of motion for the two shafts) the circ.u.mference C is less than that of D, rendering a rolling contact impossible.

[Ill.u.s.tration: Fig. 57.]

We have supposed that the diameters of the cones be equal, but the conditions will remain the same when their diameters are unequal; thus, in Fig. 58 the circ.u.mference of A is twice that of B, hence the latter will make two rotations to one of the former, and the contact will still be a rolling one. Similarly the circ.u.mference of D is one half that of C, hence D will also make two rotations to one of C, and the contact will also be a rolling one; a condition which will always exist independent of the diameters of the wheels so long as the angles of the faces, or wheels, or (what is the same thing, the line of contact between the two,) radiates from the point E, which is located where the axes of the shafts would meet.

[Ill.u.s.tration: Fig. 58.]

[Ill.u.s.tration: Fig. 59.]

The principles governing the forms of the cones on which the teeth are to be located thus being explained, we may now consider the curves of the teeth. Suppose that in Fig. 59 the cone A is fixed, and that the cone whose axis is F be rotated upon it in the direction of the arrow.

Then let a point be fixed in any part of the circ.u.mference of B (say at _d_), and it is evident that the path of this point will be as B rolls around the axis F, and at the same time around A from the centre of motion, E. The curve so generated or described by the point _d_ will be a spherical epicycloid. In this case the exterior of one cone has rolled upon the coned surface of the other; but suppose it rolls upon the interior, as around the walls of a conical recess in a solid body; then a point in its circ.u.mference would describe a curve known as the spherical hypocycloid; both curves agreeing (except in their spherical property) to the epicycloid and hypocycloid of the spur-wheel. But this spherical property renders it very difficult indeed to practically delineate or mark the curves by rolling contact, and on account of this difficulty Tredgold devised a method of construction whereby the curves may be produced sufficiently accurate for all practical purposes, as follows:--

[Ill.u.s.tration: Fig. 60.]

In Fig. 60 let A A represent the axis of one shaft, and B the axis of the other, the axes of the two meeting at W. Mark E, representing the diameter of one wheel, and F that of the other (both lines representing the pitch circles of the respective wheels). Draw the line G G pa.s.sing through the point W, and the point T, where the pitch circles E, F meet, and G G will be the line of contact between the cones. From W as a centre, draw on each side of G G dotted lines as _p_, representing the height of the teeth above and below the pitch line G G. At a right angle to G G mark the line J K, and from the junction of this line with axis B (as at Q) as a centre, mark the arc _a_, which will represent the pitch circle for the large diameter of pinion D; mark also the arc _b_ for the addendum and _c_ for the roots of the teeth, so that from _b_ to _c_ will represent the height of the tooth at that end.

[Ill.u.s.tration: Fig. 61.]

Similarly from P, as a centre, mark (for the large diameter of wheel C,) the pitch circle _g_, root circle _h_, and addendum _i_. On these arcs mark the curves in the same manner as for spur-wheels. To obtain these arcs for the small diameters of the wheels, draw M M parallel to J K.

Set the compa.s.ses to the radius R L, and from P, as a centre, draw the pitch circle _k_. To obtain the depth for the tooth, draw the dotted line _p_, meeting the circle _h_, and the point W. A similar line from circle _i_ to W will show the height of the addendum, or extreme diameter; and mark the tooth curves on _k_, _l_, _m_, in the same manner as for a spur-wheel.

Similarly for the pitch circle of the small end of the pinion teeth, set the compa.s.ses to the radius S L, and from Q as a centre, mark the pitch circle _d_, outside of _d_ mark _e_ for the height of the addendum and inside of _d_ mark _f_ for the roots of the teeth at that end. The distance between the dotted lines (as _p_) represents the full height of the teeth, hence _h_ meets line _p_, being the root of tooth for the large wheel, and to give clearance, the point of the pinion teeth is marked below, thus arc _b_ does not meet _h_ or _p_. Having obtained these arcs the curves are rolled as for a spur-wheel.

A tooth thus marked out is shown at _x_, and from its curves between _b_ _c_, a template for the large diameter of the pinion tooth may be made, while from the tooth curves between the arcs _e_ _f_, a template for the smallest tooth diameter of the pinion can be made.

Similarly for the wheel C the outer end curves are marked on the lines _g_, _h_, _i_, and those for the inner end on the lines _k_, _l_, _m_.

[Ill.u.s.tration: Fig. 62.]