Chapter I: The Teeth of Gear-Wheels (3)

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 passing 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.

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 compasses 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 compasses 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_.

Internal or annular gear-wheels have their tooth curves formed by rolling the generating circle upon the pitch circle or base circle, upon the same general principle as external or spur-wheels. But the tooth of the annular wheel corresponds with the space in the spur-wheel, as is shown in Fig. 61, in which curve A forms the flank of a tooth on a spur-wheel P, and the face of a tooth on the annular wheel W. It is obvious then that the generating circle is rolled within the pitch circle for the face of the wheel and without for its flank, or the reverse of the process for spur-wheels. But in the case of internal or annular wheels the path of contact of tooth upon tooth with a pinion having a given number of teeth increases in proportion as the number of teeth in the wheel is diminished, which is also the reverse of what occurs in spur-wheels; as will readily be perceived when it is considered that if in an internal wheel the pinion have as many teeth as the wheel the contact would exist around the whole pitch circles of the wheel and pinion and the two would rotate together without any motion of tooth upon tooth. Obviously then we have, in the case of internal wheels, a consideration as to what is the greatest number (as well as what is the least number) of teeth a pinion may contain to work with a given wheel, whereas in spur-wheels the reverse is again the case, the consideration being how few teeth the wheel may contain to work with a given pinion. Now it is found that although the curves of the teeth in internal wheels and pinions may be rolled according to the principles already laid down for spur-wheels, yet cases may arise in which internal gears will not work under conditions in which spur-wheels would work, because the internal wheels will not engage together. Thus, in Fig. 62, is a pinion of 12 teeth and a wheel of 22 teeth, a generating circle having a diameter equal to the radius of the pinion having been used for all the tooth curves of both wheel and pinion. It will be observed that teeth A, B, and C clearly overlap teeth D, E, and F, and would therefore prevent the wheels from engaging to the requisite depth. This may of course be remedied by taking the faces off the pinion, as in Fig. 63, and thus confining the arc of contact to an arc of recess if the pinion drives, or an arc of approach if the wheel drives; or the number of teeth in the pinion may be reduced, or that in the wheel increased; either of which may be carried out to a degree sufficient to enable the teeth to engage and not interfere one with the other. In Fig. 64 the number of teeth in the pinion P is reduced from 12 to 6, the wheel W having 22 as before, and it will be observed that the teeth engage and properly clear each other.

By the introduction into the figure of a segment of a spur-wheel also having 22 teeth and placed on the other side of the pinion, it is shown that the path of contact is greater, and therefore the angle of action is greater, in internal than in spur gearing. Thus suppose the pinion to drive in the direction of the arrows and the thickened arcs A B will be the arcs of approach, A measuring longer than B. The dotted arcs C D represent the arcs of receding contact and C is found longer than D, the angles of action being 66° for the spur-wheels and 72° for the annular wheel.

On referring again to Fig. 62 it will be observed that it is the faces of the teeth on the two wheels that interfere and will prevent them from engaging, hence it will readily occur to the mind that it is possible to form the curves of the pinion faces correct to work with the faces of the wheel teeth as well as with the flanks; or it is possible to form the wheel faces with curves that will work correctly with the faces, as well as with the flanks of the pinion teeth, which will therefore increase the angle of action, and Professor McCord has shown in an article in the London _Engineering_ how to accomplish this in a simple and yet exceedingly ingenious manner which may be described as follows:--

It is required to find a describing circle that will roll the curves for the flanks of the pinion and the faces of the wheels, and also a describing circle for the flanks of the wheel and the faces of the pinion; the curve for the wheel faces to work correctly with the faces as well as with the flanks of the pinion, and the curve for the pinion faces to work correctly with both the flanks and faces of the internal wheel.

In Fig. 65 let P represent the pitch circle of an annular or internal wheel whose centre is at A, and Q the pitch circle of a pinion whose centre is at B, and let R be a describing circle whose centre is at C, and which is to be used to roll all the curves for the teeth. For the flanks of the annular wheel we may roll R within P, while for the faces of the wheel we may roll R outside of P, but in the case of the pinion we cannot roll R within Q, because R is larger than Q, hence we must find some other rolling circle of less diameter than R, and that can be used in its stead (the radius of R always being greater than the radius of the axis of the wheel and pinion for reasons that will appear presently). Suppose then that in Fig. 66 we have a ring whose bore R corresponds in diameter to the intermediate describing circle R, Fig. 65 and that Q represents the pinion. Then we may roll R around and in contact with the pinion Q, and a tracing point in R will trace the curve M N O, giving a curve a portion of which may be used for the faces of the pinion. But suppose that instead of rolling the intermediate describing circle R around P, we roll the circle T around P, and it will trace precisely the same curve M N O; hence for the faces of the pinion we have found a rolling circle T which is a perfect substitute for the intermediate circle Q, and which it will always be, no matter what the diameters of the pinion and of the intermediate describing circle may be, providing that the diameter of T is equal to the difference between the diameters of the pinion and that of the intermediate describing circle as in the figure. If now we use this describing circle to roll the flanks of the annular wheel as well as the faces of the pinion, these faces and flanks will obviously work correctly together. Since this describing circle is rolled on the outside of the pinion and on the outside of the annular wheel we may distinguish it as the exterior describing circle.

Now instead of rolling the intermediate describing circle R within the annular wheel P for the face curves of the teeth upon P, we may find some other circle that will give the same curve and be small enough to be rolled within the pinion Q for its teeth flanks. Thus in Fig. 67 P represents the pitch circle of the annular wheel and R the intermediate circle, and if R be rolled within P, a point on the circumference of R will trace the curve V W. But if we take the circle S, having a diameter equal to the difference between the diameter of R and that of P, and roll it within P, a point in its circumference will trace the same curve V W; hence S is a perfect substitute for R, and a portion of the curve V W may be used for the faces of the teeth on the annular wheel. The circle S being used for the pinion flanks, the wheel faces and pinion flanks will work correctly together, and as the circle S is rolled within the pinion for its flanks and within the wheel for its faces, it may be distinguished as the interior describing circle.

To prove the correctness of the construction it may be noted that with the particular diameter of intermediate describing circle used in Fig. 65, the interior and exterior describing circles are of equal diameters; hence, as the same diameter of describing circle is used for all the faces and flanks of the pair of wheels they will obviously work correctly together, in accordance with the rules laid down for spur gearing. The radius of S in Fig. 69 is equal to the radius of the annular wheel, less the radius of the intermediate circle, or the radius from A to C. The radius of the exterior describing circle T is the radius of the intermediate circle less the radius of the pinion, or radius C B in the figure.

Now the diameter of the intermediate circle may be determined at will, but cannot exceed that of the annular wheel or be less than the pinion. But having been selected between these two limits the interior and exterior describing circles derived from it give teeth that not only engage properly and avoid the interference shown in Fig. 62, but that will also have an additional arc of action during the recess, as is shown in Fig. 68, which represents the wheel and pinion shown in Fig. 62, but produced by means of the interior and exterior describing circles. Supposing the pinion to be the driver the arc of approach will be along the thickened arc of the interior describing circle, while during the arc of recess there will be an arc of contact along the dotted portion of the exterior describing circle as in ordinary gearing. But in addition there will be an arc of recess along the dotted portion of the intermediate circle R, which arc is due to the faces of the pinion acting upon the faces as well as upon the flanks of the wheel teeth. It is obvious from this that as soon as a tooth passes the line of centres it will, during a certain period, have two points of contact, one on the arc of the exterior describing circle, and another along the arc of R, this period continuing until the addendum circle of the pinion crosses the dotted arc of the exterior describing circle at Z.

The diameters of the interior and exterior describing circles obviously depend upon the diameter of the intermediate circle, and as this may, as already stated, be selected, within certain limits, at will, it is evident that the relative diameters of the interior and exterior describing circles will vary in proportion, the interior becoming smaller and the exterior larger, while from the very mode of construction the radius of the two will equal that of the axes of the wheel and pinion. Thus in Fig. 69 the radii of S, T, equal A B, or the line of centres, and their diameters, therefore, equal the radius of the annular wheel, as is shown by dotting them in at the upper half of the figure. But after their diameters have been determined by this construction either of them may be decreased in diameter and the teeth of the wheels will clear (and not interfere as in Fig. 62), but the action will be the same as in ordinary gear, or in other words there will be no arc of action on the circle R. But S cannot be increased without correspondingly decreasing T, nor can T be increased without correspondingly decreasing S.

Fig. 70 shows the same pair of gears as in Fig. 68 (the wheel having 22 and the pinion 12 teeth), the diameter of the intermediate circle having been enlarged to decrease the diameter of S and increase that of T, and as these are left of the diameter derived from the construction there is receding action along R from the line of centres to T.

In Fig. 71 are represented a wheel and pinion, the pinion having but four teeth less than the wheel, and a tooth, J, being shown in position in which it has contact at two places. Thus at _k_ it is in contact with the flank of a tooth on the annular wheel, while at L it is in contact with the face of the same tooth.

As the faces of the teeth on the wheel do not have contact higher than point _t_, it is obvious that instead of having them 3/10 of the pitch as at the bottom of the figure, we may cut off the portion X without diminishing the arc of contact, leaving them formed as at the top of the figure. These faces being thus reduced in height we may correspondingly reduce the depth of flank on the pinion by filling in the portion G, leaving the teeth formed as at the top of the pinion. The teeth faces of the wheel being thus reduced we may, by using a sufficiently large intermediate circle, obtain interior and exterior describing circles that will form teeth that will permit of the pinion having but one tooth less than the wheel, or that will form a wheel having but one tooth more than the pinion.

The limits to the diameter of the intermediate describing circle are as follows: in Fig. 72 it is made equal in diameter to the pitch diameter of the pinion, hence B will represent the centre of the intermediate circle as well as of the pinion, and the pitch circle of the pinion will also represent the intermediate circle R. To obtain the radius for the interior describing circle we subtract the radius of the intermediate circle from the radius of the annular wheel, which gives A P, hence the pitch circle of the pinion also represents the interior circle R. But when we come to obtain the radius for the exterior describing circle (T), by subtracting the radius of the pinion from that of the intermediate circle, we find that the two being equal give O for the radius of (T), hence there could be no flanks on the pinion.

Now suppose that the intermediate circle be made equal in diameter to the pitch circle of the annular wheel, and we may obtain the radius for the exterior describing circle T; by subtracting the radius of the pinion from that of the intermediate circle, we shall obtain the radius A B; hence the radius of (T) will equal that of the pinion. But when we come to obtain the radius for the interior describing circle by subtracting the radius of the intermediate circle from that of the annular wheel, we find these two to be equal, hence there would be no interior describing circle, and, therefore, no faces to the pinion.

The action of the teeth in internal wheels is less a sliding and more a rolling one than that in any other form of toothed gearing. This may be shown as follows: In Fig. 73 let A A represent the pitch circle of an external pinion, and B B that of an internal one, and P P the pitch circle of an external wheel for A A or an internal one for B B, the point of contact at the line of centres being at C, and the direction of rotation P P being as denoted by the arrow; the two pinions being driven, we suppose a point at C, on the pitch circle P P, to be coincident with a point on each of the two pinions at the line of centres. If P P be rotated so as to bring this point to the position denoted by D, the point on the external pinion having moved to E, while that on the internal pinion has moved to F, both having moved through an arc equal to C D, then the distance from E to D being greater than from D to F, more sliding motion must have accompanied the contact of the teeth at the point E than at the point F; and the difference in the length of the arc E D and that of F D, may be taken to represent the excess of sliding action for the teeth on E; for whatever, under any given condition, the amount of sliding contact may be, it will be in the proportion of the length of E D to that of F D. Presuming, then, that the amount of power transmitted be equal for the two pinions, and the friction of all other things being equal--being in proportion to the space passed (or in this case slid) over--it is obvious that the internal pinion has the least friction.