Chapter I: The Teeth of Gear-Wheels (2)

But the form of tooth on one wheel is conformed simply to suit those on the other, hence, neither two of the wheels A, nor would two of B, work correctly together.

They may be qualified to do so, however, by simply adding to the tops of the teeth on A, teeth of the form of those on B, and adding to those on B, and within the pitch circle, teeth corresponding to those on A, as in Fig. 26, where at K´ and J´ teeth are provided on B corresponding to J and K on A, while on A there are added teeth O´, N´, corresponding to O, N, on B, with the result that two wheels such as A or two such as B would work correctly together, either being the driver or either the follower, and rotation may occur in either direction. In this operation we have simply added faces to the teeth on A, and flanks to those on B, the curves being generated or obtained by rolling the generating, or curve marking, circle C upon the pitch circles P and P´. Thus, for the flanks of the teeth of A, C is rolled upon, and within the pitch circle P of A; while for the face curves of the same teeth C is rolled upon, but without or outside of P. Similarly for the teeth of wheel B the generating circle C is rolled within P´ for the flanks and without for the faces. With the curves rolled or produced with the same diameter of generating circle the wheels will work correctly together, no matter what their relative diameter may be, as will be shown hereafter.

In this demonstration, however, the curves for the faces of the teeth being produced by an operation distinct from that employed to produce the flank curves, it is not clearly seen that the curves for the flanks of one wheel are the proper curves to insure a uniform velocity to the other. This, however, may be made clear as follows:--

In Fig. 27 let _a_ _a_ and _b_ _b_ represent the pitch circles of two wheels of equal diameters, and therefore having the same number of teeth. On the left, the wheels are shown with the teeth in, while on the right-hand side of the line of centres A B, the wheels are shown blank; _a_ _a_ is the pitch line of one wheel, and _b_ _b_ that for the other. Now suppose that both wheels are capable of being rotated on their shafts, whose centres will of course be on the line A B, and suppose a third disk, Q, be also capable of rotation upon its centre, C, which is also on the line A B. Let these three wheels have sufficient contact at their perimeters at the point _n_, that if one be rotated it will rotate both the others (by friction) without any slip or lost motion, and of course all three will rotate at an equal velocity. Suppose that there is fixed to wheel Q a pencil whose point is at _n_. If then rotation be given to _a_ _a_ in the direction of the arrow _s_, all three wheels will rotate in that direction as denoted by their respective arrows _s_.

Assume, then, that rotation of the three has occurred until the pencil point at _n_ has arrived at the point _m_, and during this period of rotation the point _n_ will recede from the line of centres A B, and will also recede from the arcs or lines of the two pitch circles _a_ _a_, _b_ _b_. The pencil point being capable of marking its path, it will be found on reaching _m_ to have marked inside the pitch circle _b_ _b_ the curve denoted by the full line _m_ _x_, and simultaneously with this curve it has marked another curve outside of _a_ _a_, as denoted by the dotted line _y_ _m_. These two curves being marked by the pencil point at the same time and extending from _y_ to _m_, and _x_ also to _m_. They are prolonged respectively to _p_ and to K for clearness of illustration only.

The rotation of the three wheels being continued, when the pencil point has arrived at O it will have continued the same curves as shown at O _f_, and O _g_, curve O _f_ being the same as _m_ _x_ placed in a new position, and O _g_ being the same as _m_ _y_, but placed in a new position. Now since both these curves (O _f_ and O _g_) were marked by the one pencil point, and at the same time, it follows that at every point in its course that point must have touched both curves at once. Now the pencil point having moved around the arc of the circle Q from _n_ to _m_, it is obvious that the two curves must always be in contact, or coincide with each other, at some point in the path of the pencil or describing point, or, in other words, the curves will always touch each other at some point on the curve of Q, and between _n_ and O. Thus when the pencil has arrived at _m_, curve _m_ _y_ touches curve K _x_ at the point _m_, while when the pencil had arrived at point O, the curves O _f_ and O _g_ will touch at O. Now the pitch circles _a_ _a_ and _b_ _b_, and the describing circle Q, having had constant and uniform velocity while the traced curves had constant contact at some point in their lengths, it is evident that if instead of being mere lines, _m_ _y_ was the face of a tooth on _a_ _a_, and _m_ _x_ was the flank of a tooth on _b_ _b_, the same uniform motion may be transmitted from _a_ _a_, to _b_ _b_, by pressing the tooth face _m_ _y_ against the tooth flank _m_ _x_. Let it now be noted that the curve _y_ _m_ corresponds to the face of a tooth, as say the face E of a tooth on _a_ _a_, and that curve _x_ _m_ corresponds to the flank of a tooth on _b_ _b_, as say to the flank F, short portions only of the curves being used for those flanks. If the direction of rotation of the three wheels was reversed, the same shape of curves would be produced, but they would lie in an opposite direction, and would, therefore, be suitable for the other sides of the teeth. In this case, the contact of tooth upon tooth will be on the other side of the line of centres, as at some point between _n_ and Q.

In this illustration the diameter of the rolling or describing circle Q, being less than the radius of the wheels _a_ _a_ or _b_ _b_, the flanks of the teeth are curves, and the two wheels being of the same diameter, the teeth on the two are of the same shape. But the principles governing the proper formation of the curve remain the same whatever be the conditions. Thus in Fig. 28 are segments of a pair of wheels of equal diameter, but the describing, rolling, or curve-generating circle is equal in diameter to the radius of the wheels. Motion is supposed to have occurred in the direction of the arrows, and the tracing point to have moved from _n_ to _m_. During this motion it will have marked a curve _y_ _m_, a portion of the _y_ end serving for the face of a tooth on one wheel, and also the line _k_ _x_, a continuation of which serves for the flank of a tooth on the other wheel. In Fig. 29 the pitch circles only of the wheels are marked, _a_ _a_ being twice the diameter of _b_ _b_, and the curve-generating circle being equal in diameter to the radius of wheel _b_ _b_. Motion is assumed to have occurred until the pencil point, starting from _n_, had arrived at _o_, marking curves suitable for the face of the teeth on one wheel and for the flanks of the other as before, and the contact of tooth upon tooth still, at every point in the path of the teeth, occurring at some point of the arc _n_ _o_. Thus when the point had proceeded as far as point _m_ it will have marked the curve _y_ and the radial line _x_, and when the point had arrived at _o_, it will have prolonged _m_ _y_ into _o_ _g_ and _x_ into _o_ _f_, while in either position the point is marking both lines. The velocities of the wheels remain the same notwithstanding their different diameters, for the arc _n_ _g_ must obviously (if the wheels rotate without slip by friction of their surfaces while the curves are traced) be equal in length to the arc _n_ _f_ or the arc _n_ _o_.

In Fig. 30 _a_ _a_ and _b_ _b_ are the pitch circles of two wheels as before, and _c_ _c_ the pitch circle of an annular or internal gear, and D is the rolling or describing circle. When the describing point arrived at _m_, it will have marked the curve _y_ for the face of a tooth on _a_ _a_, the curve _x_ for the flank of a tooth on _b_ _b_, and the curve _e_ for the face of a tooth on the internal wheel _c_ _c_. Motion being continued _m_ _y_ will be prolonged to _o_ _g_, while simultaneously _x_ will be extended into _o_ _f_ and _e_ into _h_ _v_, the velocity of all the wheels being uniform and equal. Thus the arcs _n_ _v_, _n_ _f_, and _n_ _g_, are of equal length.

In Fig. 31 is shown the case of a rack and pinion; _a_ _a_ is the pitch line of the rack, _b_ _b_ that of the pinion, A B at a right angle to _a_ _a_, the line of centres, and D the generating circle. The wheel and rack are shown with teeth _n_ on one side simply for clearness of illustration. The pencil point _n_ will, on arriving at _m_, have traced the flank curve _x_ and the curve _y_ for the face of the rack teeth.

It has been supposed that the three circles rotated together by the frictional contact of their perimeters on the line of centres, but the circumstances will remain the same if the wheels remain at rest while the generating or describing circle is rolled around them. Thus in Fig. 32 are two segments of wheels as before, _c_ representing the centre of a tooth on _a_ _a_, and _d_ representing the centre of a tooth on _b_ _b_. Now suppose that a generating or rolling circle be placed with its pencil point at _e_, and that it then be rolled around _a_ _a_ until it had reached the position marked 1, then it will have marked the curve from _e_ to _n_, a part of this curve serving for the face of tooth _c_. Now let the rolling circle be placed within the pitch circle _a_ _a_ and its pencil point _n_ be set to _e_, then, on being rolled to position 2, it will have marked the flank of tooth _c_. For the other wheel suppose the rolling wheel or circle to have started from _f_ and rolled to the line of centres as in the cut, it will have traced the curve forming the face of the tooth _d_. For the flank of _d_ the rolling circle or wheel is placed within _b_ _b_, its tracing point set at _f_ on the pitch circle, and on being rolled to position 3 it will have marked the flank curve. The curves thus produced will be precisely the same as those produced by rotating all three wheels about their axes, as in our previous demonstrations.

The curves both for the faces and for the flanks thus obtained will vary in their curvature with every variation in either the diameter of the generating circle or of the base or pitch circle of the wheel. Thus it will be observable to the eye that the face curve of tooth _c_ is more curved than that of _d_, and also that the flank curve of _d_ is more spread at the root than is that for _c_, which has in this case resulted from the difference between the diameter of the wheels _a_ _a_ and _b_ _b_. But the curves obtained by a given diameter of rolling circle on a given diameter of pitch circle will be correct for any pitch of teeth that can be used upon wheels having that diameter of pitch circle. Thus, suppose we have a curve obtained by rolling a wheel of 20 inches circumference on a pitch circle of 40 inches circumference--now a wheel of 40 inches in circumference may contain 20 teeth of 2 inch arc pitch, or 10 teeth of 4 inch arc pitch, or 8 teeth of 5 inch arc pitch, and the curve may be used for either of those pitches.

If we trace the path of contact of each tooth, from the moment it takes until it leaves contact with a tooth upon the other wheel, we shall find that contact begins at the point where the flank of the tooth on the wheel that drives or imparts motion to the other wheel, meets the face of the tooth on the driven wheel, which will always be where the point of the driven tooth cuts or meets the generating or rolling circle of the driving tooth. Thus in Fig. 33 are represented segments of two spur-wheels marked respectively the driver and the driven, their generating circles being marked at G and G´, and X X representing the line of centres. Tooth A is shown in the position in which it commences its contact with tooth B at B. Secondly, we shall find that as these two teeth approach the line of centres X, the point of contact between them moves or takes place along the thickened arc or curve C X, or along the path of the generating circle G.

Thus we may suppose tooth D to be another position of tooth A, the contact being at F, and as motion was continued the contact would pass along the thickened curve until it arrived at the line of centres X. Now since the teeth have during this path of contact approached the line of centres, this part of the whole arc of action or of the path of contact is termed the arc of approach. After the two teeth have passed the line of centres X, the path of contact of the teeth will be along the dotted arc from X to L, and as the teeth are during this period of motion receding from X this part of the contact path is termed the arc of recess.

That contact of the teeth would not occur earlier than at C nor later than at L, is shown by the dotted teeth sides; thus A and B would not touch when in the position denoted by the dotted teeth, nor would teeth I and K if in the position denoted by their dotted lines.

If we examine further into this path of contact we find that throughout its whole path the face of the tooth of one wheel has contact with the flank only of the tooth of the other wheel, and also that the flank only of the driving-wheel tooth has contact before the tooth reaches the line of centres, while the face of only the driving tooth has contact after the tooth has passed the line of centres.

Thus the flanks of tooth A and of tooth D are in driving contact with the faces of teeth B and E, while the face of tooth H is in contact with the flank of tooth I.

These conditions will always exist, whatever be the diameters of the wheels, their number of teeth or the diameter of the generating circle. That is to say, in fully developed epicycloidal teeth, no matter which of two wheels is the driver or which the driven wheel, contact on the teeth of the driver will always be on the tooth flank during the arc of approach and on the tooth face during the arc of recess; while on the driven wheel contact during the arc of approach will be on the tooth face only, and during the arc of recess on the tooth flank only, it being borne in mind that the arcs of approach and recess are reversed in location if the direction of revolution be reversed. Thus if the direction of wheel motion was opposite to that denoted by the arrows in Fig. 33 then the arc of approach would be from M to X, and the arc of recess from X to N.

It is laid down by Professor Willis that the motion of a pair of gear-wheels is smoother in cases where the path of contact begins at the line of centres, or, in other words, when there is no arc of approach; and this action may be secured by giving to the driven wheel flanks only, as in Fig. 34, in which the driver has fully developed teeth, while the teeth on the driven have no faces.

In this case, supposing the wheels to revolve in the direction of arrow P, the contact will begin at the line of centres X, move or pass along the thickened arc and end at B, and there will be contact during the arc of recess only. Similarly, if the direction of motion be reversed as denoted by arrow Q, the driver will begin contact at X, and cease contact at H, having, as before, contact during the arc of recess only.

But if the wheel W were the driver and V the driven, then these conditions would be exactly reversed. Thus, suppose this to be the case and the direction of motion be as denoted by arrow P, the contact would occur during the arc of approach, from H to X, ceasing at X.

Or if W were the driver, and the direction of motion was as denoted by Q, then, again, the path of contact would be during the arc of approach only, beginning at B and ceasing at X, as denoted by the thickened arc B X.

The action of the teeth will in either case serve to give a theoretically perfect motion so far as uniformity of velocity is concerned, or, in other words, the motion of the driver will be transmitted with perfect uniformity to the driven wheel. It will be observed, however, that by the removal of the faces of the teeth, there are a less number of teeth in contact at each instant of time; thus, in Fig. 33 there is driving contact at three points, C, F, and J, while in Fig. 34 there is driving contact at two points only. From the fact that the faces of the teeth work with the flanks only, and that one side only of the teeth comes into action, it becomes apparent that each tooth may have curves formed by four different diameters of rolling or generating circles and yet work correctly, no matter which wheel be the driver, or which the driven wheel or follower, or in which direction motion occurs. Thus in Fig. 35, suppose wheel V to be the driver, having motion in the direction of arrow P, then faces a on the teeth of V will work with flanks B of the teeth on W, and so long as the curves for these faces and flanks are obtained with the same diameter of rolling circle, the action of the teeth will be correct, no matter what the shapes of the other parts of the teeth. Now suppose that V still being the driver, motion occurs in the other direction as denoted by Q, then the faces C of the teeth on V will drive the flanks C of the teeth on W, and the motion will again be correct, providing that the same diameter (whatever it may be) of rolling circle be used for these faces and flanks, irrespective, of course, of what diameter of rolling circle is used for any other of the teeth curves. Now suppose that W is the driver, motion occurring in the direction of P, then faces E will drive flanks F, and the motion will be correct as before if the curves E and F are produced with the same diameter of rolling circle. Finally, let W be the driving wheel and motion occur in the direction of Q, and faces G will drive flanks H, and yet another diameter of rolling circle may be used for these faces and flanks. Here then it is shown that four different diameters of rolling circles may be used upon a pair of wheels, giving teeth-forms that will fill all the requirements so far as correctly transmitting motion is concerned. In the case of a pair of wheels having an equal number of teeth, so that each tooth on one wheel will always fall into gear with the same tooth on the other wheel, every tooth may have its individual curves differing from all the others, providing that the corresponding teeth on the other wheel are formed to match them by using the same size of rolling circle for each flank and face that work together.

It is obvious, however, that such teeth would involve a great deal of labor in their formation and would possess no advantage, hence they are not employed. It is not unusual, however, in a pair of wheels that are to gear together and that are not intended to interchange with other wheels, to use such sizes as will give to for the face of the teeth on the largest wheel of the pair and for the flanks of the teeth of the smallest wheel, a generating circle equal in diameter to the radius of the smallest wheel, and for the faces of the teeth of the small wheel and the flanks of the teeth of the large one, a generating circle whose diameter equals the radius of the large wheel.

It will now be evident that if we have planned a pair or a train of wheels we may find how many teeth will be in contact for any given pitch, as follows. In Fig. 36 let A, B, and C, represent three blanks for gear-wheels whose addendum circles are M, N and O; P representing the pitch circles, and Q representing the circles for the roots of the teeth. Let X and Y represent the lines of centres, and A, H, I and K the generating or rolling circle, whose centres are on the respective lines of centres--the diameter of the generating circle being equal to the radius of the pinion, as in the Willis system, then, the pinion M being the driver, and the wheels revolving in the direction denoted by the respective arrows, the arc or path of contact for the first pair will be from point D, where the generating circle G crosses circle N to E, where generating circle H crosses the circle M, this path being composed of two arcs of a circle. All that is necessary, therefore, is to set the compasses to the pitch the teeth are to have and step them along these arcs, and the number of steps will be the number of teeth that will be in contact. Similarly, for the second pair contact will begin at R and end at S, and the compasses applied as before (from R to S) along the arc of generating circle I to the line of centres, and thence along the arc of generating circle K to S, will give in the number of steps, the number of teeth that will be in contact. If for any given purpose the number of teeth thus found to be in contact is insufficient; the pitch may be made finer.

When a wheel is intended to be formed to work correctly with any other wheel having the same pitch, or when there are more than two wheels in the train, it is necessary that the same size of generating circle be used for all the faces and all the flanks in the set, and if this be done the wheels will work correctly together, no matter what the number of the teeth in each wheel may be, nor in what way they are interchanged. Thus in Fig. 37, let A represent the pitch line of a rack, and B and C the pitch circles of two wheels, then the generating circle would be rolled within B, as at 1, for the flank curves, and without it, as at 2, for the face curves of B. It would be rolled without the pitch line, as at 3, for the rack faces, and within it, as at 4, for the rack flanks, and without C, as at 5, for the faces, and within it, as at 6, for flanks of the teeth on C, and all the teeth will work correctly together however they be placed; thus C might receive motion from the rack, and B receive motion from C. Or if any number of different diameters of wheels are used they will all work correctly together and interchange perfectly, with the single condition that the same size of generating circle be used throughout. But the curves of the teeth so formed will not be alike. Thus in Fig. 38 are shown three teeth, all struck with the same size of generating circle, D being for a wheel of 12 teeth, E for a wheel of 50 teeth, and F a tooth of a rack; teeth E, F, being made wider so as to let the curves show clearly on each side, it being obvious that since the curves are due to the relative sizes of the pitch and generating circles they are equally applicable to any pitch or thickness of teeth on wheels having the same diameters of pitch circle.

In determining the diameter of a generating circle for a set or train of wheels, we have the consideration that the smaller the diameter of the generating circle in proportion to that of the pitch circle the more the teeth are spread at the roots, and this creates a pressure tending to thrust the wheels apart, thus causing the axle journals to wear. In Fig. 39, for example, A A is the line of centres, and the contact of the curves at B C would cause a thrust in the direction of the arrows D, E. This thrust would exist throughout the whole path of contact save at the point F, on the line of centres. This thrust is reduced in proportion as the diameter of the generating circle is increased; thus in Fig. 40, is represented a pair of pinions of 12 teeth and 3 inch pitch, and C being the driver, there is contact at E, and at G, and E being a radial line, there is obviously a minimum of thrust.

What is known as the Willis system for interchangeable gearing, consists of using for every pitch of the teeth a generating circle whose diameter is equal to the radius of a pinion having 12 teeth, hence the pinion will in each pitch have radial flanks, and the roots of the teeth will be more spread as the number of teeth in the wheel is increased. Twelve teeth is the least number that it is considered practicable to use; hence it is obvious that under this system all wheels of the same pitch will work correctly together.

Unless the faces of the teeth and the flanks with which they work are curves produced from the same size of generating circle, the velocity of the teeth will not be uniform. Obviously the revolutions of the wheels will be proportionate to their numbers of teeth; hence in a pair of wheels having an equal number of teeth, the revolutions will per force be equal, but the driver will not impart uniform motion to the driven wheel, but each tooth will during the path of contact move irregularly.

The velocity of a pair of wheels will be uniform at each instant of time, if a line normal to the surfaces of the curves at their point of contact passes through the point of contact of the pitch circles on the line of centres of the wheels. Thus in Fig. 41, the line A A is tangent to the teeth curves where they touch, and D at a right angle to A A, and meets it at the point of the tooth curves, hence it is normal to the point of contact, and as it meets the pitch circles on the line of centres the velocity of the wheels will be uniform.

The amount of rolling motion of the teeth one upon the other while passing through the path of contact, will be a minimum when the tooth curves are correctly formed according to the rules given. But furthermore the sliding motion will be increased in proportion as the diameter of the generating circle is increased, and the number of teeth in contact will be increased because the arc, or path, of contact is longer as the generating circle is made larger.

Thus in Fig. 42 is a pair of wheels whose tooth curves are from a generating circle equal to the radius of the wheels, hence the flanks are radial. The teeth are made of unusual depth to keep the lines in the engraving clear. Suppose V to be the driver, W the driven wheel or follower, and the direction of motion as at P, contact upon tooth A will begin at C, and while A is passing to the line of centres the path of contact will pass along the thickened line to X. During this time the whole length of face from C to R will have had contact with the length of flank from C to N, and it follows that the length of face on A that rolled on C N can only equal the length of C N, and that the amount of sliding motion must be represented by the length of R N on A, and the amount of rolling motion by the length N C. Again, during the arc of recess (marked by dots) the length of flank that will have had contact is the depth from S to Ls, and over this depth the full length of tooth face on wheel V will have swept, and as L S equals C N, the amount of rolling and of sliding motion during the arc of recess is equal to that during the arc of approach, and the action is in both cases partly a rolling and partly a sliding one. The two wheels are here shown of the same diameter, and therefore contain an equal number of teeth, hence the arcs of approach and of recess are equal in length, which will not be the case when one wheel contains more teeth than the other. Thus in Fig. 43, let A represent a segment of a pinion, and B a segment of a spur-wheel, both segments being blank with their pitch circles, the tooth height and depth being marked by arcs of circles. Let C and D represent the generating circles shown in the two respective positions on the line of centres. Let pinion A be the driver moving in the direction of P, and the arc of approach will be from E to X along the thickened arc, while the arc of recess will be as denoted by the dotted arc from X to F. The distance E X being greater than distance X F, therefore the arc of approach is longer than that of recess.

But suppose B to be the driver and the reverse will be the case, the arc of approach will begin at G and end at X, while the arc of recess will begin at X and end at H, the latter being farther from the line of centres than G is. It will be found also that, one wheel being larger than the other, the amount of sliding and rolling contact is different for the two wheels, and that the flanks of the teeth on the larger wheel B, have contact along a greater portion of their depths than do the flanks of those on the smaller, as is shown by the dotted arc I being farther from the pitch circle than the dotted arc J is, these two dotted arcs representing the paths of the lowest points of flank contact, points F and G, marking the initial lowest contact for the two directions of revolution.

Thus it appears that there is more sliding action upon the teeth of the smaller than upon those of the larger wheel, and this is a condition that will always exist.

In Fig. 44 is represented portion of a pair of wheels corresponding to those shown in Fig. 42, except that in this case the diameter of the generating circle is reduced to one quarter that of the pitch diameter of the wheels. V is the driver in the direction the teeth of V that will have contact is C N, which, the wheels, being of equal diameter, will remain the same whichever wheel be the driver, and in whatever direction motion occurs. The amount of rolling motion is, therefore, C N, and that of sliding is the difference between the distance C N and the length of the tooth face.

If now we examine the distance C N in Fig. 42, we find that reducing the diameter of generating circle in Fig. 44 has increased the depth of flank that has contact, and therefore increased the rolling motion of the tooth face along the flank, and correspondingly diminished the sliding action of the tooth contact. But at the same time we have diminished the number of teeth in contact. Thus in Fig. 42 there are three teeth in driving contact, while in Fig. 44 there are but two, viz., D and E.

In an article by Professor Robinson, attention is called to the fact that if the teeth of wheels are not formed to have correct curves when new, they cannot be improved by wear; and this will be clearly perceived from the preceding remarks upon the amount of rolling and sliding contact. It will also readily appear that the nearer the diameter of the generating to that of the base circle the more the teeth wear out of correct shape; hence, in a train of gearing in which the generating circle equals the radius of the pinion, the pinion will wear out of shape the quickest, and the largest wheel the least; because not only does each tooth on the pinion more frequently come into action on account of its increased revolutions, but furthermore the length of flank that has contact is less, while the amount of sliding action is greater. In Fig. 45, for example, are a wheel and pinion, the latter having radial flanks and the pinion being the driver, the arc of approach is the thickened arc from C to the line of centres, while the arc of recess is denoted by the dotted arc. As contact on the pinion flank begins at point C and ends at the line of centres, the total depth of flank that suffers wear from the contact is that from C to N; and as the whole length of the wheel tooth face sweeps over this depth C N, the pinion flanks must wear faster than the wheel faces, and the pinion flanks will wear underneath, as denoted by the dotted curve on the flanks of tooth W. In the case of the wheel, contact on its tooth flanks begins at the line of centres and ends at L, hence that flank can only wear between point L and the pitch line L; and as the whole length of pinion face sweeps on this short length L S, the pinion flank will wear most, the wear being in the direction of the dotted arc on the left-hand side V of the tooth. Now the pinion flank depth C N, being less than the wheel flank depth S L, and the same length of tooth face sweeping (during the path of contact) over both, obviously the pinion tooth will wear the most, while both will, as the wear proceeds, lose their proper flank curve. In Fig. 46 the generating arcs, G and G´, and the wheel are the same, but the pinion is larger. As a result the acting length C N, of pinion flank is increased, as is also the acting length S L, of wheel flank; hence, the flanks of both wheels would wear better, and also better preserve their correct and original shapes.

It has been shown, when referring to Figs. 42 and 44, when treating of the amount of sliding and of rolling motion, that the smaller the diameter of rolling circle in proportion to that of pitch circle, the longer the acting length of flank and the more the amount of rolling motion; and it follows that the teeth would also preserve their original and true shape better. But the wear of the teeth, and the alteration of tooth form by reason of that wear, will, in any event, be greater upon the pinion than upon the wheel, and can only be equal when the two wheels are of equal diameter, in which case the tooth curves will be alike on both wheels, and the acting depths of flank will be equal, as shown in Fig. 47, the flanks being radial, and the acting depths of flank being shown at J K. In Fig. 48 is shown a pair of wheels with a generating circle, G and G´, of one quarter the diameter of the base circle or pitch diameter, and the acting length of flank is shown at L M. The wear of the teeth would, therefore, in this latter case, cause it in time to assume the form shown in Fig. 49. But it is to be noted that while the acting depth of flank has been increased the arcs of contact have been diminished, and that in Fig. 47 there are two teeth in contact, while in Fig. 48 there is but one, hence the pressure upon each tooth is less in proportion as the diameter of the generating circle is increased. If a train of wheels are to be constructed, or if the wheels are to be capable of interchanging with other combinations of wheels of the same pitch, the diameter of the generating circle must be equal to the smallest wheel or pinion, which is, under the Willis system, a pinion of 12 teeth; under the Pratt and Whitney, and Brown and Sharpe systems, a pinion of 15 teeth.

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.

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.

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.

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 circumstances 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.

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 circumstances will permit_, by making the pitch finer, giving to a wheel, of a given diameter, a greater number of teeth.

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.

[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.

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.

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 spaces, 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 circumferences will be equal, and the angular and velocity ratios will therefore be equal.

Let C and D represent two circles about the respective cones, being equidistant from E, and therefore of equal diameters and circumferences, 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 circumferences 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 circumference C is less than that of D, rendering a rolling contact impossible.

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 circumference 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 circumference 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.

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 circumference 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 circumference 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:--