Fig. 35Fig. 35.
Fig. 35.
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, inFig. 33there is driving contact at three points,c,f, andj, while inFig. 34there 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 inFig. 35, suppose wheelvto be the driver, having motion in the direction of arrowp, then faces a on the teeth ofvwill work with flanksbof the teeth onw, 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 matterwhat the shapes of the other parts of the teeth. Now suppose thatvstill being the driver, motion occurs in the other direction as denoted byq, then the facescof the teeth onvwill drive the flankscof the teeth onw, 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 thatwis the driver, motion occurring in the direction ofp, then facesewill drive flanksf, and the motion will be correct as before if the curveseandfare produced with the same diameter of rolling circle. Finally, letwbe the driving wheel and motion occur in the direction ofq, and facesgwill drive flanksh, 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 tofor 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.
Fig. 36Fig. 36.
Fig. 36.
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. InFig. 36leta,b, andc, represent three blanks for gear-wheels whose addendum circles arem,nando;prepresenting the pitch circles, andqrepresenting the circles for the roots of the teeth. Letxandyrepresent the lines of centres, anda,h,iandkthe 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 pinionmbeing 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 pointd, where the generating circlegcrosses circlentoe, where generating circlehcrosses the circlem, 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 atrand end ats, and the compasses applied as before (fromrtos) along the arc of generating circleito the line of centres, and thence along the arc of generating circlektos, 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.
Fig. 37Fig. 37.
Fig. 37.
Fig. 38Fig. 38.
Fig. 38.
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 inFig. 37, letarepresent the pitch line of a rack, andbandcthe pitch circles of two wheels, then the generating circle would be rolled withinb, as at 1, for the flank curves, and without it, as at 2, for the face curves ofb. 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 withoutc, as at 5, for the faces, and within it, as at 6, for flanks of the teeth onc, and all the teeth will work correctly together however they be placed; thuscmight receive motion from the rack, andbreceive motion fromc. 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 inFig. 38are shown three teeth, all struck with the same size of generating circle,dbeing for a wheel of 12 teeth,efor a wheel of 50 teeth, andfa tooth of a rack; teethe,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.
Fig. 39Fig. 39.
Fig. 39.
Fig. 40Fig. 40.
Fig. 40.
In determining the diameter of a generating circle for a set ortrain 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. InFig. 39, for example,a ais the line of centres, and the contact of the curves atb cwould cause a thrust in the direction of the arrowsd,e. This thrust would exist throughout the whole path of contact save at the pointf, on the line of centres. This thrust is reduced in proportion as the diameter of the generating circle is increased; thus inFig. 40, is represented a pair of pinions of 12 teeth and 3 inch pitch, andcbeing the driver, there is contact ate, and atg, andebeing 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.
Fig. 41Fig. 41.
Fig. 41.
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 inFig. 41, the linea ais tangent to the teeth curves where they touch, anddat a right angle toa 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 thearc, or path, of contact is longer as the generating circle is made larger.
Fig. 42Fig. 42.
Fig. 42.
Fig. 43Fig. 43.
Fig. 43.
Thus inFig. 42is 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. Supposevto be the driver,wthe driven wheel or follower, and the direction of motion as atp, contact upon toothawill begin atc, and whileais passing to the line of centres the path of contact will pass along the thickened line tox. During this time the whole length of face fromctorwill have had contact with the length of flank fromcton, and it follows that the length of face onathat rolled onc ncan only equal the length ofc n, and that the amount of sliding motion must be represented by the length ofr nona, and the amount of rolling motion by the lengthn c. Again, during the arc of recess (marked by dots) the length of flank that will have had contact is the depth fromstols, and over this depth the full length of tooth face on wheelvwill have swept, and asl sequalsc 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 inFig. 43, letarepresent a segment of a pinion, andba segment of a spur-wheel, both segments being blank with their pitch circles, the tooth height and depth being marked by arcs of circles. Letcanddrepresent the generating circles shown in the two respective positions on the line of centres. Let pinionabe the driver moving in the direction ofp, and the arc of approach will be frometoxalong the thickened arc, while the arc of recess will be as denoted by the dotted arc fromxtof. The distancee xbeing greater than distancex f, therefore the arc of approach is longer than that of recess.
But supposebto be the driver and the reverse will be the case, the arc of approach will begin atgand end atx, while the arc of recess will begin atxand end ath, the latter being farther from the line of centres thangis. 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 wheelb, have contact along a greater portion of their depths than do the flanks of those on the smaller, as is shown by the dotted arcibeing farther from the pitch circle than the dotted arcjis, these two dotted arcs representing the paths of the lowest points of flank contact, pointsfandg, 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.
Fig. 44Fig. 44.
Fig. 44.
InFig. 44is represented portion of a pair of wheels corresponding to those shown inFig. 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.vis the driver in the directionthe teeth ofvthat will have contact isc 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 distancec nand the length of the tooth face.
If now we examine the distancec ninFig. 42, we find that reducing the diameter of generating circle inFig. 44has 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 inFig. 42there are three teeth in driving contact, while inFig. 44there are but two, viz.,dande.
Fig. 45Fig. 45.
Fig. 45.
Fig. 46Fig. 46.
Fig. 46.
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. InFig. 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 fromcto the line of centres, while the arc of recess is denoted by the dotted arc. As contact on the pinion flank begins at pointcand ends at the line of centres, the total depth of flank that suffers wear from the contact is that fromcton; and as the whole length of the wheel tooth face sweeps over this depthc 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 toothw. In the case of the wheel, contact on its tooth flanks begins at the line of centres and ends atl, hence that flank can only wear between pointland the pitch linel; and as the whole length of pinion face sweeps on this short lengthl s, the pinion flank will wear most, the wear being in the direction of the dotted arc on the left-hand sidevof the tooth. Now the pinion flank depthc n, being less than the wheel flank depths 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. InFig. 46the generating arcs,gandg′, and the wheel are the same, but the pinion is larger. As a result the acting lengthc n, of pinion flank is increased, as is also the acting lengths l, of wheel flank; hence, the flanks of both wheels would wear better, and also better preserve their correct and original shapes.
Fig. 47Fig. 47.
Fig. 47.
Fig. 48Fig. 48.
Fig. 48.
Fig. 49Fig. 49.
Fig. 49.
It has been shown, when referring toFigs. 42and44, 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 thewheel, 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 inFig. 47, the flanks being radial, and the acting depths of flank being shown atj k. InFig. 48is shown a pair of wheels with a generating circle,gandg′, of one quarter the diameter of the base circle or pitch diameter, and the acting length of flank is shown atl m. The wear of the teeth would, therefore, in this latter case, cause it in time to assume the form shown inFig. 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 inFig. 47there are two teeth in contact, while inFig. 48there 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.
Fig. 50Fig. 50.
Fig. 50.
InFig. 50, for example, are a wheelaand pinionb, the teeth on the wheel being extended to a point. Suppose that the wheelais the driver, and contact will begin between the two teethdandfon the dotted arc. Now suppose toothdto have moved to positionc, andfwill have been moved to positionh. The degrees of angle the pinion has been moved through are therefore denoted byi, whereas the degrees of angle the arcs of contact contain are therefore denoted byj.
The degrees of angle that the wheelahas moved through are obviously denoted bye, because the point of toothdhas during the arcs of contact moved from positiondto positionc. The degrees of angle contained in its path of contact are denoted byk, and are less thane, hence, in the case of teeth terminating in a point as toothd, the excess of angle of action over path of contact is as many degrees as are contained in one-half the thicknessof 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 atp.
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 toFig. 33or34, 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.
Fig. 51Fig. 51.
Fig. 51.
InFig. 51, for example, is shown a section of two pinions of equal diameters, and it will be observed that if the full lineadetermined the height of the addendum there would be contact either atcorbonly (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 todthere would be contact either ateor atf, 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 giventime, 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.
Fig. 52Fig. 52.
Fig. 52.
Fig. 53Fig. 53.
Fig. 53.
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 inFig. 52. But the minimum of friction is attained when the two flanks for the tooth are drawn into one common hypocycloid, as inFig. 53. The difference between the form of tooth shown inFig. 52and that shown inFig. 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.
Fig. 54Fig. 54.
Fig. 54.
[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 inFig. 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. Supposeba tracing point onb, then asbrolls onait will describe the epicycloida b. A parallel linec dwill work at a constant distance as atc dfroma b, and this distance may be the radius of that part ofdthat 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.
[1]From an article by Professor Robinson.
ComparingFigs. 53and54, we see that the bases in53are flattest, and that the contact of faces upon them must range nearer the pitch line than in54. Hence,53presents 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. 53may 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.
Fig. 55Fig. 55.
Fig. 55.
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 inFig. 55is 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 centree, where the axes of the two shafts would meet if produced. Hence the depth, thickness, and height of the tooth decreases asthe pointeis 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 asfollows:—
Fig. 56Fig. 56.
Fig. 56.
InFig. 56letfandgrepresent two shafts, rotating about their respective axes; and having cones whose greatest diameters are ataandb, and whose points are ate. The diameterabeing equal to that ofbtheir circumferences will be equal, and the angular and velocity ratios will therefore be equal.
Fig. 57Fig. 57.
Fig. 57.
Letcanddrepresent two circles about the respective cones, being equidistant frome, 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 pointe. If such, however, is not the case, then the contact will no longer be a rolling one. Thus, inFig. 57the diameters or circumferences ataandbbeing equal, the surfaces would roll upon each other, but on account of the line of contact not radiating frome(which is the common centre of motion for the two shafts) the circumferencecis less than that ofd, rendering a rolling contact impossible.
Fig. 58Fig. 58.
Fig. 58.
We have supposed that the diameters of the cones be equal, but the conditions will remain the same when their diameters are unequal; thus, inFig. 58the circumference ofais twice that ofb, hence the latter will make two rotations to one of the former, and the contact will still be a rolling one. Similarly the circumference ofdis one half that ofc, hencedwill also make two rotations to one ofc, 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 pointe, which is located where the axes of the shafts would meet.
Fig. 59Fig. 59.
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 inFig. 59the coneais fixed, and that the cone whose axis isfbe rotated upon it in the direction of the arrow. Then let a point be fixed in any part of the circumference ofb(say atd), and it is evident that the path of this point will be asbrolls around the axisf, and at the same time aroundafrom the centre of motion,e. The curve so generated or described by the pointdwill 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, asfollows:—
Fig. 60Fig. 60.
Fig. 60.
InFig. 60leta arepresent the axis of one shaft, andbthe axis of the other, the axes of the two meeting atw. Marke,representing the diameter of one wheel, andfthat of the other (both lines representing the pitch circles of the respective wheels). Draw the lineg gpassing through the pointw, and the pointt, where the pitch circlese,fmeet, andg gwill be the line of contact between the cones. Fromwas a centre, draw on each side ofg gdotted lines asp, representing the height of the teeth above and below the pitch lineg g. At a right angle tog gmark the linej k, and from the junction of this line with axisb(as atq) as a centre, mark the arca, which will represent the pitch circle for the large diameter of piniond; mark also the arcbfor the addendum andcfor the roots of the teeth, so that frombtocwill represent the height of the tooth at that end.
Similarly fromp, as a centre, mark (for the large diameter of wheelc,) the pitch circleg, root circleh, and addendumi. 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, drawm mparallel toj k. Set the compasses to the radiusr l, and fromp, as a centre, draw the pitch circlek. To obtain the depth for the tooth, draw the dotted linep, meeting the circleh, and the pointw. A similar line from circleitowwill show the height of the addendum, or extreme diameter; and mark the tooth curves onk,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 radiuss l, and fromqas a centre, mark the pitch circled, outside ofdmarkefor the height of the addendum and inside ofdmarkffor the roots of the teeth at that end. The distance between the dotted lines (asp) represents the full height of the teeth, hencehmeets linep, being the root of tooth for the large wheel, and to give clearance, the point of the pinion teeth is marked below, thus arcbdoes not meethorp. Having obtained these arcs the curves are rolled as for a spur-wheel.
A tooth thus marked out is shown atx, and from its curves betweenb c, a template for the large diameter of the pinion tooth may be made, while from the tooth curves between the arcse f, a template for the smallest tooth diameter of the pinion can be made.
Similarly for the wheelcthe outer end curves are marked on the linesg,h,i, and those for the inner end on the linesk,l,m.