Fig. 13Fig. 13.
Fig. 13.
Suppose thata(Fig. 13) makes 1 revolution per minute, how many willcmake,ahaving 60 teeth,b30 teeth, andc40 teeth? In this case we have but one driving wheela, and one driven wheelb, the driver having 60 teeth, the driven 30, hence 60 ÷ 30 = 2, equals revolutions ofband also ofc, the two latter being on the same shaft.
It will be observed then that the revolutions are in the same proportion as the numbers of the teeth or the radii of the wheels, or what is the same thing, in the same proportion as their diameters. The number of teeth, however, is usually taken as being easier obtained than the diameter of the pitch circles, and easier to calculate, because the teeth will be represented by a whole number, whereas the diameter, radius, or circumference, will generally contain fractions.
Fig. 14Fig. 14.
Fig. 14.
Suppose that the 4 wheels inFig. 14have the respective numbers of teeth marked beside them, and that the upper one having 40 teeth makes 60 revolutions per minute, then we may obtain the revolutions of the others asfollows:—
and a remainder of the reciprocating decimals. We may now prove this by reversing the question, thus. Suppose the 120 wheel to make 666⁄100revolutions per minute, how many will the 40 wheel make?
revolutions of the 40 wheel, the discrepancy of1⁄100being due to the 6.66 leaving a remainder and not therefore being absolutely correct.
That the amount of power transmitted by gearing, whether compounded or not, is equal throughout every wheel in the train, may be shown asfollows:—
Referring again toFig. 10, it has been shown that with a 50 lb. weight suspended from a 4 inch shafte, there would be 3033⁄100lbs. at the perimeter ofa. Now suppose a rotation be made, then the 50 lb. weight would fall a distance equal to the circumference of the shaft, which is (3.1416 × 4 = 1256⁄100) 1256⁄100inches. Now the circumference of the wheel is (60 dia. × 3.1416 = 18849⁄100cir.) 18849⁄100inches, which is the distance through which the 333⁄100lbs. would move during one rotation ofa. Now 3.33 lbs. moving through 188.49 inches represents the same amount of power as does 50 lbs. moving through a distance of 12.56 inches, as may be found by converting the two into inch lbs. (that is to say, into the number of inches moved by 1 lb.), bearing in mind that there will be a slight discrepancy due to the fact that the fractions .33 in the one case, and .56 in the other are not quite correct. Thus:
Taking the next wheels inFig. 12, it has been shown that the 3.33 lbs. delivered fromato the perimeter ofb, becomes 2.49 lbs. at the perimeter ofc, and it has also been shown thatcmakes two revolutions to one ofa, and its diameter being 40 inches, the distance this 2.49 lbs. will move through in one revolution ofawill therefore be equal to twice its circumference, which is (40 dia. × 3.1416 = 125.666 cir., and 125.666 × 2 = 251.332) 251.332 inches. Now 2.49 lbs. moving through 251.332 gives when brought to inch lbs. 627.67 inch lbs., thus 251.332 × 2.49 = 627.67. Hence the amount of power remains constant, but is altered in form, merely being converted from a heavy weight moving a short distance, into a lighter one moving a distance exactly as much greater as the weight or force is lessened or lighter.
Gear-wheels therefore form a convenient method of either simply transmitting motion or power, as when the wheels are all of equal diameter, or of transmitting it and simultaneously varying its velocity of motion, as when the wheels are compounded either to reduce or increase the speed or velocity in feet per second of the prime mover or first driver of the train or pair, as the case may be.
Fig. 15Fig. 15.
Fig. 15.
In considering the action of gear-teeth, however, it sometimes is more convenient to denote their motion by the number of degrees of angle they move through during a certain portion of a revolution, and to refer to their relative velocities in terms of the ratio or proportion existing between their velocities. The first of these is termed the angular velocity, or the number of degrees of angle the wheel moves through during a given period, while the second is termed the velocity ratio of the pair of wheels. Let it be supposed that two wheels of equal diameter have contact at their perimeters so that one drives the other by friction without any slip, then the velocity of a point on the perimeter of one will equal that of a point on the other. Thus inFig. 15letaandbrepresent the pitch circles of two wheels, andcan imaginary line joining the axes of the two wheels and termed the line of centres. Now the point of contact of the two wheels will be on the line ofcentres as atd, and if a point or dot be marked atdand motion be imparted fromatob, then when each wheel has made a quarter revolution the dot onawill have arrived atewhile that onbwill have arrived atf. As each wheel has moved through one quarter revolution, it has moved through 90° of angle, because in the whole circle there is 360°, one quarter of which is 90°, hence instead of saying that the wheels have each moved through one quarter of a revolution we may say they have moved through an angle of 90°, or, in other words, their angular velocity has, during this period, been 90°. And as both wheels have moved through an equal number of degrees of angle their velocity ratio or proportion of velocity has been equal.
Obviously then the angular velocity of a wheel represents a portion of a revolution irrespective of the diameter of the wheel, while the velocity ratio represents the diameter of one in proportion to that of the other irrespective of the actual diameter of either of them.
Fig. 16Fig. 16.
Fig. 16.
Now suppose that inFig. 16ais a wheel of twice the diameter ofb; that the two are free to revolve about their fixed centres, but that there is frictional contact between their perimeters at the line of centres sufficient to cause the motion of one to be imparted to the other without slip or lost motion, and that a point be marked on both wheels at the point of contactd. Now let motion be communicated toauntil the mark that was made atdhas moved one-eighth of a revolution and it will have moved through an eighth of a circle, or 45°. But during this motion the mark onbwill have moved a quarter of a revolution, or through an angle of 90° (which is one quarter of the 360° that there are in the whole circle). The angular velocities of the two are, therefore, in the same ratio as their diameters, or two to one, and the velocity ratio is also two to one. The angular velocity of each is therefore the number of degrees of angle that it moves through in a certain portion of a revolution, or during the period that the other wheel of the pair makes a certain portion of a revolution, while the velocity ratio is the proportion existing between the velocity of one wheel and that of the other; hence if the diameter of one only of the wheels be changed, its angular velocity will be changed and the velocity ratio of the pair will be changed. The velocity ratio may be obtained by dividing either the radius, pitch, diameter, or number of teeth of one wheel into that of the other.
Conversely, if a given velocity ratio is to be obtained, the radius, diameter, or number of teeth of the driver must bear the same relation to the radius, diameter, or number of teeth of the follower, as the velocity of the follower is desired to bear to that of the driver.
Fig. 17Fig. 17.
Fig. 17.
If a pair of wheels have an equal number of teeth, the same pairs of teeth will come into action at every revolution; but if of two wheels one is twice as large as the other, each tooth on the small wheel will come into action twice during each revolution of the large one, and will work during each successive revolution with the same two teeth on the large wheel; and an application of the principle of the hunting tooth is sometimes employed in clocks to prevent the overwinding of their springs, the device being shown inFig. 17, which is from “Willis’ Principles of Mechanism.”
For this purpose the winding arborchas a pinionaof 19 teeth fixed to it close to the front plate. A pinionbof 18 teeth is mounted on a stud so as to be in gear with the former. A radial platecdis fixed to the face of the upper wheela, and a similar platefeto the lower wheelb. These plates terminate outward in semicircular nosesd,e, so proportioned as to cause their extremities to abut against each other, as shown in the figure, when the motion given to the upper arbor by the winding has brought them into the position of contact. The clock being now wound up, the winding arbor and wheelawill begin to turn in the opposite direction. When its first complete rotation is effected the wheelbwill have gained one tooth distance from the line of centres, so as to place the stopdin advance ofeand thus avoid a contact withe, which would stop the motion. As each turn of the upper wheel increases the distance of the stops, it follows from the principle of the hunting cog, that after eighteen revolutions ofaand nineteen ofbthe stops will come together again and the clock be prevented from running down too far. The winding key being applied, the upper wheelawill be rotated in the opposite direction, and the winding repeated as above.
Thus the teeth on one wheel will wear to imbed one upon the other. On the other hand the teeth of the two wheels may be of such numbers that those on one wheel will not fall into gear with the same teeth on the other except at intervals, and thus an inequality on any one tooth is subjected to correction by all the teeth in the other wheel. When a tooth is added to the number of teeth on a wheel to effect this purpose it is termed a hunting cog, or hunting tooth, because if one wheel have a tooth less, then any two teeth which meet in the first revolution are distant, one tooth in the second, two teeth in the third, three in the fourth, and so on. The odd tooth is on this account termed a hunting tooth.
It is obvious then that the shape or form to be given to the teeth must, to obtain correct results, be such that the motion of the driver will be communicated to the follower with the velocity due to the relative diameters of the wheels at the pitch circles, and since the teeth move in the arc of a circle it is also obvious that the sides of the teeth, which are the only parts that come into contact, must be of same curve. The nature of this curve must be such that the teeth shall possess the strength necessary to transmit the required amount of power, shall possess ample wearing surface, shall be as easily produced as possible for all the varying conditions, shall give as many teeth in constant contact as possible, and shall, as far as possible, exert a pressure in a direction to rotate the wheels without inducing undue wear upon the journals of the shafts upon which the wheels rotate. In cases, however, in which some of these requirements must be partly sacrificed to increase the value of the others, or of some of the others, to suit the special circumstances under which the wheels are to operate, the selection is left to the judgment of the designer, and the considerations which should influence his determinations will appear hereafter.
Fig. 18Fig. 18.
Fig. 18.
Fig. 19Fig. 19.
Fig. 19.
Modern practice has accepted the curve known in general terms as the cycloid, as that best filling all the requirements of wheel teeth, and this curve is employed to produce two distinct forms of teeth, epicycloidal and involute. In epicycloidal teeth the curve forming the face of the tooth is designated an epicycloid, and that forming the flank an hypocycloid. An epicycloid may be traced or generated, as it is termed, by a point in the circumference of a circle that rolls without slip upon the circumference of another circle. Thus, inFig. 18,aandbrepresent two wooden wheels,ahaving a pencil atp, to serve as a tracing or marking point. Now, if the wheels are laid upon a sheet of paper and while holdingbin a fixed position, rollain contact withband let the tracing point touch the paper, the pointpwill trace the curvec c. Suppose now the diameter of the base circlebto be infinitely large, a portion of its circumference may be represented by a straight line, and the curve traced by a point on the circumference of the generating circle as it rolls along the base linebis termed a cycloid. Thus, inFig. 19,bis the base line,athe rolling wheel or generating circle, andc cthe cycloidal curve traced or marked by the pointdwhenais rolled alongb. If now we suppose the base linebto represent the pitch line of a rack, it will be obvious that part of the cycloid at one end is suitable for the face on one side of the tooth, and a part at the other end is suitable for the face of the other side of the tooth.
Fig. 20Fig. 20.
Fig. 20.
A hypocycloid is a curve traced or generated by a point on the circumference of a circle rolling within and in contact (without slip) with another circle. Thus, inFig. 20,arepresents a wheel in contact with the internal circumference ofb, and a point on its circumference will trace the two curves,c c, both curves starting from the same point, the upper having been traced by rolling the generating circle or wheelain one direction and the lower curve by rolling it in the opposite direction.
Fig. 21Fig. 21.
Fig. 21.
To demonstrate that by the epicycloidal and hypocycloidal curves, forming the faces and flanks of what are known as epicycloidal teeth, motion may be communicated from one wheel to another with as much uniformity as by frictional contact of their circumferential surfaces, leta,b, inFig. 21, represent two plain wheel disks at liberty to revolve about their fixed centres, and letc crepresent a margin of stiff white paper attached to the face ofbso as to revolve with it. Now suppose thataandbare in close contact at their perimeters at the pointg, and that there is no slip, and that rotary motion commenced when the pointe(where as tracing point a pencil is attached), in conjunction with the pointf, formed the point of contact of the two wheels, and continued until the pointseandfhad arrived at their respective positions as shown in the figure; the pencil atewill have traced upon the margin of white paper the portion of an epicycloid denoted by the curvee f; and as the movement of the two wheelsa,b, took place by reason of the contact of their circumferences, it is evident that the length of the arcegmust be equal to that of the arcgf, and that the motion ofa(supposing it to be the driver) would be communicated uniformly tob.
Fig. 22Fig. 22.
Fig. 22.
Now suppose that the wheels had been rotated in the opposite direction and the same form of curve would be produced, but it would run in the opposite direction, and these two curves may be utilized to form teeth, as inFig. 22, the points on the wheelaworking against the curved sides of the teeth onb.
Fig. 23Fig. 23.
Fig. 23.
To render such a pair of wheels useful in practice, all that is necessary is to diminish the teeth onbwithout altering thenature of the curves, and increase the diameter of the points ona, making them into rungs or pins, thus forming the wheels into what is termed a wheel and lantern, which are illustrated inFig. 23.
arepresents the pinion (or lantern), andbthe wheel, andc,c, the primitive teeth reduced in thickness to receive the pins ona. This reduction we may make by setting a pair of compasses to the radius of the rung and describing half-circles at the bottom of the spaces inb. We may then set a pair of compasses to the curve ofc, and mark off the faces of the teeth ofbto meet the half-circles at the pitch line, and reduce the teeth heights so as to leave the points of the proper thickness; having in this operation maintained the same epicycloidal curves, but brought them closer together and made them shorter. It is obvious, however, that such a method of communicating rotary motion is unsuited to the transmission of much power; because of the weakness of, and small amount of wearing surface on, the points or rungs ina.
Fig. 24Fig. 24.
Fig. 24.
In place of points or rungs we may have radial lines, these lines, representing the surfaces of ribs, set equidistant on the radial face of the pinion, as inFig. 24. To determine the epicycloidal curves for the faces of teeth to work with these radial lines, we may take a generating circlec, of half the diameter ofa, and cause it to roll in contact with the internal circumference ofa, and a tracing point fixed in the circumference ofcwill draw the radial lines shown upona. The circumstances will not be altered if we suppose the three circles,a,b,c, to be movable about their fixed centres, and let their centres be in a straight line; and if, under these circumstances, we suppose rotation to be imparted to the three circles, through frictional contact of their perimeters, a tracing point on the circumference ofcwould trace the epicycloids shown uponband the radial lines shown upona, evidencing the capability of one to impart uniform rotary motion to the other.
Fig. 25Fig. 25.
Fig. 25.
To render the radial lines capable of use we must let them be the surfaces of lugs or projections on the face of the wheel, as shown inFig. 25atd,e, &c., or the faces of notches cut in the wheel as atf,g,h, &c., the metal betweenfandgforming a toothj, having flanks only. The wheelbhas the curves of each tooth brought closer together to give room for the reception of the teeth upona. We have here a pair of gears that possess sufficient strength and are capable of working correctly in either direction.
But the form of tooth on one wheel is conformed simply to suit those on the other, hence, neither two of the wheelsa, nor would two ofb, work correctly together.
Fig. 26Fig. 26.
Fig. 26.
They may be qualified to do so, however, by simply adding tothe tops of the teeth ona, teeth of the form of those onb, and adding to those onb, and within the pitch circle, teeth corresponding to those ona, as inFig. 26, where atk′andj′teeth are provided onbcorresponding tojandkona, while onathere are added teetho′,n′, corresponding too,n, onb, with the result that two wheels such asaor two such asbwould 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 ona, and flanks to those onb, the curves being generated or obtained by rolling the generating, or curve marking, circlecupon the pitch circlespandp′. Thus, for the flanks of the teeth ofa,cis rolled upon, and within the pitch circlepofa; while for the face curves of the same teethcis rolled upon, but without or outside ofp. Similarly for the teeth of wheelbthe generating circlecis rolled withinp′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 asfollows:—
Fig. 27Fig. 27.
Fig. 27.
InFig. 27letaaandbbrepresent 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 centresab, the wheels are shown blank;aais the pitch line of one wheel, andbbthat for the other. Now suppose that both wheels are capable of being rotated on their shafts, whose centres will of course be on the lineab, and suppose a third disk,q, be also capable of rotation upon its centre,c, which is also on the lineab. Let these three wheels have sufficient contact at their perimeters at the pointn, 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 wheelqa pencil whose point is atn. If then rotation be given toaain the direction of the arrows, all three wheels will rotate in that direction as denoted by their respective arrowss.
Assume, then, that rotation of the three has occurred until the pencil point atnhas arrived at the pointm, and during this period of rotation the pointnwill recede from the line of centresab, and will also recede from the arcs or lines of the two pitch circlesaa,bb. The pencil point being capable of marking its path, it will be found on reachingmto have marked inside the pitch circlebbthe curve denoted by the full linemx, and simultaneously with this curve it has marked another curve outside ofaa, as denoted by the dotted lineym. These two curves being marked by the pencil point at the same time and extending fromytom, andxalso tom. They are prolonged respectively topand tokfor clearness of illustration only.
The rotation of the three wheels being continued, when the pencil point has arrived atoit will have continued the same curves as shown atof, andog, curveofbeing the same asmxplaced in a new position, andogbeing the same asmy, but placed in a new position. Now since both these curves (ofandog) 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 circleqfromntom, 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 ofq, and betweennando. Thus when the pencil has arrived atm, curvemytouches curvekxat the pointm, while when the pencil had arrived at pointo, the curvesofandogwill touch ato. Now the pitch circlesaaandbb, and the describing circleq, 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,mywas the face of a tooth onaa, andmxwas the flank of a tooth onbb, the same uniform motion may be transmitted fromaa, tobb, by pressing the tooth facemyagainstthe tooth flankmx. Let it now be noted that the curveymcorresponds to the face of a tooth, as say the faceeof a tooth onaa, and that curvexmcorresponds to the flank of a tooth onbb, as say to the flankf, 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 betweennandq.
Fig. 28Fig. 28.
Fig. 28.
Fig. 29Fig. 29.
Fig. 29.
In this illustration the diameter of the rolling or describing circleq, being less than the radius of the wheelsaaorbb, 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 inFig. 28are 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 fromntom. During this motion it will have marked a curveym, a portion of theyend serving for the face of a tooth on one wheel, and also the linekx, a continuation of which serves for the flank of a tooth on the other wheel. InFig. 29the pitch circles only of the wheels are marked,aabeing twice the diameter ofbb, and the curve-generating circle being equal in diameter to the radius of wheelbb. Motion is assumed to have occurred until the pencil point, starting fromn, had arrived ato, 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 arcno. Thus when the point had proceeded as far as pointmit will have marked the curveyand the radial linex, and when the point had arrived ato, it will have prolongedmyintoogandxintoof, while in either position the point is marking both lines. The velocities of the wheels remain the same notwithstanding their different diameters, for the arcngmust obviously (if the wheels rotate without slip by friction of their surfaces while the curves are traced) be equal in length to the arcnfor the arcno.
Fig. 30Fig. 30.
Fig. 30.
InFig. 30aaandbbare the pitch circles of two wheels as before, andccthe pitch circle of an annular or internal gear, anddis the rolling or describing circle. When the describing point arrived atm, it will have marked the curveyfor the face of a tooth onaa, the curvexfor the flank of a tooth onbb, and the curveefor the face of a tooth on the internal wheelcc.Motion being continuedmywill be prolonged toog, while simultaneouslyxwill be extended intoofandeintohv, the velocity of all the wheels being uniform and equal. Thus the arcsnv,nf, andng, are of equal length.
Fig. 31Fig. 31.
Fig. 31.
InFig. 31is shown the case of a rack and pinion;aais the pitch line of the rack,bbthat of the pinion,abat a right angle toaa, the line of centres, anddthe generating circle. The wheel and rack are shown with teethnon one side simply for clearness of illustration. The pencil pointnwill, on arriving atm, have traced the flank curvexand the curveyfor the face of the rack teeth.
Fig. 32Fig. 32.
Fig. 32.
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 inFig. 32are two segments of wheels as before,crepresenting the centre of a tooth onaa, anddrepresenting the centre of a tooth onbb. Now suppose that a generating or rolling circle be placed with its pencil point ate, and that it then be rolled aroundaauntil it had reached the position marked 1, then it will have marked the curve frometon, a part of this curve serving for the face of toothc. Now let the rolling circle be placed within the pitch circleaaand its pencil pointnbe set toe, then, on being rolled to position 2, it will have marked the flank of toothc. For the other wheel suppose the rolling wheel or circle to have started fromfand rolled to the line of centres as in the cut, it will have traced the curve forming the face of the toothd. For the flank ofdthe rolling circle or wheel is placed withinbb, its tracing point set atfon 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 toothcis more curved than that ofd, and also that the flank curve ofdis more spread at the root than is that forc, which has in this case resulted from the difference between the diameter of the wheelsaaandbb. 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.
Fig. 33Fig. 33.
Fig. 33.
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 inFig. 33are represented segments of two spur-wheels marked respectively the driver and the driven, their generating circles being marked atgandg′, andxxrepresenting the line of centres. Toothais shown in the position in which it commences its contact with toothbatb. Secondly, we shall find that as these two teeth approach the line of centresx, the point of contact between them moves or takes place along the thickened arc or curvec x, or along the path of the generating circleg.
Thus we may suppose toothdto be another position of tootha, the contact being atf, and as motion was continued the contact would pass along the thickened curve until it arrived at the line of centresx. 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 centresx, the path of contact of the teeth will be along the dotted arc fromxtol, and as the teeth are during this period of motion receding fromxthis part of the contact path is termed the arc of recess.
That contact of the teeth would not occur earlier than atcnor later than atl, is shown by the dotted teeth sides; thusaandbwould not touch when in the position denoted by the dotted teeth, nor would teethiandkif 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 toothaand of toothdare in driving contact with the faces of teethbande, while the face of toothhis in contact with the flank of toothi.
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 inFig. 33then the arc of approach would be frommtox, and the arc of recess fromxton.
Fig. 34Fig. 34.
Fig. 34.
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 inFig. 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 arrowp, the contact will begin at the line of centresx, move or pass along the thickened arc and end atb, and there will be contact during the arc of recess only. Similarly, if the direction of motion be reversed as denoted by arrowq, the driver will begin contact atx, and cease contact ath, having, as before, contact during the arc of recess only.
But if the wheelwwere the driver andvthe driven, then these conditions would be exactly reversed. Thus, suppose this to be the case and the direction of motion be as denoted by arrowp, the contact would occur during the arc of approach, fromhtox, ceasing atx.
Or ifwwere the driver, and the direction of motion was as denoted byq, then, again, the path of contact would be during the arc of approach only, beginning atband ceasing atx, as denoted by the thickened arcb x.