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Originally appearing in Volume V17, Page 1003 of the 1911 Encyclopedia Britannica.
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CASE I. If n is a divisor of N, a=N; b=N/n; c--el. (20) CASE 2. If the greatest common divisor of N and n be d, a number less than n, so that n=md, N = Md; then a=mN=Mn=Mmd; b=M; c=m. (21) CASE 3. If N and n be prime to each other, a=nN; b=N; c=n. (22) It is considered desirable by millwrights, with a view to the preservation of the uniformity of shape of the teeth of a pair of wheels, that each given tooth in one wheel should work with as many different teeth in the other wheel as possible. They there-fore study that the numbers of teeth in each pair of wheels which work together shall either be prime to each other, or shall have their greatest common divisor as small as is consistent with a velocity ratio suited for the purposes of the machine. § 45. Sliding Contact: Forms of the Teeth of Spur-wheels and Racks.--A line of connexion of two pieces in sliding contact is a line perpendicular to their surfaces at a point whet* they touch. Bearing this in mind, the principle of the comparative motion of a pair of teeth belonging to a pair of spur-wheels, or to a spur-wheel and a rack, is found by applying the principles stated generally in §§ 36 and 37 to the case of parallel axes for a pair of spur-wheels, and to the case of an axis perpendicular to the direction of shifting for a wheel and a rack. In fig. 101, let Ci, C2 be the centres of a pair of spur-wheels; B1IB1', B2IB2' portions of their pitch-circles, touching at I, the pitch-point. Let the wheel I be the driver, and the wheel 2 the follower. Flo. 100. (al+a2),IT; (26) so that it is greater the farther the point of contact is from the line of centres; and at the instant when that point passes the line of centres, and coincides with the pitch-point, the velocity of sliding is null, and the action of the teeth is, for the instant, that of rolling contact. IV. The path of contact is the line traversing the various positions of the point T. If the line of connexion preserves always the same position, the path of contact coincides with it, and is straight; in other cases the path of contact is curved. It is divided by the pitch-point I into two .parts—the arc or line of approach described by T in approaching the line of centres, and the arc or line of recess described by T after having passed the line of centres. During the approach, the flank D1BI of the driving tooth drives the face D2B2 of the following tooth, and the teeth are sliding towards each other. During the recess (in which the position of the teeth is exemplified in the figure by curves marked with accented letters), the face B1'A1' of the driving tooth drives the flank B2'A2' of the following tooth; and the teeth are sliding from each other. The path of contact is bounded where the approach commences by the addendum-circle of the follower, and where the recess terminates by the addendum-circle of the driver. The length of the path of contact should be such that there shall always be at least one pair of teeth in contact; and it is better still to make it so long that there shall always be at least two pairs of teeth in contact. V. The obliquity of the action of the teeth is the angle EIT= ICI Pi =1C2P2. In practice it is found desirable that the mean value of the obliquity of action during the contact of teeth should not exceed 15°, nor the maximum value 30°. It is unnecessary to give separate figures and demonstrations for inside gearing. The only modification required in the formulae is, that in equation (26) the difference of the angular velocities should be substituted for their sum. § 46. Involute Teeth.—The simplest form of tooth which fulfils the conditions of § 45 is obtained in the following manner (see fig. 102). Let Cl, C1 be the centres of two wheels, BIIB1', B2IB2' their pitch-circles, I the pitch-point; let the obliquity of action of theteeth be constant, so that the same straight line PjIPz shall represent at once the constant line of connexion of teeth and the path of contact. Draw CIP1, C2P2 perpendicular to PIIP2, and with those lines as radii describe about the centres of the wheels the circles DIDI', D2D2', called base-circles. It is evident that the radii of the base-circles bear to each other the same proportions as the radii of the pitch-circles, and also that C1P1= ICI . cos obliquity C2P2=IC2. cos obliquity (The obliquity which is found to answer best in practice is about 142°; its cosine is about , and its sine about ,3,. These values though not absolutely exact, are near enough to the truth for practical purposes.) Suppose the base-circles to be a pair of circular pulleys connected by means of a cord whose course from pulley to pulley is P1IP2. As the line of connexion of those pulleys is the same as that of the proposed teeth, they will rotate with the required velocity ratio. Now, suppose a tracing point T to be fixed to the cord, so as to be carried along the path of con-tact PIIP2, that point will trace on a plane rotating along with the wheel I part of the involute of the base-circle D1DI', and on a plane rotating along with the wheel 2 part of the involute of the base-circle D2D2'; and the two curves so traced will always touch .each other in the required point of contact T, and will therefore fulfil the condition required by Principle I. of § 45. Consequently, one of the forms suitable for the teeth of wheels is the involute of a circle; and the obliquity of the action of such teeth is the angle whose cosine is the ratio of the radius of their base-circle to that of the pitch-circle of the wheel. All involute teeth of the same pitch work smoothly together. To find the length of the path of contact on either side of the pitch-point I, it is to be observed that the distance between the fronts of two successive teeth, as measured along P1IP2, is less than the pitch in the ratio of cos obliquity : 1; and consequently that, if distances equal to- the pitch be marked off either way from I towards PI and P2 respectively, as the extremities of the path of contact, and if, according to Principle IV. of § 45, the addendum-circles be described through the points so found, there will always be at least two pairs of teeth in action at once. In practice it is usual to make the path of contact somewhat longer, viz. about 2.4 times the pitch; and with this length of path, and the obliquity already mentioned of 142°, the addendum is about 3•I of the pitch. The teeth of a rack, to work correctly with wheels having involute teeth, should have plane surfaces perpendicular to the line of connexion, and consequently making with the direction of motion of the rack angles equal to the complement of the obliquity of action. § 47. Teeth for a given Path of Contact: Sang's Method.—In the preceding section the form of the teeth is found by assuming a figure for the path of contact, viz. the straight line. Any other convenient figure may be assumed for the path of contact, and the corresponding forms of the teeth found by determining what curves a point T, moving along the assumed path of contact, will trace on two disks rotating round the centres of the wheels with angular velocities bearing that relation to the component velocity of T along TI, which is given by Principle II. of § 45, and'by equation (25). This method of finding the forms of the teeth of wheels forms the subject of an elaborate and most interesting treatise by Edward Sang. All wheels having teeth of the same pitch, traced from the same path of contact, work correctly together, and are said to belong to the same set. § 48. Teeth traced by Rolling Curves.—If any curve R (fig. 103) be rolled on the inside of the pitch-circle BB of a wheel, it appears, from § 30, that the instan- taneous axis of the rolling curve at any instant will be at the point I, where it touches the pitch-circle for the moment, and that consequently the line AT, traced by a tracing-point T, fixed to the rolling curve upon the plane of the wheel, will be every-where perpendicular to the straight line TI; so that the traced curve AT will be suitable for the flank of a tooth, in which T is the point of contact corresponding to the position I of the pitch-point. If the Let DITBIA1, D2TB2A2 be the positions, at a given instant, of the acting surfaces of a pair of teeth in the driver and follower respectively, touching each other at T; the line of connexion of, those teeth is PIPi, perpendicular to their surfaces at T. Let C1 P1, C2P2 be perpendiculars let fall from the centres of the wheels on the line of contact. Then, by § 36, the angular velocity-ratio is a2/ai=CIP1/C2P2. (23) The following principles regulate the forms of the teeth and their relative motions: I. The angular velocity ratio due to the sliding contact of the teeth will be the same with that due to the rolling contact of the pitch-circles, if the line of connexion of the teeth cuts the line of centres at the pitch-point. For, let P1P2 cut the line of centres at I; then, by similar a1 : a2 : C2P2 : CIPI : : IC2 : : ICI; (24) which is also the angular velocity ratio due to the rolling contact of the circles BIIB1', B2IB2'. This principle determines the forms of all teeth of spur-wheels. It also determines the forms of the teeth of straight racks, if one of the centres be removed, and a straight line EIE', parallel to the direction of motion of the rack, and perpendicular to C1IC2, be substituted for a pitch-circle. II. The component of the velocity of the point of contact of the teeth T along the line of connexion is aI . CjPI = a2 • C2Pz. (25) triangles, (27) same rolling curve R, with the same tracing-point T, be rolled on the outside of any other pitch-circle, it will have the face of a tooth suitable to work with the flank AT. In like manner, if either the same or any other rolling curve R' be rolled the opposite way, on the outside of the pitch-circle BB, so that the tracing point T' shall start from A, it will trace the face AT' of a tooth suitable to work with a flank traced by rolling the same curve R' with the same tracing-point T' inside any other pitch-circle. The figure of the path of contact is that traced on a fixed plane by the tracing-point, when the rolling curve is rotated in such a manner as always to touch a fixed straight line EIE (or E'I'E', as the case may be) at a fixed point I (or I'). If the same rolling curve and tracing-point be used to trace both the faces and the flanks of the teeth of a number of wheels of different sizes but of the same pitch, all those wheels will work correctly together, and will form a set. The teeth of a rack, of the same set, are traced by rolling the rolling curve on both sides of a straight line. The teeth of wheels of any figure, as well as of circular wheels, may be traced by rolling curves on their pitch-surfaces; and all teeth of the same pitch, traced by the same rolling curve with the same tracing-point, will work together correctly if their pitch-surfaces are in rolling contact. § 49. Epicycloidal Teeth.—The most convenient rolling curve is the circle. The path of contact which it traces is identical with itself ; and the flanks of the teeth are internal and their faces external epicycloids for wheels, and both flanks and faces are cycloids for a rack. For a pitch-circle of twice the radius of the rolling or describing circle (as it is called) the internal epicycloid is a straight line, being, in fact, a diameter of the pitch-circle, so that the flanks of the teeth for such a pitch-circle are planes radiating from the axis. For a smaller pitch-circle the flanks would be convex and in-curved or under-cut, which would be inconvenient; therefore the smallest wheel of a set should have its pitch-circle of twice the radius of the describing circle, so that the flanks may be either straight or concave. In fig. 104 let BB' be part of the pitch-circle of a wheel with epicycloidal teeth; CIC' the line of centres; I the pitch-point; EIE'. a straight tangent to the pitch-circle at that point; R the internal and R' the equal external describing circles, so placed as to touch the pitch-circle and each other at I. Let DID' be the path of con-tact, consisting of the arc of approach DI and the arc of recess ID'. In order that there may always be at least two pairs of teeth in action, each of those arcs should be equal to the pitch. The obliquity of the action in passing the line of centres is nothing; the maximum obliquity is the angle EID=E'ID; and the mean obliquity is one-half of that angle. It appears from experience that the mean obliquity should not exceed 15°; therefore the maximum obliquity should be about 30°; therefore the equal arcs DI and ID' should each be one-sixth of a circumference; therefore the circumference of the describing circle should be six times the pitch. It follows that the smallest pinion of a set in which pinion the flanks are straight should have twelve teeth. § 5o. Nearly Epicycloidal Teeth: Willis's Method.—To facilitate the drawing of epicycloidal teeth in practice, Willis showed how to approximate to their figure by means of two circular arcs—one concave, for the flank, and the other convex, for the face—and each having for its radius the mean radius of curvature of the epicycloidal arc. Willis's formulae are founded on the following properties of epicycloids: Let R be the radius of the pitch-circle; r that of the describing circle; B the angle made by the normal TI to the epicycloid at a given point T, with a tangent-to the circle at I—that is, the obliquity of the action at T. Then the radius of curvature of the epicycloid at T is For an internal epicycloid, p =4r sin OR_ ~ ( 28) For an external epicycloid, p'=4rsinoR+2r Also, to find the position of the centres of curvature relatively to the pitch-circle, we have, denoting the chord of the describing circle TI by c, c=2r sin 0; and therefore For the flank, R p —c=2r sin o R—2r R (29) For the face, p'—c=2r sin o R+ 2r For the proportions approved of by Willis, sin 0=i nearly ; r = p (the pitch) nearly; c=Jp nearly; and, if N be the number of teeth in the wheel, r/R =6/N nearly; therefore, approximately, p P —c=a' N-12 p —c=2' N+i2 Hence the following construction (fig. 105). Let BB be part of the pitch-circle, and a the point where a tooth is to cross it. Set off ab=ac=ip. Draw radii bd, ce; draw fb, cg, making angles of 75; ° with those radii. Make bf=p'—c, cg=p —c. From f, with the radius fa, draw the circular arc ah; from g, with the radius ga, draw the circular arc ak. Then ah is the face and ak the flank of the tooth required. To facilitate the application of this rule, Willis published tables of p —c and p' —c, and invented an instrument called the"odontograph." § 51. Trundles and Pin-Wheels.—If a wheel or trundle have cylindrical pins or staves for teeth, the faces of the teeth of a wheel suitable for driving it are described by first tracing external epicycloids, by rolling the pitch-circle of the pin-wheel or trundle on the pitch-circle of the driving-wheel, with the centre of a stave for a tracing-point, and then drawing curves parallel to, and within the epicycloids, at a distance from them equal to the radius of a stave. Trundles having only six staves will work with large wheels. § 52. Backs of Teeth and Spaces.—Toothed wheels being in general intended to rotate either way, the backs of the teeth are made similar to the fronts. The space between two teeth, measured on the pitch-circle, is made about Ith part wider than the thickness of the tooth on the pitch-circle--that is to say, Thickness of tooth pitch; Width of space =1 pitch. The difference of Itr of the pitch is called the back-lash. The clearance allowed between the points of teeth and the bottoms of the spaces between the teeth of the other wheel is about one-tenth of the pitch. § 53. Stepped and Helical Teeth.—R. J. Hooke invented the making of the fronts of teeth in a series of steps with a view to increase the smoothness of action. A wheel thus formed resembles in shape a series of equal and similar toothed disks placed side by s de, with the teeth of each a little behind those of the preceding c isk. He also invented, with the same object, teeth whose fronts, instead of being parallel to the line of contact of the pitch-circles, cross it obliquely, so as to be of a screw-like or helical form. In wheel-work of this kind the contact of each pair of teeth commences at the foremost end of the helical front, and terminates at the aftermost end; and the helix is of such a pitch that the contact of one pair of teeth shall not terminate until that of the next pair has commenced. Stepped and helical teeth have the desired effect of increasing the smoothness of motion, but they require more difficult and expensive workmanship than common teeth; and helical teeth are, besides, open to the objection that they exert a laterally oblique pressure, which tends to increase resistance, and unduly strain the machinery. § 54. Teeth of Bevel-Wheels.—The acting surfaces of the teeth of bevel-wheels are of the conical kind, generated by the motion of a line passing through the common apex of the pitch-cones, while its extremity is carried round the outlines of the cross section of the teeth made by a sphere described about that apex. The operations of describing the exact figures of the teeth of bevel-wheels, whether by involutes or by rolling curves, are in every respect analogous to those for describing the figures of the teeth of spur-wheels, except that in the case of bevel-wheels all those operations are to be performed on the surface of a sphere described about the apex instead of on a plane, sub- stituting poles for centres, and 0 great circles for straight lines. In consideration of the practical difficulty, especially in the case of large wheels, of obtaining an accurate spherical surface, and of drawing upon it when obtained, the following approximate method, proposed originally by T r e d g old, is generally used: Let 0(fig.106) be the common apex of a pair of bevel-wheels;
End of Article: CASE I
JOHN CASE (d. 1600)

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